# How to determine sensitivity in a dose-response curve?

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In a dose-response curve the % inhibition can be plotted against concentration antagonist. In our case, the effect of the antagonist is tested in different genetic variants of a microbe.

When trying to determine the potency of a drug on the different variants, I compared the IC50 values. The lower the concentration required for 50% inhibition, the more potent the drug.

When comparing drug efficacy, I compare the concentrations required to reach Emax or the maximum inhibition that each drug can exert - which turns out to be 100% inhibition.

To compare sensitivity, how does one proceed?

I think you are after the slope of the dose-response curve:

Dose-response curve. Souce: Merck

The slope indicates the change in response per unit dose, which translates to sensitivity.

A few notes on these calculations:

• Although investigators sometimes say they compute the area under the dose-response &ldquocurve&rdquo, it really only makes sense to compute area under the dose-response data, not under a fit curve. If you have enough quality data to fit a dose-response curves and determine the EC50 (or IC50), slope and maximum effect with reasonably narrow confidence intervals, then compare those parameters. Comparing area under the dose response relationship is a strategy to use when you are not able to determine those parameters with precision.
• If the X values are equally spaced, then Prism&rsquos AUC calculations are equivalent to taking a weighted average of all the Y values, giving the Y values corresponding to the lowest and highest X values half the weight of the other points. If the X values are not equally spaced, The area is still a weighted average of the responses, but the weights account for the unequal spacing with the responses corresponding the the widely spaced doses having more weight than responses from doses that are more closely spaced.
• The area under the curve reported by Prism is the area under the &ldquocurve&rdquo created by connecting the responses with straight lines. Its units are the X units (usually log concentration) times the Y units. It is not really possible to interpret these units, but all you&rsquoll care about is comparing the area determined under different conditions or with different cell lines.
• You can only compare areas computed with exactly the same set of X values. If the data sets use different doses or concentrations, so have a different set of X values, any comparison of area would be meaningless.
• Prism's AUC analysis asks you to specify a baseline for the area. You can enter Y=0 or Y equal to some other baseline value measured using appropriate controls.
• If you entered replicate (say triplicate) Y values for each X, Prism's AUC analysis averages these and only the mean is entered into the calculations of the AUC, but the variation among replicates lets Prism compute the SE of the AUC and its 95% CI.
• If you want to compare areas under dose-response data from two data sets, focus on the difference rather than the ratio. You'll get a different ratio if you define the baseline differently.

All these tests described here are measures of accuracy in some sense or another. In is disturbing that most people, when confronted with the task of defining accuracy, will usually be unable to give more than the common household definition (i.e. it is "the quality of being correct or true to some objective standard"). In statistics, the definition of accuracy is governed by the ISO, who define as follows:

• Accuracy is the proximity of measurement results to the true value
• Precision is the repeatability, or reproducibility of the measurement

Accuracy is occasionally referred to as "diagnostic accuracy" or "diagnostic effectiveness" and is expressed as the proportion of correctly classified subjects among all subjects:

## How to determine sensitivity in a dose-response curve? - Biology

Little Pro on 2019-06-06 Views:

Species Sensitivity Distributions (SSD) are a very important technique used in ecological risk assessment. It is primarily used to derive predicted no effect concentrations (PNECs) for environment risk assessment.

### Principles of SSD

Different species show different sensitivities to the same chemical substance and the variation between those species can be described by a statistical distribution. If you have conducted many eco-toxicity tests on the same substance using multiple species (fish, invertebrates and plants) and obtained many eco-toxicity endpoints, you can use SSD, the statistical approach, to draw a curve such as the following one and derive HC5 (hazardous concentration for 5% of species).

Derived HC5 can then be used to calculate PNEC by dividing it with an additional safety factor (1-5).

If you only have a small dataset (i.e. algae, daphnid and fish), the SSD approach will not be applicable. In that case, you usually need to divide the lowest NOEC value with larger safety factor (typically 10).

## DISCUSSION

GnRH agonist administration suppressed endogenous LH and testosterone secretion therefore, circulating testosterone concentrations during treatment were proportional to the administered dose of testosterone enanthate. This strategy of combined administration of GnRH agonist and graded doses of testosterone enanthate was effective in establishing different levels of serum testosterone concentrations among the five treatment groups. The different levels of circulating testosterone concentrations created by this regimen were associated with dose- and concentration-dependent changes in fat-free mass, fat mass, thigh and quadriceps muscle volume, muscle strength, leg power, hemoglobin, circulating IGF-I, and plasma HDL cholesterol. Serum PSA levels, sexual desire and activity, and spatial cognition did not change significantly at any dose. The changes in fat-free mass, muscle volume, leg press strength and power, hemoglobin, and IGF-I were positively correlated, whereas changes in plasma HDL cholesterol and fat mass were negatively correlated with testosterone dose and total and free testosterone concentrations during treatment.

The compliance with the treatment regimen was high. The participants received 100% of their scheduled GnRH agonist, and 99% of testosterone injections. Serum LH levels were suppressed in all men, demonstrating the effectiveness of GnRH agonist treatment. The treatment regimen was well tolerated. There were no significant changes in PSA or liver enzymes at any dose. However, long-term effects of androgen administration on the prostate, cardiovascular risk, and behavior are unknown.

Serum testosterone levels were measured 7 days after previous injection they reflect the lowest testosterone levels after an injection. Testosterone concentrations were higher at other time points. Weekly injections of testosterone enanthate are associated with fluctuations in testosterone levels (44). Although nadir testosterone concentrations were highly correlated with testosterone enanthate dose, it is possible that sustained testosterone delivery by a patch or gel might reveal different dose-response relationships, particularly with respect to hemoglobin and HDL cholesterol (19).

There were no significant changes in overall sexual activity or sexual desire in any group, including those receiving the 25-mg dose. Testosterone replacement of hypogonadal men improves frequency of sexual acts and fantasies, sexual desire, and response to visual erotic stimuli (3, 13, 15, 17, 31, 41). Our data demonstrate that serum testosterone concentrations at the lower end of male range can maintain some aspects of sexual function (3, 13). Testosterone has been shown to regulate nitric oxide synthase activity in the cavernosal smooth muscle (32), and it is possible that optimum penile rigidity might require higher testosterone levels than those produced by the 25-mg dose.

This study demonstrates that an increase in circulating testosterone concentrations results in dose-dependent increases in fat-free mass, muscle size, strength, and power. The relationships between circulating testosterone concentrations and changes in fat-free mass and muscle size conform to a single log-linear dose-response curve. Our data do not support the notion of two separate dose-response curves reflecting two independent mechanisms of testosterone action on the muscle. Forbes et al. (22) predicted 25 years ago that the muscle mass accretion during androgen administration is related to the cumulative androgen dose, the product of daily dose and treatment duration. Our data are consistent with Forbes's hypothesis of a linear relationship between testosterone dose and lean mass accretion however, we do not know whether increasing the treatment duration would lead to further gains in muscle mass.

In addition, we do not know whether responsiveness to testosterone is attenuated in older men. Testosterone dose-response relationships might be modulated by other muscle growth regulators, such as nutritional status, exercise and activity level, glucocorticoids, thyroid hormones, and endogenous growth hormone secretory status.

Serum PSA levels decrease after androgen withdrawal, and testosterone replacement of hypogonadal men increases PSA levels into the normal range (16, 34). We did not find significant changes in PSA at any dose, indicating that the lowest dose of testosterone maintained PSA levels. We did not measure prostate volume in this study therefore, we do not know whether prostate volume exhibits the same relationship with testosterone dose as PSA levels.

Hemoglobin levels changed significantly in relation to testosterone dose and concentration. Testosterone regulates erythropoiesis through its effects on erythropoietin and stem cell proliferation (14,35, 40). Although modest increments in hemoglobin might be beneficial in androgen-deficient men with chronic illness who are anemic, marked increases in hemoglobin levels could increase the risk of cerebrovascular events (25) and hypertension (42).

Although men, on average, perform better on tests of spatial cognition than women, testosterone replacement has not been consistently shown to improve spatial cognition in hypogonadal men (1, 29, 48). We did not find changes in spatial cognition at any dose. The effect size of gender differences in spatial cognition is small it is possible that our study did not have sufficient power to detect small differences. We cannot exclude the possibility that gender differences in spatial cognition might be due to organizational effects of testosterone and might not respond to changes in testosterone levels in adult men.

Although mean change in fat-free mass and muscle size correlated with testosterone dose and concentration, there was considerable heterogeneity in response to testosterone administration within each group. These individual differences in response to androgen administration might reflect differences in activity level, testosterone metabolism, nutrition, or polymorphisms in androgen receptor, myostatin, 5-α-reductase, or other muscle growth regulators.

Our data demonstrate that different androgen-dependent processes have different testosterone dose-response relationships. Some aspects of sexual function and spatial cognition, and PSA levels, were maintained by relatively low doses of testosterone in GnRH agonist-treated men and did not increase further with administration of higher doses of testosterone. In contrast, graded doses of testosterone were associated with dose and testosterone concentration-dependent changes in fat-free mass, fat mass, muscle volume, leg press strength and power, hemoglobin, IGF-I, and plasma HDL cholesterol. The precise mechanisms for the tissue- and function-specific differences in testosterone dose dependence are not well understood (36). Although only a single androgen receptor protein is expressed in all androgen-responsive tissues, tissue specificity of androgen action might be mediated through combinatorial recruitment of tissue-specific coactivators and corepressors (36).

Testosterone doses associated with significant gains in fat-free mass, muscle size, and strength were associated with significant reductions in plasma HDL concentrations. Further studies are needed to determine whether clinically significant anabolic effects of testosterone can be achieved without adversely affecting cardiovascular risk. Selective androgen receptor modulators that preferentially augment muscle mass and strength, but only minimally affect prostate and cardiovascular risk factors, are desirable (36).

This study was supported primarily by National Institutes of Health (NIH) Grant 1RO1-AG-14369 additional support was provided by Grants 1RO1-DK-49296, 1RO1-DK-59297–01, Federal Drug Administration Grant ODP 1397, a General Clinical Research Center Grant MO-00425, NIH-National Center for Research Resources-00954, RCMI Grants P20-RR-11145–01 (RCMI Clinical Research Initiative) and G12-RR-03026. BioTechnology General (Iselin, NJ) provided testosterone enanthate, and R. P. Debio (Martigny, Switzerland) provided the GnRH agonist (Decapeptyl).

## Phase Response Curves:

This paper was originally published in Circadian Clocks from Cell to Human , ed. T. Hiroshige & K. Honma (Hokkaido Univ. Press, Sapporo), 1992, pp. 209-249. The paper is reprinted here with a few slight revisions by permission from Drs. Ken-ichi and Sato Honma.

Few chronobiologists have attempted to review the topic of Phase Response Curves (PRCs)--reviews by Aschoff (1) and Pittendrigh (55) are the notable examples. Probably the task has seemed too daunting to undertake. Compiling the PRC Atlas (31) forced me to study all the published PRCs. This project led to some generalizations that I believe are worthwhile to summarize herein. This paper does not attempt to comprehensively review all PRCs--the Atlas is itself the most comprehensive review possible--but to discuss generalizations based on the PRC Atlas. The topics to be addressed are (A) PRCs as a reflection of entrainment mechanisms (resetting by light, dark, and temperature stimuli) and ecological strategies thereof (B) phototransduction pathways (C) PRCs as probes for the mechanism of circadian pacemakers (D) PRCs as phase markers for the oscillator and (E) PRCs as gauges of the amplitude of circadian oscillators. This paper shows figures, most of which come from the PRC Atlas, and refers to the PRC numbers of the Atlas format (e.g., A/Gp-2, C/Nc-10).

A PRC is a plot of phase-shifts as a function of circadian phase of a stimulus. Stimuli include light pulses, temperature pulses, or pulses of drugs or chemicals. As shown in Figs. 1A, 1D, and 1G, representative PRCs of circadian oscillators for light pulses exhibit delay phase shifts in the early subjective night and advance phase shifts in the late subjective night, with little phase shifting occurring during the subjective day (hence, the subjective day portion of the PRC is often referred to as the "dead zone"). As will be discussed later, some of these topological features of light PRCs are crucial in determining the ability of circadian pacemakers to entrain to the daily light/dark cycle.

As illustrated in Fig. 1, there are two so-called "types" of PRCs--Type 1 and Type O (72). Type 1 displays relatively small phase-shifts (e.g., usually less than 6-hour phase-shifts) and has a continuous transition between delays and advances (Fig. 1A), whereas Type 0 PRCs show large phase-shifts (Fig. 1D). If the phase shifts of a Type 0 PRC are plotted as advances and delays, a discontinuity (the "breakpoint"--see below and in Fig. 2) often appears at the transition between delay and advance phase-shifts. The terms Type "0" and "1" refer to the average slope of the curve when plotted as "new phase" vs. "old phase"-- a so-called "phase transition curve" or PTC (whereas a PRC plots "phase-shift" vs. "old phase"). As shown in Fig. 1B, Type 1 resetting can be visualized as a PTC with an average slope of 1 (45 0 angle), whereas Fig. 1E depicts a Type 0 PTC, which has an average slope of 0 (0 0 angle).

Whether Type 1 or Type 0 resetting is exhibited often depends upon the strength of the stimulus. For example, increasing the light dose (Culex: 7.5-min vs. 120-min light pulses, F/Cq-1-2) or drug dosage (Gonyaulax: anisomycin 0.1 vs. 0.3 m M, F/Gp-19-20) converts Type 1 into Type 0 resetting. Other factors, however, can also cause the Type 1 to Type 0 conversion: e.g., genetic mutation (Drosophila melanogaster: F/Dm-1-2) and background light quality and/or intensity (Gonyaulax: A/Gp-4-5). These "other factors" probably affect the sensitivity of the clock to stimuli, so that they affect the perceived stimulus strength.

The "limit-cycle" interpretation of Type 1 vs. Type 0 resetting is also compared in Figures 1C and 1F. Phase-shifting stimuli are posited to change the state variables from the "limit cycle" (the circle of Figs. 1C, 1F, 1I) to another area of the phase plane, labelled the "resetting contour" (the heavy dashed line of Figs. 1C, 1F, 1I). If this change moves the state variables to another isochron on the phase plane, a steady-state phase shift will be observed. Type 1 resetting results if the resetting contour is not moved beyond the singularity (resetting contour on "near side" of singularity), whereas Type 0 resetting occurs if the stimulus is strong enough to move the variables beyond the singular region (resetting contour on "far side" of singularity).
Figs. 1G, 1H, 1I also illustrate an interesting case of "critical stimulus" resetting in Gonyaulax (A/Gp-5). Light pulses given early in the subjective night provoke Type 1 resetting, while light pulses given later yield Type 0 resetting, yielding a highly asymmetrical PRC and resetting contour.

It might be assumed that stimuli presented during the dead zone (e.g., ct 3-10) do not modify the state variables. While this can be true for some specific models, it is not a necessity of a limit-cycle model. The other, equally plausible, alternative is that stimuli presented during the dead zone induce changes of the state variables, but these altered values do not move the variables to a different isochron (see Fig. 1F). Therefore, no phase-shift results. Consequently, state variables of the pacemaker are not necessarily insensitive to the stimulus during the dead zone--in fact, the stimulus could induce large changes of the state variables, but these changes do not move the pacemaker to a different isochron.

The breakpoint discontinuity of PRCs is in some cases merely a plotting convention of arbitrarily assigning phase-shifts in one half-cycle (12 hours) as delays and the other half-cycle as advances. To avoid these arbitrary distinc tions, sometimes Type 0 PRCs are plotted monotonically--that is, all phase-shifts are plotted as delays from 0-24 hours, as shown in Fig. 1D. When plotting those Type 0 PRCs which happen to be asymmetric (e.g., Fig. 1G), the breakpoint is not an arbitrary convention, because the PRC has a discontinuity no matter how the PRC is plotted.

The advantage of plotting phase-shifts monotonically (or as a PTC) is that such plots do not mislead the reader into assuming that advance vs. delay resetting are mechanistically different, e.g., that advance phase-shifts result from a pacemaker's state variable being changed in one direction, while delay phase-shifts change the variable in an opposite direction. In fact, limit-cycle models usually suggest no mechanistic difference between advances and delays in Type O resetting. Moreover, the limit-cycle models usually interpret the transition from Type 1 to Type O resetting to be merely dependent upon whether the magnitude of the stimulus is sufficient to shift the resetting contour beyond the "singularity" (72).

In retrospect, I regret that I did not also plot phase-resetting behavior as PTCs in the Atlas, for two reasons. First, PTCs avoid the misleading advance vs. delay distinction discussed above. Second, PTCs may encode other information about the pacemaker. For example, Cote (8) recently replotted a PRC published by Tamponnet and Edmunds (69) into a PTC format and may have discovered a heretofore unknown phenomenon: Type 2 resetting ! Winfree (72) claims that Type 2 resetting implies three or more state variables must be intrinsic to the pacemaker's mechanism. This startling insight was obtained by merely replotting the data as a PTC. Therefore, PTCs can give us additional information. (Please note that the computer version of the Atlas allows one to switch quickly between PRC and PTC formats.)

II. How can a PRC be measured ?

In principle, PRCs can be determined by a number of different protocols, as described by Aschoff (1). Four of the most commonly used protocols are described below (see ref. 1 for a discussion of other protocols):

(1) The stimulus (pulse) is applied while the oscillator is freerunning (e.g., a light pulse to an organism freerunning in DD). In this case, the individual organism serves as its own control, and accurate assignment of the circadian time of the stimulus depends upon knowledge of the circadian time of f r in a freerun. Usually the circadian time of f r is assessed by its phase in a LD 12: 12 cycle, but it is important to make sure that no "masking" of f r occurs in LD. The possibility of masking can be evaluated by releasing the organism from LD to a freerun and confirming that f r in the freerun extrapolates back to f r in LD.

(2) The stimulus (pulse) is applied in a freerun shortly after release from entraining conditions (e.g., a light pulse to an organism in DD within a few cycles after release from LD 12:12). This is a good method when a population of organisms is being tested it requires a few control organisms (cultures) which do not receive a stimulus with which to compare the treated ones. This is probably the best method for estimating entrainment behavior, as the PRC shape soon after release from entrainment should be more reflective of its shape during entrainment than will its shape after a long exposure to free-running conditions.

(3) The stimulus is a "step" from one continuous condition to another (e.g., DD to LL).

(4) The PRC can be estimated from the phase angle assumed by the rhythm to different T-cycles of the stimulus (e.g., T-cycles of light pulses). An example of this method is that of Eskin (13), who compared the PRCs derived by method #1 above (see H/Pd-1) and method #4 and found them to be equivalent. Method #4 cannot give a complete PRC because the phase angle will not be stable around the breakpoint region of the PRC during entrainment to T-cycles.

III. How should a PRC be plotted ?

For compilation of PRCs into the Atlas, a standardized format was chosen so that different PRCs could be easily compared. The comments below refer to the format used in the PRC Atlas.

(1) General information: The abscissa is the circadian time (ct) of the stimulus, from ct 0 to ct 24. Circadian time 0 to ct 24 is the duration of the endogenous period ( t , "tau"). Because LD 12:12 is taken to be standard entrainment conditions, circadian time 0 is defined as the beginning of the subjective daytime (therefore, subjective dawn or "lights-on"), and ct 12 as the beginning of the subjective night (therefore subjective dusk). The ordinate is the magnitude of the phase shift in circadian hours (see below). Advances are plotted above the abscissa as positive values, while delays are plotted below as negative values. For Type 0 resetting, all the phase-shifts may be plotted monotonically, as in Fig. 1D.

(2) Circadian time: Because circadian pacemakers have different endogenous frequencies, PRCs among different organisms cannot be directly compared unless their time scales are standardized to "circadian time." The first aspect of circadian time is that the scales for both the circadian time of stimulus (abscissa) and the magnitude of phase shift (ordinate) are expressed in "circadian hours." PRCs are scaled in circadian hours so that both the horizontal and vertical axes of PRCs from different organisms may be directly compared. A circadian hour is equal to 1/24 of the endogenous period, t (therefore, a circadian hour = tau/24 hours). To convert "real" hours to "circadian" hours, the number of real hours (e.g., of the phase shift) is multiplied by 24/tau.

(3) Definition of circadian time zero: The second aspect of circadian time is that PRCs must be plotted along the abscissa relative to some defined time, e.g., circadian time zero (ct 0). In general, the definition of ct 0 has been the least standardized variable of PRCs in the literature and yet, it is crucial for being able to compare the phase-shifting responses among organisms.

The standard definition of ct 0 is as the phase in the freerun which extrapolates back to the last "dawn" (i.e., "lights-on" of the last-seen LD 12:12 cycle prior to release into constant conditions-DD or LL). In many cases, however, alternative definitions are necessary. PRCs which are measured in LL often use the beginning of LL as the extrapolated "dawn" rather than "lights-on" of the final light cycle. In addition, many PRCs have been measured from organisms which have been in constant conditions for a long time so that it is inaccurate or inconvenient to extrapolate to the final lights-on signal. For these PRCs, circadian time is usually defined from the phase reference point ( f r ). First, the phase angle in LD 12:12 of the phase reference point ( f r ) of the rhythm is measured. Then, this f r is assumed to define the same phase of the oscillator in freerunning conditions, and ct 0 becomes a certain number of circadian hours before or after f r in the freerun. For example, activity onset ( f r ) of nocturnal rodents occurs at dusk in LD 12:12, and is consequently defined as occurring at ct 12. Therefore, ct 0 becomes that time which is twelve circadian hours before or after f r in a freerun. As discussed above (method #1 of PRC measurement), determining f r in LD can be complicated by the problem of masking. Consequently, in all cases, the circadian time of f r in LD should be determined by releasing the organism into freerunning conditions and extrapolating f r back to its phase angle in the last cycle of LD.

(4) Estimation of the magnitude and direction of the phase-shift: There are two major problems to be taken into account when estimating the phase-shift: (1) frequently the period (tau) changes after a stimulus ("aftereffects"), and (2) often--especially in the case of advance phase-shifts--there can be transient cycles of little or no phase resetting before the steady-state phase-shift is established. The best way to avoid both of these problems is to extrapolate f r for many cycles before and after the stimulus, preferably by a least-squares linear regression. Obvious transient cycles should be excluded from this regression. Then the phase-shift is calculated by the difference on the day of the stimulus between the extrapolated f r before and after the stimulus. In the case of method #2 of PRC measurement, the control f r 's are extrapolated back to the day of the stimulus and used to compare with the extrapolations from experimental organisms.

For Type 1 resetting, it is usually easy to determine whether the phase-shift should be plotted as an advance or delay on a PRC, but when the large phaseshifts of Type 0 resetting are encountered, it is often difficult to unambiguously assign the direction of the phase-shift. One approach is merely to plot the PRC monotonically--from a limit cycle perspective, the distinction between advances and delays in Type 0 resetting is arbitrary anyway (see Fig. 1D).

Another approach to operationally distinguish between advances and delays while using a PRC-type presentation is to perform dose response experiments, thereby generating dose response curves (DRCs). DRCs assay the response at a given circadian phase of the clock to varying intensities/concentrations of the stimulus. Reducing the stimulus intensity will switch Type 0 resetting to Type 1 resetting, at which point the distinction between advances and delays becomes more obvious.

(5) Definition of stimulus phase: No matter what type of stimulus is considered, be it a light, temperature, or chemical pulse, the onset of the stimulus pulse has been plotted as the "stimulus phase" on the abscissa of the PRCs in the Atlas. In their original papers, many authors have plotted PRCs using other conventions for stimulus time--often the midpoint was used, and sometimes even the end of the pulse. There is no a priori reasoning which favors any of these criteria as the "stimulus phase." All are arbitrary. Aschoff urged in 1965 (1) that the stimulus midpoint be used as stimulus time. His argument--that PRCs plotted that way "line up" better--makes sense for many light PRCs. When one considers all the types of stimuli and PRC shapes, however, such a plotting convention can cause problems. In particular, I will argue later that for many chemical/drug stimuli, the effective duration of the pulse is unknown, since the time of recovery from a drug does not always coincide with the time of wash-out. If the effective duration is unknown, the midpoint is unknown.

The beginning of the pulse was therefore chosen as the standard marker for "stimulus phase." The phase-shifting response of an oscillator is likely to be a characteristic of the phase when the stimulus begins, that is, of the first unperturbed phase to be presented with a stimulus. If one uses the midpoint or end of the stimulus as the marker, then one is choosing a phase which has already been perturbed. And using the onset as the marker for stimulus phase circumvents the complications of recovery time.

IV. What are PRCs good for ?

(A) PRCs as gauges of entrainment mechanism

PRCs for light and temperature stimuli have been most valuable in understanding how circadian pacemakers are entrained to the daily cycle by environmental cues of light and temperature (55). Briefly, phase-resetting compensates for the fact that the freerunning period t of circadian oscillators is not equal to 24 hours--therefore, entraining stimuli (e.g., light) reset the clock so as to equalize the period of the entrained oscillator (the circadian clock) to the period of the entraining oscillator (the daily rotation of the earth). This basic principle is summarized by the following equation for a circadian oscillator under steady-state entrainment: t - T = Df

Light pulse resetting
Light is usually the most important Zeitgeber for entraining circadian oscillators. Therefore, PRCs for light stimuli have special interest and indeed, have been studied most extensively. Light-pulse PRCs usually have similar characteristics: delay phase-shifts in the early subjective night, advance phase-shifts in the late subjective night, and little phase-shifting during the subjective daytime. This generalization holds true whether or not the overt rhythm peaks in the day, night, or at twilight (55). Therefore, the PRCs of nocturnal vs. diurnal organisms are similarly phased to the light/dark cycle, even though their rhythms are not.

The magnitude of phase-shifting exhibited by the clock is a gauge to the "limits of entrainment." Obviously, PRCs with large phase-shifts can permit synchronization to light-dark T-cycles of a broader range as compared with low amplitude PRCs. The magnitude of phase-shifting by light is dependent upon the intensity and duration of the stimulus (among other factors). As the intensity and/or duration is increased, light PRCs of limit-cycle pacemakers go through two transitions. As mentioned above, the phase-shifting first changes from Type 1 to Type 0 resetting (Figs. 1A, 1D), so the magnitude of phaseshifting increases, but the circadian time of the transition between delay and advance shifts remains fixed. There are several good examples of this type of PRC transition (Type 1 to Type 0) with increasing stimulus strength: Chlamydomonas (A/Cr-1-4), Euglena (A/Eg-3,4,7), Kalanchoe (D/Kb-1-11), Leucophaea (F/Lm-6-10), Culex (F/Cq-1-2), Nauphoeta (F/Nc-1-2), Sarcophaga (F/Sa- 13), Drosophila (52), and Rattus (H/Re-1-3).

As the duration (and possibly intensity) of the light pulse is increased further, the second transition occurs: the circadian time of the break point begins to shift to earlier times (Fig. 2). This second transition has been interpreted as the clock "stopping" at ct 12 until the light pulse is terminated, but other data suggests that the clock continues to oscillate during the light pulse on another limit cycle which is near an isochron of ct 12 of the limit cycle in DD (51). Consequently, the clock always returns to the original limit cycle at ct 12 at the end of the light pulse. Fewer examples of the second type of PRC transition (i.e., shifting breakpoint) exist for light PRCs-- the best are Drosophila (52), and Sarcophaga (F/Sa-1-11, Fig. 2).

In addition to the changes of the PRC shape mentioned in the preceding paragraph, Page has recently discovered developmental plasticity of t and PRC shape (49, 50). Page has found that raising cockroach larvae in various illumination regimes (LD, LL, DD and T-cycles) transforms the light PRCs of the mature cockroaches--i.e., the resulting PRCs become mostly-advance, mostly-delay, or symmetric PRCs, depending upon the developmental conditions (F/ Lm-1-5, Fig. 3). This unexpected result demonstrates that PRCs are not developmentally immutable.

Stable entrainment does not necessarily require a PRC which has essentially the same symmetrical advance vs. delay topology as do the PRCs depicted in Figs. 1A and 1D. In fact, in order for entrainment to occur, a circadian oscillator's PRC need only have (1) a region of negative slope which is greater than - 2, and (2) a point on the PRC where the phase shift equals t -T. In particular, it is not necessary to have a PRC with both delays and advances. If the freerunning period is longer than 24 hours, a PRC that exhibits only advance resetting will allow stable entrainment (this would be an example of a highly asymmetric PRC). A specific example is Gonyaulax cells under red light illumination (Fig. 1G, A/Gp-5): the period is 25 hours and the PRC for blue or white light pulses is essentially all advances (up to 12 hours advance). In this case, Gonyaulax will entrain to a light/dark cycle (or white/red light cycle) with the onset of the light pulse (dawn) occurring at that circadian time which results in a one hour phase advance (33). Therefore, highly asymmetric PRCs can allow stable entrainment.

Dark pulse resetting
The apparently opposite stimulus of light pulses is to give dark pulses to organisms in LL (Fig. 4). The most simplistic model would predict that dark pulse PRCs will be the mirror-image of light pulse PRCs, which is nearly true in Paramecium (Fig. 4). Although this is an approximately valid description of some dark pulse resetting, dark pulse PRCs are sometimes not the exact mirror image of light pulse PRCs. One rationale for explaining why the two types of PRCs may not be mirror images is that the dark pulse stimulus in most PRCs is often longer than the corresponding light pulse stimulus. The PRC Atlas includes dark pulse PRCs for Acetabularia, Euglena, Gonyaulax, Paramecium, Lemna, chicken pineals, hamsters, sparrows, and bats (Taphozous).

Reebs and Mrosovsky (60) discovered an interesting "artifact" in hamsters with regard to dark pulses and pulses of some drugs. They noticed that dark pulses and some antidepressant drugs (e.g., the benzodiazepine triazolam) stimulate wheel-running activity in hamsters. They subsequently tested whether the stimulation of locomotor activity would alone mimic the phase-shifting action of dark pulses. It did! Furthermore, van Reeth and Turek (71) found that stimulation of activity was also the means by which phase-shifting by triazolam was accomplished. Therefore, in hamsters, dark pulses appear to reset by feedback of the overt rhythm back onto the pacemaker. It will be interesting to find other examples of feedback of overt rhythms onto pacemakers.

Temperature pulse resetting
Temperature pulse PRCs have been measured in a variety of organisms those included in the Atlas test the clock response in Euglena, Gonyaulax, Oedogonium, Neurospora, Bryophyllum, Kalanchoe, Lemna, Phaseolus, Hemideina, Leucophaea, Uca, Perognathus, and hamsters (in this final case, pulses of hypothermia). Although temperature can undoubtedly function as a zeitgeber, it apparently plays a supporting role to the light/dark cycle. In entrainment studies of conflicting light and temperature cycles, the light/dark cycle predominates in Euglena (5), Drosophila, cockroaches (52), and Pectinophora (57). Comparisons of the amplitudes of the light vs. temperature PRCs were not done in these studies. It would be interesting to repeat these types of experiments using light and temperature stimuli which elicit PRCs of equivalent amplitude, and then determine whether light still predominates.

Ecological strategies of PRC shape
The case of light-pulse PRCs which are asymmetric is especially interesting from an ecological perspective. These are PRCs which exhibit both delays and advances, but in which the area under either the advance or the delay portion of the PRC predominates (referred to below as the ratio of advance area to delay area, or A/D). Fig. 5 illustrates PRCs with various A/D shapes. Pittendrigh and Daan (53, 55, 56) have pointed out that an appropriate combination of tau values and asymmetric PRC shapes can give stable entrainment of pacemakers to various photoperiods such that f r of the oscillator will always occur at a given phase angle to either dawn or dusk of the various light/dark cycles. This phase angle will be independent of the length of the photoperiod, so that it is compensated for seasonal changes in the photoperiod. For example, a tau of less than 24 hours in combination with a PRC which has relatively more delay area than advance area (= small A/D) will allow ct 12 of the pacemaker's cycle to coincide with dusk on light/dark cycles which have a variety of ecologically relevant daylengths (e.g., photoperiods from 6 to 18 hours). This is a strategy which may be adaptive for a nocturnal animal (e.g., a mouse). The converse example--long tau and large advance/delay PRC--yields an oscillator whose ct 0 phase will coincide with dawn irrespective of the photoperiod's duration--hence, an optimal strategy for an organism which is active in the daytime (53, 55, 56).

Pittendrigh's and Daan's idea assumes that dusk is more important than dawn for night-active organisms, and that dawn is more important for day-active organisms. Obviously, this is a simplification of the ecology of y conservation. The crucial point is whether the organism's ecology demands adjustment of the clock to dawn versus dusk, not whether the organism is active in day or night. For example, one might imagine a nocturnal rodent which becomes active at an indeterminate phase sometime in the middle of the night, but must return to its burrow before dawn to escape predation. For this hypothetical rodent, a large A/D PRC and long t might be the optimal clock system. Conversely, when a constant phase-angle to dusk is adaptive, a small A/D and short t would be a good clock design (a common strategy for nocturnal rodents).

Another question about organisms exhibiting multiple rhythms controlled by a (presumably) single pacemaker is: which rhythm's phase is most important to conserve ? A Gonyaulax cell has rhythms of both photosynthesis peaking in the day and bioluminescence peaking at night. Which y strategy the cell picks may depend upon which rhythm's phase angle is more important. As its pacemaker has a long t and large A/D PRC, Gonyaulax seems to conserve its phase-relationship to dawn (33).

A different, but related, issue is whether the organism is exposed to the complete photoperiod during the day under natural conditions. (This "daylight-exposure" criterion is not the same as "day-activity"--for example, nocturnal predators such as cats may "see" the complete photoperiod during the day, but hunt at night.) Whether or not an organism is exposed to a more-or-less complete photoperiod (PP c ) is important from the perspective of entrainment. We might suppose that clocks which are exposed to a complete photoperiod (PP c ) might depend upon "continuous (parametric) entrainment" instead of "discrete (non-parametric) entrainment" (55). Moreover, clocks exposed to the full photoperiod could afford to be less light-sensitive than clocks which see only brief light pulses at dawn and/or dusk. These differences in photoperiod exposure might be correlated with differences in PRC shape (A/D), t , and sensitivity.

To determine if PRC shape, t , and sensitivity are correlated with PP c -exposure and/or activity patterns, information from the light PRCs in the Atlas has been distilled into Tables 1, 2, and 3. The first conclusion which is obvious from Table 1 is that it is impossible to draw any conclusion about the relative sensitivities between "PP c -exposed" and "non-PP c -exposed" clocks. Whether one defines "sensitivity" on the basis of (1) threshold intensity, (2) threshold duration, and/or (3) amplitude of PRC, few PRCs can be directly compared.

Many different light intensities and durations ("stimulus" column) have been used, and few attempts to establish threshold sensitivities have been done. Although many of the studies on "PP c -exposed" organisms used long duration and/or high intensity light pulses--suggesting low sensitivity--some of the most sensitive clocks are "PP c -exposed" (e.g., Neurospora, Samanea).

Furthermore, the PRC can be "history-dependent," as shown for hamsters' and sparrows' response to prior photoperiod and T-cycles (e.g., H/Ma-6-13 H/ Pd-2-4), and for cockroaches after larval growth in various T-cycles (Fig. 3, F/ Lm-1-5). Because these studies have used widely varying conditions of light pulse duration and/or intensity, it is presently impossible to compare the sensitivity to light of oscillators in "PP c -exposed" and "non-PP c -exposed" organisms.

Tables 2 and 3 summarize the data of Table 1 from the perspectives of activity patterns and PPc-exposure. Table 2 shows the correlations between PRC shape, t , and activity patterns in animals. Pittendrigh's prediction that night-active animals might prefer A/D < 1 and t < 24 is weakly supported by the data. The prediction that day-active animals might prefer A/D >1 and t >24 is not supported by the data--no significant correlations between PRC shape and t are obvious for day-active animals. The data for crepuscular animals are too scanty to make any conclusions.

Table 3 summarizes PRC shape and t data for all representative organisms on the basis of PP c -exposure. PP c -exposed organisms exhibit A/D shapes of all kinds, with slightly more symmetrical PRCs . PP c -exposed organisms seem to avoid t values close to 24 hours. Non-PP c -exposed organisms favor A/D <1 and Type 1 PRCs . When the data of Table 3 are reanalysed on the basis of plants vs. animals or whether the PRC was measured in background illumination (DD vs. LL), no new correlations emerge. (Note, however, that Aschoff's extensive analysis of t vs. intensity of LL does indicate significant differences for this response between "day-active" and "night-active" organisms--see ref. 2. Unfortunately, PRC shape is not known for most of those organisms.) The data of Table 3 demonstrate that some correlations between PRC shape and photoperiod-exposure may exist.

The data discussed above may prompt some revision of our concepts of entrainment mechanism. Our current entrainment models successfully explain the discrete entrainment mechanism of non-PP c -exposed clocks. It is in these cases that PRCs for brief light signals have helped us to understand entrainment. But organisms which are exposed to a complete photoperiod are in a quite different situation, which is not modelled well by PRCs to brief stimuli. In particular, t may be changed by the light during a complete photoperiod (9, 55), the y -jump does not occur (56), and the clock can afford to be much less sensitive to light.

Pittendrigh (55) has discussed potential mechanisms of "continuous" entrainment in PP c -exposed organisms, but this mechanism(s) remains a fertile field to be tilled by future investigations, both by modelling and direct experimentation. In particular, limit-cycle modelling of continuous entrainment will undoubtedly lead to new insights (51). Additionally, some experimental questions which could be addressed in this context include:

(1) Compare threshold sensitivity (intensity and/or duration, but especially intensity) between PP c -exposed and non-PP c -exposed organisms (so far, only the hamster's threshold sensitivity has been carefully measured--see ref. 67).

(2) Test intensity/duration reciprocity for PP c -exposed vs. non-PP c -exposed organisms. So far, reciprocity has only been tested in hamsters (67) and Chlamydomonas (42).

(3) Measure PRC for long light pulses (8-16 hours) as the best gauge for predicting entrainment properties of PP c -exposed organisms.

(4) Test which aspect of a LD cycle is used as a zeitgeber in PP c - vs. non-PP c - exposed organisms. For example, how does the clock respond to the gradual changes of light intensity at twilight ? In plants using phytochrome as clock photoreceptor, it may be the changing R:FR at dawn and dusk which is the most important zeitgeber. Also, can one predict phase-shifting by light pulses from the phase-shifts elicited by steps-LL to DD, or DD to LL ?

(5) How does behavior modify the exposure of the organism to light and thereby modulate entrainment ? DeCoursey (10) has shown that behavior can be a crucial component of entrainment in nocturnal rodents. Contrasting the behavioral components of entrainment in PP c -exposed vs. non-PP c -exposed organisms might be enlightening.

( B) Phototransduction pathways

Photopigments involved in light pulse r esetting
Apparently, the circadian clock became linked early in its evolution with (or had as a component) a photosensitive process which allowed the entrainment of the clock to the light/dark cycle of the sun. It might be supposed that this linkage could lead to valuable clues about the conservation or diversity of the oscillator's biochemical mechanism if the circadian pacemaker originated once during evolution and its mechanism was subsequently conserved, then one scenario would predict that the pigment(s) involved in the photosensitive process might also be conserved.

Unfortunately, this is clearly not true. The action spectra described below show that the clock photopigments in various organisms are quite different amongst each other. This means either (a) that a conserved clock mechanism has switched its photopigment a number of times during evolution (more proximal steps in the phototransduction pathway might still be conserved), or (b) if the connection between clock mechanism and photopigment has been conserved during evolution, then the clock must have originated independently many times, and therefore the biochemical mechanisms of the clocks in various organisms could be quite different.

A first step towards characterizing the phototransduction pathway is to identify the clock's photopigment. In practice, identifying a photopigment requires measuring some specific characteristic of the photopigment and comparing it with the characteristics of known photopigments. Usually this entails action spectroscopy.

Action spectroscopy is dosimetry with light. To measure an action spectrum, one measures the photoresponse (in this case, the phase shift) at different wavelengths. For each wavelength, a range of fluences are used ("fluence" is number of photons per unit area "fluence rate" is fluence per unit time, and is equivalent to the less precise term "intensity"). In the ideal case, where "univariance" holds, the slope of the fluence response curve (plotted as phase shift vs. log fluence) is the same for each wavelength, and the sensitivity at each wavelength is evaluated as the fluence necessary to achieve a given arbitrary photoresponse. On the other hand, if the shape or linearity of the fluence response is different at various wavelengths, it can mean that a screening pigment is interfering with the spectral response.

It is beyond the scope of this paper (see ref. 19 for a review of action spectroscopy) to go into greater detail about general action spectroscopy. But a specific issue about action spectra for resetting of limit cycle oscillators needs to be addressed. For Type 1 resetting, the resetting curve does not cross the singularity (Fig. 1C), and so the fluence response should match that of an ideal univariant case. This is in fact the result obtained by Takahashi et al. (67) for phase resetting of the hamster clock.

For Type 0 resetting, however, the expected response can be quite different even with an ideal univariant photoreceptor. At the phase(s) where light pulses yield maximal phase resetting, increasing the pulse fluence can cause the clock to be reset to regions close to or beyond the singularity. This can cause a discontinuity in the fluence response curve. This type of response to varying fluence has been observed in Gonyaulax (33), Neurospora (Fig. 32 in ref. 11), and Chlamydomonas (35). These discontinuous fluence response curves are reminiscent of the dose response of the Gonyaulax clock to anisomycin, which is another case of singular behavior (70).

A discontinuous fluence response curve means that some process "downstream" from the photopigment's absorption of light is converting the initially continuous photochemical response into a discontinuous biological response. In the case of clock photoreceptors, it is the limit-cycle organization of the circadian oscillator which is responsible for converting the initially monotonic response into a discontinuous response as the light pulse moves the pacemaker past the singular region.

How then should the spectral sensitivity of clock photoreceptors be measured? The following three tactics should be valid procedures to construct accurate action spectra for clock photopigments (see ref. 35 for more discussion and experimental examples). The first tactic is to use fluences and/or durations which elicit only Type 1 resetting (for a recent example, see ref. 67). In this case, the critical response is defined as an arbitrary "percent response" along a continuous fluence response curve. If Type 0 resetting is required by the experiment, the second tactic is to measure the action spectrum at a circadian phase which is not close to the PRC's "breakpoint." If this is done, the likelihood that the pacemaker will be moved through the singularity is reduced, and therefore the fluence response will probably be continuous. As in the first tactic, the critical response will be selected by the experimenter as an arbitrary "percent response ?" Finally, in the case of Type 0 resetting, one can plot on the ordinate of the action spectrum the fluence at which singular behavior is elicited (i.e., arhythmicity or a discontinuity in the fluence response curve). This third tactic depends upon the pacemaker itself setting a critical threshold from which the action spectrum is derived in lieu of the experimenter selecting a "percent response" level from a continuous function (35).

Photobiology of clock photoreceptors is summarized in Tables 4 and 5. Most of these studies were performed before the complications of Type 0 resetting were fully appreciated. For example, several of the action spectra listed in Table 4 use only a single light fluence for each wavelength (an "equal-intensity" action spectrum) instead of the range of fluences required for proper action spectroscopy. This means that neither univariance nor the continuity of the fluence response was tested. Therefore, the conclusions of these studies may be compromised by artifacts. This criticism is true for the studies of Coleus, Kalanchoe, and Bryophyllum (possibly also Paramecium). In the case of Gonyaulax, only two fluences were checked, so this action spectrum may also have been compromised by this problem.

Light-induced resetting in Chlamydomonas exhibits a remarkable feature--the action spectra for cells in DD vs. LL are different! The clocks of cells in LL respond to red and blue light and photosynthetic inhibitors prevent light-induced phase- resetting (34). Thus, components of photosynthesis appear to mediate clock resetting of cells in LL. On the other hand, green and red light resets the clock in DD, and photosynthetic inhibitors are ineffectual (42). The identity of this photoreceptor is unknown. The sensitivity of the two photosystems are also very different: cells in DD respond to 1/2000 of the fluence needed to reset cells in LL.

While the identification of most photopigments require action spectra to be measured throughout the visible range of wavelengths, one photopigment is exceptional: phytochrome, a common photopigment in plants, is characterized by red absorption (about 660 nm) and whose photoresponse is reversed by subsequent illumination with far-red light (about 730 nm). (Phytochrome can also exhibit the so-called "high irradiance response," or HIR, which is not reversed by far-red light, but is in fact potentiated by simultaneous irradiance with red and far-red light.) For simple phytochrome response (not HIR), far-red reversibility is usually considered to be a sufficient diagnostic and a complete action spectrum is often not done. Examples of circadian clock resetting in which phytochrome is implied are shown in Table 5.

In both Gonyaulax (A/Gp-8) and Paramecium (A/Pb-1,2), brief pulses of ultraviolet light can cause significant phase-shifting. The cells respond differ ently to ultraviolet light than to visible light: (1) the magnitude of the phase shifting is not very phase dependent, and (2) the shifts are all advances (Gonyaulax) or all delays (Paramecium). Phase resetting by ultraviolet light (UV-C) might not be relevant to entrainment in a natural setting, but it may imply something about the biochemical mechanism of circadian oscillators.

So far, the identification of some of these clock photopigments is not conclusive. Nevertheless, it is clear that no single photopigment is used by all circadian pacemakers. Some green plants use a phytochrome, while some animals use a rhodopsin. More studies of this type are warranted. (See ref. 47 for a more thorough, but outdated, discussion of clock photoreceptors.)

It may also be worthwhile to test the intensity vs. duration reciprocity of clock photoreceptors. Just as "HIR" photoreceptors show quite different reciprocity characteristics than that of phytochrome acting in its classic red/ far-red mode, clock photoreceptors may likewise exhibit features not found in other photoreceptive systems. For example, Takahashi et al. (67) found an unusually long reciprocity (for rhodopsin) in the phase-shifting response of nocturnal hamsters. Logically, it "makes sense" for a clock photopigment to integrate light signals over long durations, but quantitative reciprocity studies have only rarely been done in the clock field. Reciprocity should also be tested in PPc-exposed organisms, which might ignore reciprocity--perhaps these organisms pay attention to the duration of the light pulse rather than to the total number of photons absorbed.

Characterization of phototransduction pathways: Chemical/drug PRCs have been used to trace the pathway of phase-shifting information from some type of receptor (e.g., photo-receptor) to the clockwork. This approach has been championed by Eskin (14). For example, Eskin has shown that the clock of the Aplysia eye responds to increases of cyclic GMP in much the same way as it does to light (E/Ac-1,14), suggesting that--as in light transduction within the vertebrate eye--cyclic GMP mediates the effect of light transduction within this clock's photoreceptor (16). This cGMP-induced resetting may be mediated by new protein synthesis (59). Moreover, Eskin has discovered that the neurotransmitter serotonin causes phase-shifting within this eye (E/Ac-11), and this effect seems to be mediated by increases of the intracellular concentration of cyclic AMP because--in addition to other evidence--the adenylate cyclase activator forskolin elicits a PRC which is the same as that for serotonin (E/Ac-13 ref. 15).

Johnson and Nakashima (32) used a similar approach to study light-induced phase resetting in Neurospora. We found that inhibition of protein synthesis prevented light-induced phase-shifts in a dose-dependent manner. As a control, we showed that phase-shifting by light was not inhibited by the drug in mutants whose protein synthetic mechanism was resistant to the drug.

These studies exemplify some of the ways in which PRCs for chemical/drug stimuli can be used to study the transduction of phase-shifting information (either by light or other stimuli). In the case of blocking treatments which do not themselves cause phase-resetting, the interpretation of the results is relatively straightforward--if the blocking treatment inhibits the phase-shift by the tested stimulus, then the process affected by the blocking treatment may be involved in the transduction/transmission of the tested stimulus. The approach of using chemicals/drugs to block phase-shifting by another stimulus can, however, be difficult to interpret if the blocking treatment also causes phase resetting. These complications are discussed elsewhere (32).

Spectral Influences on Rhythm Expression and tau: In some plants, frequent phytochrome stimulation appears to be necessary for the persistence of circadian rhythmicity. This has been observed in Albizzia (64), Lemna (39, 40), and in transgenic tobacco (37). In two other plants, red vs. blue light has been found to differentially affect tau : in coleus' red LL shortens t , while blue LL lengthens t (23) in Gonyaulax, the phenomenon is reversed: red LL lengthens t , while blue LL shortens t (62). These results have been interpreted to suggest that two photopigments are coupled to the circadian pacemakers in these organisms.

(C) PRCs as probes for the clock mechanism: chemical/drug resetting

Chemical stimuli have also been extensively tested for phase-resetting action. Early studies (e.g., ref. 25) suggested that the circadian clock was relatively resistant to drugs and chemicals, but now many pharmacological treatments have been discovered which reset the clock.

In general, the motivation for studying the phase response of circadian pacemakers to light or temperature stimuli has been to understand how the entrainment of the pacemaker to the solar day is accomplished. The motive for studying the pacemaker's response to chemicals and drugs is different. The hope is to unveil the biochemical mechanism of the pacemaker by assessing its pharmacological sensitivity. The impact of chemicals upon the pacemaker has been assayed by their effect upon both period and phase.

What can chemical/drug PRCs tell us about the pacemaker? PRCs for pulses of chemicals/drugs are usually interpreted to mean that the presumed biochemical target(s) affected by the chemical is either a state variable or a state parameter of the pacemaker. In the discussion that follows, I will discuss chemical-induced clock resetting in the context of changes of state variables. In a very simple oscillator composed only of a single biochemical component with two state variables (e.g., concentration and rate of change of concentration), chemicals which increase vs. decrease the component's level should evoke PRCs which are 180' apart and should have a predictable phase relationship to the phase of the oscillation of the component (see ref. 61 for an example).

The situation is considerably more complicated for an oscillator which is composed of multiple components with multiple state variables, which will probably be found to be true for circadian oscillators. For multi-dimensional oscillators, the PRCs for perturbation of state variables cannot be predicted by the oscillation of any single state variable. Can PRCs still then be used to test whether biochemical entities are potential state variables ? Yes--but accurate prediction of PRC shape depends upon modelling of all or most of the specific state variables and parameters in the oscillator and the interactions between these components. In the absence of such a specific model, the only unassailable prediction that can be made is that perturbation of the level of a state variable should provoke phase resetting. The shape or phase angle of the resulting PRC is not diagnostic in the absence of a specific multicomponent model. (See ref. 22 for an attempt to use chemical/drug PRCs to distinguish hands from state variables/ parameters and to build a model of the pacemaker's mechanism.)

If this is true, is there any value for measuring complete PRCs for drugs/ chemicals? Does measuring phase responsiveness at phases throughout the circadian cycle tell us anything more than data from a single phase point ? I think the answer is yes, for several reasons. First, observing phase-shifts at various phases reassures us that the result is not an artifact. Second, knowing the phase responsiveness throughout the cycle will be useful for later modelling of the pacemaker or for designing future experiments. Finally, responsiveness must be measured at many phases in order to detect discontinuities or to distinguish Type 1 from Type 0 resetting this information will undoubtably be crucial when the time comes to model the pacemaker's biochemistry.

Furthermore, note that the state variables of a limit cycle oscillator may be changed by phase-resetting stimuli, even if the stimuli are presented at phases of the "dead zone." As discussed previously in this article, the pacemaker is not necessarily insensitive to resetting stimuli presented during the dead zone the state variables can be changed at these phases, but this change does not move the pacemaker to a different isochron. This phenomenon is relevant to methods of testing whether a specific biochemical substance is a state variable: phase-resetting stimuli presented during dead zone phases may modify state variables, even though no phase shift is elicited.

Trends in chemical/drug PRCs: Do different organisms show similar responses to pharmacological treatments? If so, would it suggest that the pacemaker's biochemical mechanism has been conserved during evolution? Not necessarily. For example, the fact that deuterium oxide lengthens the circadian period of so many dissimilar organisms has been invoked to suggest that the clock's biochemistry has been conserved. Unfortunately, however, the action of deuterium oxide is so non-specific and its slowing effects upon biological processes is so universal that this data has not been helpful in unraveling the clock mechanism.

Other chemicals/drugs might be more useful. The class of drugs whose phase-shifting action has been best characterized in a variety of organisms is that of protein synthesis inhibitors on 80S ribosomes: cycloheximide, puromycin, anisomycin, and streptimidone. These drugs reset the clocks in a wide range of organisms: the algae Acetabularia (A/Am-2,3), and Gonyaulax (A/Gp-13-24, also see Fig. 10), the fungus Neurospora (C/Nc-19, 20), the angiosperms Phaseolus (D/ Pc-18,19) and Lemna (D/Lg-7), the eye of the mollusk Aplysia (E/Ac-15-18), chick pineal cells (Fig. 6F see ref. 68), and hamsters (ref. 29). Fig. 6 illustrates that most of these PRCs show strong Type 0 resetting. Amino acid analogs also have potent resetting effects in Lemna (D/Lg-10-17 ref. 41). Cycloheximide is also known to lengthen the period of Euglena (17) and Chlamydomonas (21), but PRCs have not been reported for these organisms. Finally, it is interesting to note that inhibitors of protein synthesis on 70S ribosomes (e.g., chloramphenicol) have little or no effect on circadian rhythms in eukaryotes. Therefore, protein synthesis on mitochondrial or chloroplast ribosomes seems unnecessary for circadian precession in eukaryotic cells.

Another class of chemical/drug stimuli which have been applied to several types of organisms is that of inhibitors of metabolism. Again, the clocks of many organisms are reset by metabolic inhibitors: Euglena (nitrogen A/Eg 9), Neurospora (azide, cyanide, antimycin A C/Nc-26-28), Lemna (azide, cyanide D/ Lg-8,9), Phaseolus (cyanide, azide D/Pc-20,21), Aplysia eyes (cyanide, dinitrophenol, E/Ac-7,8), and Drosophila (nitrogen F/Dp-11). Some of these PRCs are shown in Fig. 7. In general, the responses to metabolic inhibitors are more variable amongst different species than the responses to protein synthesis inhibitors.

Drugs which affect cyclic AMP (cAMP) have also been applied to a variety of organisms: Euglena (theophylline A/Eg-12), Phaseolus (theophylline ref. 44), Trifolium (cAMP, theophylline D/Tr-1,3), Aplysia (forskolin E/Ac-13), rats (theophylline H/R-1), and rat SCN in vitro (cAMP analog ref. 58). Four of these PRCs are illustrated in Fig. 8.

Some other chemicals/drugs have been tested in various organisms, e.g., drugs which affect (1) intracellular Ca ++ or calmodulin (A23187, chloropromazine, trifluoperazine, verapamil, EGTA, and theophylline see Fig. 9), (2) membrane potential (K+, Li+, strophanthidin, electrical stimulation), or (3) other membrane properties and ion fluxes (fusaric acid, the anaesthetic quinidine, and the ionophores valinomycin, CCCP, CCmP, and A23187) but of these stimuli, no single agent has been applied to enough different organisms to allow meaningful comparison.

Interpretive problems with chemical/drug resetting: Of course, there are many caveats for the interpretation of chemical/drug PRCs. Most obvious is that one must be cautious about assigning the site of action of the chemical/drug. Side effects of pharmacological treatments abound, and are notorious for misleading researchers' conclusions. Four kinds of controls have been used as evidence for specificity of drug action. The first control is to measure the concentration dependence of the drug's effect upon the presumed site of action (e.g., protein synthesis) and compare it with the concentration dependence of the drug's phase-resetting efficacy. If the two do not correlate closely, then the drug's impact on phase is likely to be via a different site of action. The second way to assess the possibility of side-effects is to test mutants whose presumed site of action is resistant to the drug. If phase-shifting is concomitantly reduced in these mutants, one may be more confident that side-effects are not responsible for phase-resetting action (46). The third method is to check whether derivatives of a drug which are inactive at the presumed biochemical target are nonetheless able to phase-shift the clock, as tested by Jacklet (30) for derivatives of the protein synthesis inhibitor anisomycin. Finally, a fourth control is to test various drugs which inhibit the same overall biochemical process, but by different mechanisms. A prime example of this control is the testing of various drugs which inhibit protein synthesis at different sites: cycloheximide, anisomycin, puromycin, streptimidone et al. (e.g., A/Gp-13-24, E/Ac-15-18, see Fig. 6).

Another problem with pulse application of chemicals/drugs is that the organism may not recover quickly from the inhibition or stimulation (whereas, the recovery from light pulses is usually considered to be rapid). The times at which (a) a drug/chemical has begun to significantly affect a targeted biochemical process and (b) the targeted process has significantly recovered from the effects of the drug/chemical may be quite different from the times of drug/ chemical (a) addition and (b) washout. These are important considerations in comparing PRCs for different drugs/chemicals whose penetration/recovery kinetics may differ, or even in comparing PRCs for the same drug/chemical in different organisms whose permeability characteristics may differ. Although this problem may seem obvious, it has been scarcely discussed in the literature and seems to be little appreciated.

The best solution would be to measure the penetration/recovery times for each drug/chemical and for each organism. One of the very few examples where such measurements have been made is that of the inhibition of protein synthesis by anisomycin in Gonyaulax . In this cell, protein synthesis is significantly inhibited by anisomycin within 5 minutes after the cells are exposed to the drug. But the recovery of protein synthesis after the drug is removed can require much more time, depending upon the concentration used. We have found that the recovery of Gonyaulax cells from anisomycin can require hours (perhaps even days) and that this recovery is a function of the concentration of the inhibitor which was administered to the cells: the higher the concentration, the longer the effective duration of the pulse (48). This is significant because increasing the duration of phase-shifting stimuli usually results in an increase of the magnitude of phase resetting and, for prolonged stimuli, often shifts the circadian time of the PRC's breakpoint.

This effect may help to explain the shifting breakpoints of PRCs to various doses of anisomycin in Gonyaulax (70). As shown in Fig. 10, one hour pulses of anisomycin at increasing concentrations to Gonyaulax cells yields a Type 1 to Type 0 transition (Fig. 10A to 10B), then a progressive shift of the breakpoint to the left (A/Gp-19-23), or as displacement of the PRC downwards, depending upon your perspective (70). In Fig. 10D, the breakpoint is seen to have come through almost a full cycle. Presumably, the state variables are being pushed farther and farther out on the phase plane as the concentration is progressively increased. The effects of anisomycin persist after washout and the duration of that persistence is determined by the dose originally presented as the dose of the anisomycin pulse is increased, the amount of time required for full recovery increases. Thus, a dose response curve for anisomycin in Gonyaula:r is simultaneously a duration response curve. Consequently, PRCs for various doses of a drug/chemical can exhibit analogous results to that of PRCs for various durations of light stimuli, which exhibit shifting breakpoints if the duration of the light pulse is long enough, as mentioned above for the series of Sarcophaga PRCs (Fig. 2).

Furthermore, the ambient temperature might also affect recovery kinetics and may account for the shift in the PRC to cycloheximide at 20 C vs. 25 C that has been observed in Acetabularia (A/Am-2,3 ref. 4). Thus, one can never be completely certain when a chemical/drug pulse ends unless the recovery of the biochemical process which is affected is directly measured. It is for this reason that I favor the use of stimulus onset as the standard stimulus marker for PRCs. (In truth, the effective onset may also be uncertain since the kinetics of drug penetration can vary. In general, however, recovery usually depends upon more factors than penetration and is therefore, likely to be more variable.)

(D) PRCs as Phase Markers for the Oscillator

PRCs have been used frequently to probe phase position of the pacemaker underlying rhythmic behavior. This technique has led to several insights into essential features of circadian organization in multicellular organisms. One example is that many multicellular organisms display "transient" cycles after a light pulse given in the late subjective night (advance phase-shift), whereas light pulses given in the early subjective night (delay phase-shifts) usually provoke few if any transients (54). This observation prompted the hypothesis of multiple, hierarchically coupled oscillators within multicellular organisms: overt rhythms are directly controlled by "slave" (B) oscillators which are synchronized to a "master" (A) circadian oscillator. This master oscillator is, in turn, entrained to the solar cycle by the resetting mechanism reflected by the PRC (54).

This hypothesis relies upon evidence from a special kind of PRC experiment, the elegant 2-pulse PRC, which shows that the pacemaker resets rapidly after a light pulse. These results are interpreted to mean that the transient cycles observed after an advance phase-shift are a reflection of the resynchronization of the intermediary slave (B) oscillator, not of the master (A) oscillator (54). At the present time, transient cycles after phase resetting by light have not been documented for unicellular organisms. This may mean that rhythms in single cells are controlled directly by light-sensitive pacemakers without any intermediary slave oscillators. Two-pulse light PRC experiments have been performed in Drosophila (54, 55), hamsters (H/Ma-12-13), and sparrows (3).

In addition, Hobohm et al. (28) have used the two-pulse PRC paradigm to answer a different question: is the oscillator of Gonyaulax precessing under some conditions in which the overt rhythm is not expressed ? The answer was "yes," as indicated by two-pulse PRCs using pulses of the protein synthesis inhibitor anisomycin as stimuli.

In addition to using chemical/drug PRCs to identify biochemical processes which act as state variables/parameters in the clockwork, they have also been used for mapping phases of the circadian pacemaker. This information can then be used to determine whether mutations or pharmacological treatments which affect tau exert their effects throughout the cycle or during only a fraction of the pacemaker's time course. The first use of a PRC for this purpose was the study of Daan and Pittendrigh which used the light PRC to map the tau-lengthening effect of deuterium oxide through the circadian cycle of the mouse (see H/Mm-1,2). Another example has been to determine whether the PRC has been deformed by exposure to various photoperiods or T-cycles (H/ Ma-6-11, H/Pd-2-4). More recently, Nakashima (45) has used light, temperature, and chemical/drug PRCs to map the phases affected by tau-mutations in Neurospora, and Kondo (41) has used amino acid analogs as resetting stimuli to map pacemaker phases in the duckweed Lemna. The use of PRCs to these various stimuli which have breakpoints spread throughout the pacemaker's cycle has allowed Nakashima and Kondo to do a much finer mapping of the circadian phases effected by the mutations than a single PRC would have permitted.

Mapping the pacemaker with chemical/drug PRCs can be fraught with problems, however, for the same reason that various concentrations of anisomycin shift the phase of the breakpoint in Gonyaulax (Fig. 10). Imagine for a moment using anisomycin to map the pacemaker in Gonyaulex under two different conditions (e.g., strain, mutation, temperature, background illumination). If the cells' sensitivity to anisomycin (or recovery from anisomycin) is altered by the differing conditions, then the cells can respond differently to the same concentration of anisomycin. This means that the clocks of cells under two different conditions might be perfectly in phase, but the phase of the PRCs elicited by Type 0 resetting by anisomycin could be quite different. This is a potential pitfall of using chemical/drug PRCs to map any pacemaker. Of course, recovery from some drugs may be much faster than as in the case of anisomycin in Gonyaulax. Nevertheless, it is obvious in retrospect that many chemicals/drugs will have effects long after the drug's washout. Therefore this is a trap which must be avoided whenever a chemical/drug is used to map the pacemaker.

If this is true, is there any value for measuring complete PRCs for drugs/ chemicals? Does measuring phase responsiveness at phases throughout the circadian cycle tell us anything more than data from a single phase point ? I think the answer is yes, for several reasons. First, observing phase-shifts at various phases reassures us that the result is not an artifact. Second, knowing the phase responsiveness throughout the cycle will be useful for later modelling of the pacemaker or for designing future experiments. Finally, responsiveness must be measured at many phases in order to detect discontinuities or to distinguish Type 1 from Type 0 resetting this information will undoubtably be crucial when the time comes to model the pacemaker's biochemistry.

Can this problem be circumvented ? Yes. First, low concentrations of a chemical/drug which elicit only Type 1 resetting should yield PRCs which are an accurate reflection of the pacemaker's phase. If the larger magnitude of Type 0 resetting is helpful towards precise assignment of PRC phase, then the concentration used must be just above threshold for Type 0 resetting before any shifting of breakpoint occurs. This Type 1 to Type 0 transition must be independently measured for each condition which is considered. The ultimate conclusion is that phase resetting to many concentrations must be measured to ensure that a chemical/drug PRC can be used to map the pacemaker under Type 0 resetting conditions. These controls have been done by Nakashima in various publications and also by Kondo (41).

(E) PRCs as Gauges of Oscillator Amplitude

Nakashima's light PRCs at different ambient temperatures (C/Nc-11-15) provide an example of how reinterpretating PRC data in terms of limit cycles can be interesting. The data, depicted in Figs. 11A, B, C, shows that the amplitude of the PRC to light decreases as the ambient temperature is increased. This might simply mean that the phototransduction mechanism becomes less efficient at higher temperatures. On the other hand, if one assumes that the light pulses move the state variables the same distance at all temperatures, a more interesting model emerges. Lakin-Thomas et al. (43) hypothesized that these data indicate that the amplitude (= diameter) of the limit cycle may increase as the temperature is raised. Thus, the same stimulus strength can provoke Type 1 resetting at high temperature (large diameter limit cycle) or Type 0 resetting at lower temperature (smaller diameter limit cycle) (shown in Fig. 11D). After proposing this interesting idea of amplitude resetting, LakinThomas et al. (43) expand their model to suggest that changes in the amplitude of limit cycles may also explain period changes of mutants, where the circumference of the limit cycle is proportional to tau.

But I think that the idea of amplitude changes is even more interesting if one hypothesizes that tau does not change as the amplitude of the limit cycle changes. Indeed, such an assumption allows a limit cycle model of temperature compensation of tau. If the amplitude of the limit cycle is larger at higher temperatures, then the state parameters can be specified so that the angular velocity (tau) is the same at different limit-cycle amplitudes. If the angular velocity is conserved at different temperatures, then the linear velocity along the circumference must be faster at higher temperature (= larger circumference). In that case, the biochemical reactions which modulate the state variables can be temperature dependent, as indicated by the larger circumference at higher temperature. By the very nature of the cycle inherent in a limit cycle, temperature-dependent reactions (linear velocity increases with temperature) can become temperature independent rhythms (angular velocity = tau that does not change with temperature) if the amplitude of the limit cycle increases as the temperature increases.

In addition to Nakashima's light PRCs at different temperatures, we have observed a similar temperature-dependency using pulses of protein-synthesis inhibitors as stimuli (4). Unfortunately, temperature dependency of light induced phase-resetting has not been universally observed. Pittendrigh did not find the light PRCs of Drosophila to be temperature dependent (F/Dp-2,7,9,10). However, the Drosophila PRCs were measured with a saturating light stimulus, which could be so large that it would not detect significant changes in the amplitude of the limit cycle. In Chlamydomonas, I have measured PRCs with sub-saturating light fluence at 18ºC and 25ºC, and have yet not detected any significant differences (unpublished observations). At present, this intriguing hypothesis needs more testing.

In this paper, I have discussed what PRCs are, how they can be measured, and my opinion as to how they should be plotted. I have also mentioned why plotting phase resetting data in a PTC format is also advantageous in some circumstances. Finally, I have described research topics in which phase-resetting data has provided crucial insights: entrainment, phototransduction, pacemaker mechanism, phase markers of the pacemaker, and gauges of oscillator amplitude. Clearly, PRCs/PTCs have enlightened us, and will continue to do so.

Although their opinions are not necessarily those expressed in this paper, I thank the following for invaluable discussions about PRCs: Drs. Woody Hastings, Takao Kondo, Richard Kronauer, Terry Page, Steven Strogatz, and Joseph Takahashi. I also thank Drs. Van Gooch and Takao Kondo for making a computer version of the PRC Atlas possible, which has facilitated the analysis and plotting of PRCs from the Atlas, and Grace Monty, for her expert assistance in preparing the manuscript. I am especially indebted to Drs. Colin Pittendrigh and Patricia DeCoursey for initiating and encouraging the compilation of PRCs into an Atlas.

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A/D = ratio of area of advance phase shifts to area of delay phase shifts in a PRC

pacemaker = oscillator = clock = master (A) oscillator

t = tau = free-running period

y = phase relationship of one cyclic phenomenon to another phase reference point

( f r ) = a phase of the rhythm used as a reference point

PP c = "complete" photoperiod

PP s = "skeleton" photoperiod

PRC = phase response curve = a plot of circadian time of stimulus ("old phase") versus phase shift

## Discussion

### Sensitivity Analysis

#### What is a sensitivity analysis in clinical research?

Sensitivity Analysis (SA) is defined as “a method to determine the robustness of an assessment by examining the extent to which results are affected by changes in methods, models, values of unmeasured variables, or assumptions” with the aim of identifying “results that are most dependent on questionable or unsupported assumptions” [2]. It has also been defined as “a series of analyses of a data set to assess whether altering any of the assumptions made leads to different final interpretations or conclusions” [3]. Essentially, SA addresses the “what-if-the-key-inputs-or-assumptions-changed”-type of question. If we want to know whether the results change when something about the way we approach the data analysis changes, we can make the change in our analysis approach and document the changes in the results or conclusions. For more detailed coverage of SA, we refer the reader to these references [4–7].

#### Why is sensitivity analysis necessary?

The design and analysis of clinical trials often rely on assumptions that may have some effect, influence or impact on the conclusions if they are not met. It is important to assess these effects through sensitivity analyses. Consistency between the results of primary analysis and the results of sensitivity analysis may strengthen the conclusions or credibility of the findings. However, it is important to note that the definition of consistency may depend in part on the area of investigation, the outcome of interest or even the implications of the findings or results.

It is equally important to assess the robustness to ensure appropriate interpretation of the results taking into account the things that may have an impact on them. Thus, it imperative for every analytic plan to have some sensitivity analyses built into it.

The United States (US) Food and Drug Administration (FDA) and the European Medicines Association (EMEA), which offer guidance on Statistical Principles for Clinical Trials, state that “it is important to evaluate the robustness of the results and primary conclusions of the trial.” Robustness refers to “the sensitivity of the overall conclusions to various limitations of the data, assumptions, and analytic approaches to data analysis” [8]. The United Kingdom (UK) National Institute of Health and Clinical Excellence (NICE) also recommends the use of sensitivity analysis in “exploring alternative scenarios and the uncertainty in cost-effectiveness results” [9].

#### How often is sensitivity analysis reported in practice?

To evaluate how often sensitivity analyses are used in medical and health research, we surveyed the January 2012 editions of major medical journals (British Medical Journal, New England Journal of Medicine, the Lancet, Journal of the American Medical Association and the Canadian Medical Association Journal) and major health economics journals (Pharmaco-economics, Medical Decision making, European Journal of Health Economics, Health Economics and the Journal of Health Economics). From every article that included some form of statistical analyses, we evaluated: i) the percentage of published articles that reported results of some sensitivity analyses and ii) the types of sensitivity analyses that were performed. Table 1 provides a summary of the findings. Overall, the point prevalent use of sensitivity analyses is about 26.7% (36/135) —which seems very low. A higher percentage of papers published in health economics than in medical journals (30.8% vs. 20.3%) reported some sensitivity analyses. Among the papers in medical journals, 18 (28.1%) were RCTs, of which only 3 (16.6%) reported sensitivity analyses. Assessing robustness of the findings to different methods of analysis was the most common type of sensitivity analysis reported in both types of journals. Therefore despite their importance, sensitivity analyses are under-used in practice. Further, sensitivity analyses are more common in health economics research—for example in conducting cost-effectiveness analyses, cost-utility analyses or budget-impact analyses—than in other areas of health or medical research.

### Types of sensitivity analyses

In this section, we describe scenarios that may require sensitivity analyses, and how one could use sensitivity analyses to assess the robustness of the statistical analyses or findings of RCTs. These are not meant to be exhaustive, but rather to illustrate common situations where sensitivity analyses might be useful to consider (Table 2). In each case, we provide examples of actual studies where sensitivity analyses were performed, and the implications of these sensitivity analyses.

#### Impact of outliers

An outlier is an observation that is numerically distant from the rest of the data. It deviates markedly from the rest of the sample from which it comes [14, 15]. Outliers are usually exceptional cases in a sample. The problem with outliers is that they can deflate or inflate the mean of a sample and therefore influence any estimates of treatment effect or association that are derived from the mean. To assess the potential impact of outliers, one would first assess whether or not any observations meet the definition of an outlier—using either a boxplot or z-scores [16]. Second, one could perform a sensitivity analysis with and without the outliers.

In a cost–utility analysis of a practice-based osteopathy clinic for subacute spinal pain, Williams et al. reported lower costs per quality of life year ratios when they excluded outliers [17]. In other words, there were certain participants in the trial whose costs were very high, and were making the average costs look higher than they probably were in reality. The observed cost per quality of life year was not robust to the exclusion of outliers, and changed when they were excluded.

A primary analysis based on the intention-to-treat principle showed no statistically significant differences in reducing depression between a nurse-led cognitive self-help intervention program compared to standard care among 218 patients hospitalized with angina over 6 months. Some sensitivity analyses in this trial were performed by excluding participants with high baseline levels of depression (outliers) and showed a statistically significant reduction in depression in the intervention group compared to the control. This implies that the results of the primary analysis were affected by the presence of patients with baseline high depression [18].

#### Impact of non-compliance or protocol deviations

In clinical trials some participants may not adhere to the intervention they were allocated to receive or comply with the scheduled treatment visits. Non-adherence or non-compliance is a form of protocol deviation. Other types of protocol deviations include switching between intervention and control arms (i.e. treatment switching or crossovers) [19, 20], or not implementing the intervention as prescribed (i.e. intervention fidelity) [21, 22].

Protocol deviations are very common in interventional research [23–25]. The potential impact of protocol deviations is the dilution of the treatment effect [26, 27]. Therefore, it is crucial to determine the robustness of the results to the inclusion of data from participants who deviate from the protocol. Typically, for RCTs the primary analysis is based on an intention-to-treat (ITT) principle—in which participants are analyzed according to the arm to which they were randomized, irrespective of whether they actually received the treatment or completed the prescribed regimen [28, 29]. Two common types of sensitivity analyses can be performed to assess the robustness of the results to protocol deviations: 1) per-protocol (PP) analysis—in which participants who violate the protocol are excluded from the analysis [30] and 2) as-treated (AT) analysis—in which participants are analyzed according to the treatment they actually received [30]. The PP analysis provides the ideal scenario in which all the participants comply, and is more likely to show an effect whereas the ITT analysis provides a “real life” scenario, in which some participants do not comply. It is more conservative, and less likely to show that the intervention is effective. For trials with repeated measures, some protocol violations which lead to missing data can be dealt with alternatively. This is covered in more detail in the next section.

A trial was designed to investigate the effects of an electronic screening and brief intervention to change risky drinking behaviour in university students. The results of the ITT analysis (on all 2336 participants who answered the follow-up survey) showed that the intervention had no significant effect. However, a sensitivity analysis based on the PP analysis (including only those with risky drinking at baseline and who answered the follow-up survey n = 408) suggested a small beneficial effect on weekly alcohol consumption [31]. A reader might be less confident in the findings of the trial because of the inconsistency between the ITT and PP analyses—the ITT was not robust to sensitivity analyses. A researcher might choose to explore differences in the characteristics of the participants who were included in the ITT versus the PP analyses.

A study compared the long-term effects of surgical versus non-surgical management of chronic back pain. Both the ITT and AT analyses showed no significant difference between the two management strategies [32]. A reader would be more confident in the findings because the ITT and AT analyses were consistent—the ITT was robust to sensitivity analyses.

#### Impact of missing data

Missing data are common in every research study. This is a problem that can be broadly defined as “missing some information on the phenomena in which we are interested” [33]. Data can be missing for different reasons including (1) non-response in surveys due to lack of interest, lack of time, nonsensical responses, and coding errors in data entry/transfer (2) incompleteness of data in large data registries due to missing appointments, not everyone is captured in the database, and incomplete data and (3) missingness in prospective studies as a result of loss to follow up, dropouts, non-adherence, missing doses, and data entry errors.

The choice of how to deal with missing data would depend on the mechanisms of missingness. In this regard, data can be missing at random (MAR), missing not at random (MNAR), or missing completely at random (MCAR). When data are MAR, the missing data are dependent on some other observed variables rather than any unobserved one. For example, consider a trial to investigate the effect of pre-pregnancy calcium supplementation on hypertensive disorders in pregnancy. Missing data on the hypertensive disorders is dependent (conditional) on being pregnant in the first place. When data are MCAR, the cases with missing data may be considered a random sample drawn from all the cases. In other words, there is no “cause” of missingness. Consider the example of a trial comparing a new cancer treatment to standard treatment in which participants are followed at 4, 8, 12 and 16 months. If a participant misses the follow up at the 8th and 16th months and these are unrelated to the outcome of interest, in this case mortality, then this missing data is MCAR. Reasons such as a clinic staff being ill or equipment failure are often unrelated to the outcome of interest. However, the MCAR assumption is often challenging to prove because the reason data is missing may not be known and therefore it is difficult to determine if it is related to the outcome of interest. When data are MNAR, missingness is dependent on some unobserved data. For example, in the case above, if the participant missed the 8th month appointment because he was feeling worse or the 16th month appointment because he was dead, the missingness is dependent on the data not observed because the participant was absent. When data are MAR or MCAR, they are often referred to as ignorable (provided the cause of MAR is taken into account). MNAR on the other hand, is nonignorable missingness. Ignoring the missingness in such data leads to biased parameter estimates [34]. Ignoring missing data in analyses can have implications on the reliability, validity and generalizability of research findings.

The best way to deal with missing data is prevention, by steps taken in the design and data collection stages, some of which have been described by Little et al. [35]. But this is difficult to achieve in most cases. There are two main approaches to handling missing data: i) ignore them—and use complete case analysis and ii) impute them—using either single or multiple imputation techniques. Imputation is one of the most commonly used approaches to handling missing data. Examples of single imputation methods include hot deck, cold deck method, mean imputation, regression technique, last observation carried forward (LOCF) and composite methods—which uses a combination of the above methods to impute missing values. Single imputation methods often lead to biased estimates and under-estimation of the true variability in the data. Multiple imputation (MI) technique is currently the best available method of dealing with missing data under the assumption that data are missing at random (MAR) [33, 36–38]. MI addresses the limitations of single imputation by using multiple imputed datasets which yield unbiased estimates, and also accounts for the within- and between-dataset variability. Bayesian methods using statistical models that assume a prior distribution for the missing data can also be used to impute data [35].

It is important to note that ignoring missing data in the analysis would be implicitly assuming that the data are MCAR, an assumption that is often hard to verify in reality.

There are some statistical approaches to dealing with missing data that do not necessarily require formal imputation methods. For example, in studies using continuous outcomes, linear mixed models for repeated measures are used for analyzing outcomes measured repeatedly over time [39, 40]. For categorical responses or count data, generalized estimating equations [GEE] and random-effects generalized linear mixed models [GLMM] methods may be used [41, 42]. In these models it is assumed that missing data are MAR. If this assumption is valid, then the complete-case analysis by including predictors of missing observations will provide consistent estimates of the parameter.

The choice of whether to ignore or impute missing data, and how to impute it, may affect the findings of the trial. Although one approach (ignore or impute, and if the latter, how to impute) should be made a priori, a sensitivity analysis can be done with a different approach to see how “robust” the primary analysis is to the chosen method for handling missing data.

A 2011 paper reported the sensitivity analyses of different strategies for imputing missing data in cluster RCTs with a binary outcome using the community hypertension assessment trial (CHAT) as an example. They found that variance in the treatment effect was underestimated when the amount of missing data was large and the imputation strategy did not take into account the intra-cluster correlation. However, the effects of the intervention under various methods of imputation were similar. The CHAT intervention was not superior to usual care [43].

In a trial comparing methotrexate with to placebo in the treatment of psoriatic arthritis, the authors reported both an intention-to-treat analysis (using multiple imputation techniques to account for missing data) and a complete case analysis (ignoring the missing data). The complete case analysis, which is less conservative, showed some borderline improvement in the primary outcome (psoriatic arthritis response criteria), while the intention-to-treat analysis did not [44]. A reader would be less confident about the effects of methotrexate on psoriatic arthritis, due to the discrepancy between the results with imputed data (ITT) and the complete case analysis.

#### Impact of different definitions of outcomes (e.g. different cut-off points for binary outcomes)

Often, an outcome is defined by achieving or not achieving a certain level or threshold of a measure. For example in a study measuring adherence rates to medication, levels of adherence can be dichotomized as achieving or not achieving at least 80%, 85% or 90% of pills taken. The choice of the level a participant has to achieve can affect the outcome—it might be harder to achieve 90% adherence than 80%. Therefore, a sensitivity analysis could be performed to see how redefining the threshold changes the observed effect of a given intervention.

In a trial comparing caspofungin to amphotericin B for febrile neutropoenic patients, a sensitivity analysis was conducted to investigate the impact of different definitions of fever resolution as part of a composite endpoint which included: resolution of any baseline invasive fungal infection, no breakthrough invasive fungal infection, survival, no premature discontinuation of study drug, and fever resolution for 48 hours during the period of neutropenia. They found that response rates were higher when less stringent fever resolution definitions were used, especially in low-risk patients. The modified definitions of fever resolution were: no fever for 24 hours before the resolution of neutropenia no fever at the 7-day post-therapy follow-up visit and removal of fever resolution completely from the composite endpoint. This implies that the efficacy of both medications depends somewhat on the definition of the outcomes [45].

In a phase II trial comparing minocycline and creatinine to placebo for Parkinson’s disease, a sensitivity analysis was conducted based on another definition (threshold) for futility. In the primary analysis a predetermined futility threshold was set at 30% reduction in mean change in Unified Parkinson’s Disease Rating Scale (UPDRS) score, derived from historical control data. If minocycline or creatinine did not bring about at least a 30% reduction in UPDRS score, they would be considered as futile and no further testing will be conducted. Based on the data derived from the current control (placebo) group, a new threshold of 32.4% (more stringent) was used for the sensitivity analysis. The findings from the primary analysis and the sensitivity analysis both confirmed that that neither creatine nor minocycline could be rejected as futile and should both be tested in Phase III trials [46]. A reader would be more confident of these robust findings.

#### Impact of different methods of analysis to account for clustering or correlation

Interventions can be administered to individuals, but they can also be administered to clusters of individuals, or naturally occurring groups. For example, one might give an intervention to students in one class, and compare their outcomes to students in another class – the class is the cluster. Clusters can also be patients treated by the same physician, physicians in the same practice center or hospital, or participants living in the same community. Likewise, in the same trial, participants may be recruited from multiple sites or centers. Each of these centers will represent a cluster. Patients or elements within a cluster often have some appreciable degree of homogeneity as compared to patients between clusters. In other words, members of the same cluster are more likely to be similar to each other than they are to members of another cluster, and this similarity may then be reflected in the similarity or correlation measure, on the outcome of interest.

There are several methods of accounting or adjusting for similarities within clusters, or “clustering” in studies where this phenomenon is expected or exists as part of the design (e.g., in cluster randomization trials). Therefore, in assessing the impact of clustering one can build into the analytic plans two forms of sensitivity analyses: i) analysis with and without taking clustering into account—comparing the analysis that ignores clustering (i.e. assumes that the data are independent) to one primary method chosen to account for clustering ii) analysis that compares several methods of accounting for clustering.

Correlated data may also occur in longitudinal studies through repeat or multiple measurements from the same patient, taken over time or based on multiple responses in a single survey. Ignoring the potential correlation between several measurements from an individual can lead to inaccurate conclusions [47].

Here are a few references to studies that compared the outcomes that resulted when different methods were/were not used to account for clustering. Noteworthy, is the fact that the analytical approaches for cluster-RCTs and multi-site RCTs are similar.

Ma et al. performed sensitivity analyses of different methods of analysing cluster RCTs [48]. In this paper they compared three cluster-level methods (un-weighted linear regression, weighted linear regression and random-effects meta-regression) to six individual level analysis methods (standard logistic regression, robust standard errors approach, GEE, random effects meta-analytic approach, random-effects logistic regression and Bayesian random-effects regression). Using data from the CHAT trial, in this analysis, all nine methods provided similar results, re-enforcing the hypothesis that the CHAT intervention was not superior to usual care.

Peters et al. conducted sensitivity analyses to compare different methods—three cluster-level (un-weighted regression of practice log odds, regression of log odds weighted by their inverse variance and random-effects meta-regression of log odds with cluster as a random effect) and five individual-level methods (standard logistic regression ignoring clustering, robust standard errors, GEE, random-effects logistic regression and Bayesian random-effects logistic regression.)—for analyzing cluster randomized trials using an example involving a factorial design [13]. In this analysis, they demonstrated that the methods used in the analysis of cluster randomized trials could give varying results, with standard logistic regression ignoring clustering being the least conservative.

Cheng et al. used sensitivity analyses to compare different methods (six models for clustered binary outcomes and three models for clustered nominal outcomes) of analysing correlated data in discrete choice surveys [49]. The results were robust to various statistical models, but showed more variability in the presence of a larger cluster effect (higher within-patient correlation).

A trial evaluated the effects of lansoprazole on gastro-esophageal reflux disease in children from 19 clinics with asthma. The primary analysis was based on GEE to determine the effect of lansoprazole in reducing asthma symptoms. Subsequently they performed a sensitivity analysis by including the study site as a covariate. Their finding that lansoprazole did not significantly improve symptoms was robust to this sensitivity analysis [50].

In addition to comparing the performance of different methods to estimate treatment effects on a continuous outcome in simulated multicenter randomized controlled trials [12], the authors used data from the Computerization of Medical Practices for the Enhancement of Therapeutic Effectiveness (COMPETE) II [51] to assess the robustness of the primary results (based on GEE to adjust for clustering by provider of care) under different methods of adjusting for clustering. The results, which showed that a shared electronic decision support system improved care and outcomes in diabetic patients, were robust under different methods of analysis.

#### Impact of competing risks in analysis of trials with composite outcomes

A competing risk event happens in situations where multiple events are likely to occur in a way that the occurrence of one event may prevent other events from being observed [48]. For example, in a trial using a composite of death, myocardial infarction or stroke, if someone dies, they cannot experience a subsequent event, or stroke or myocardial infarction—death can be a competing risk event. Similarly, death can be a competing risk in trials of patients with malignant diseases where thrombotic events are important. There are several options for dealing with competing risks in survival analyses: (1) to perform a survival analysis for each event separately, where the other competing event(s) is/are treated as censored the common representation of survival curves using the Kaplan-Meier estimator is in this context replaced by the cumulative incidence function (CIF) which offers a better interpretation of the incidence curve for one risk, regardless of whether the competing risks are independent (2) to use a proportional sub-distribution hazard model (Fine & Grey approach) in which subjects that experience other competing events are kept in the risk set for the event of interest (i.e. as if they could later experience the event) (3) to fit one model, rather than separate models, taking into account all the competing risks together (Lunn-McNeill approach) [13]. Therefore, the best approach to assessing the influence of a competing risk would be to plan for sensitivity analysis that adjusts for the competing risk event.

A previously-reported trial compared low molecular weight heparin (LMWH) with oral anticoagulant therapy for the prevention of recurrent venous thromboembolism (VTE) in patients with advanced cancer, and a subsequent study presented sensitivity analyses comparing the results from standard survival analysis (Kaplan-Meier method) with those from competing risk methods—namely, the cumulative incidence function (CIF) and Gray's test [52]. The results using both methods were similar. This strengthened their confidence in the conclusion that LMWH reduced the risk of recurrent VTE.

For patients at increased risk of end stage renal disease (ESRD) but also of premature death not related to ESRD, such as patients with diabetes or with vascular disease, analyses considering the two events as different outcomes may be misleading if the possibility of dying before the development of ESRD is not taken into account [49]. Different studies performing sensitivity analyses demonstrated that the results on predictors of ESRD and death for any cause were dependent on whether the competing risks were taken into account or not [53, 54], and on which competing risk method was used [55]. These studies further highlight the need for a sensitivity analysis of competing risks when they are present in trials.

#### Impact of baseline imbalance in RCTs

In RCTs, randomization is used to balance the expected distribution of the baseline or prognostic characteristics of the patients in all treatment arms. Therefore the primary analysis is typically based on ITT approach unadjusted for baseline characteristics. However, some residual imbalance can still occur by chance. One can perform a sensitivity analysis by using a multivariable analysis to adjust for hypothesized residual baseline imbalances to assess their impact on effect estimates.

A paper presented a simulation study where the risk of the outcome, effect of the treatment, power and prevalence of the prognostic factors, and sample size were all varied to evaluate their effects on the treatment estimates. Logistic regression models were compared with and without adjustment for the prognostic factors. The study concluded that the probability of prognostic imbalance in small trials could be substantial. Also, covariate adjustment improved estimation accuracy and statistical power [56].

In a trial testing the effectiveness of enhanced communication therapy for aphasia and dysarthria after stroke, the authors conducted a sensitivity analysis to adjust for baseline imbalances. Both primary and sensitivity analysis showed that enhanced communication therapy had no additional benefit [57].

#### Impact of distributional assumptions

Most statistical analyses rely on distributional assumptions for observed data (e.g. Normal distribution for continuous outcomes, Poisson distribution for count data, or binomial distribution for binary outcome data). It is important not only to test for goodness-of-fit for these distributions, but to also plan for sensitivity analyses using other suitable distributions. For example, for continuous data, one can redo the analysis assuming a Student-T distribution—which is symmetric, bell-shaped distribution like the Normal distribution, but with thicker tails for count data, once can use the Negative-binomial distribution—which would be useful to assess the robustness of the results if over-dispersion is accounted for [52]. Bayesian analyses routinely include sensitivity analyses to assess the robustness of findings under different models for the data and prior distributions [58]. Analyses based on parametric methods—which often rely on strong distributional assumptions—may also need to be evaluated for robustness using non-parametric methods. The latter often make less stringent distributional assumptions. However, it is essential to note that in general non-parametric methods are less efficient (i.e. have less statistical power) than their parametric counter-parts if the data are Normally distributed.

Ma et al. performed sensitivity analyses based on Bayesian and classical methods for analysing cluster RCTs with a binary outcome in the CHAT trial. The similarities in the results after using the different methods confirmed the results of the primary analysis: the CHAT intervention was not superior to usual care [10].

A negative binomial regression model was used [52] to analyze discrete outcome data from a clinical trial designed to evaluate the effectiveness of a pre-habilitation program in preventing functional decline among physically frail, community-living older persons. The negative binomial model provided an improved fit to the data than the Poisson regression model. The negative binomial model provides an alternative approach for analyzing discrete data where over-dispersion is a problem [59].

Q: Do I need to adjust the overall level of significance for performing sensitivity analyses?

A: No. Sensitivity analysis is typically a re-analysis of either the same outcome using different approaches, or different definitions of the outcome—with the primary goal of assessing how these changes impact the conclusions. Essentially everything else including the criterion for statistical significance needs to be kept constant so that we can assess whether any impact is attributable to underlying sensitivity analyses.

Q: Do I have to report all the results of the sensitivity analyses?

A: Yes, especially if the results are different or lead to different a conclusion from the original results—whose sensitivity was being assessed. However, if the results remain robust (i.e. unchanged), then a brief statement to this effect may suffice.

Q: Can I perform sensitivity analyses posthoc?

A: It is desirable to document all planned analyses including sensitivity analyses in the protocol a priori. Sometimes, one cannot anticipate all the challenges that can occur during the conduct of a study that may require additional sensitivity analyses. In that case, one needs to incorporate the anticipated sensitivity analyses in the statistical analysis plan (SAP), which needs to be completed before analyzing the data. Clear rationale is needed for every sensitivity analysis. This may also occur posthoc.

Q: How do I choose between the results of different sensitivity analyses? (i.e. which results are the best?)

A: The goal of sensitivity analyses is not to select the “best” results. Rather, the aim is to assess the robustness or consistency of the results under different methods, subgroups, definitions, assumptions and so on. The assessment of robustness is often based on the magnitude, direction or statistical significance of the estimates. You cannot use the sensitivity analysis to choose an alternate conclusion to your study. Rather, you can state the conclusion based on your primary analysis, and present your sensitivity analysis as an example of how confident you are that it represents the truth. If the sensitivity analysis suggests that the primary analysis is not robust, it may point to the need for future research that might address the source of the inconsistency. Your study cannot answer the question which results are best? To answer the question of which method is best and under what conditions, simulation studies comparing the different approaches on the basis of bias, precision, coverage or efficiency may be necessary.

Q: When should one perform sensitivity analysis?

A: The default position should be to plan for sensitivity analysis in every clinical trial. Thus, all studies need to include some sensitivity analysis to check the robustness of the primary findings. All statistical methods used to analyze data from clinical trials rely on assumptions—which need to either be tested whenever possible, with the results assessed for robustness through some sensitivity analyses. Similarly, missing data or protocol deviations are common occurrences in many trials and their impact on inferences needs to be assessed.

Q: How many sensitivity analyses can one perform for a single primary analysis?

A: The number is not an important factor in determining what sensitivity analyses to perform. The most important factor is the rationale for doing any sensitivity analysis. Understanding the nature of the data, and having some content expertise are useful in determining which and how many sensitivity analyses to perform. For example, varying the ways of dealing with missing data is unlikely to change the results if 1% of data are missing. Likewise, understanding the distribution of certain variables can help to determine which cut points would be relevant. Typically, it is advisable to limit sensitivity analyses to the primary outcome. Conducting multiple sensitivity analysis on all outcomes is often neither practical, nor necessary.

Q: How many factors can I vary in performing sensitivity analyses?

A: Ideally, one can study the impact of all key elements using a factorial design—which would allow the assessment of the impact of individual and joint factors. Alternatively, one can vary one factor at a time to be able to assess whether the factor is responsible for the resulting impact (if any). For example, in a sensitivity analysis to assess the impact of the Normality assumption (analysis assuming Normality e.g. T-test vs. analysis without assuming Normality e.g. Based on a sign test) and outlier (analysis with and without outlier), this can be achieved through 2x2 factorial design.

Q: What is the difference between secondary analyses and sensitivity analyses?

A: Secondary analyses are typically analyses of secondary outcomes. Like primary analyses which deal with primary outcome(s), such analyses need to be documented in the protocol or SAP. In most studies such analyses are exploratory—because most studies are not powered for secondary outcomes. They serve to provide support that the effects reported in the primary outcome are consistent with underlying biology. They are different from sensitivity analyses as described above.

Q: What is the difference between subgroup analyses and sensitivity analyses?

A: Subgroup analyses are intended to assess whether the effect is similar across specified groups of patients or modified by certain patient characteristics [60]. If the primary results are statistically significant, subgroup analyses are intended to assess whether the observed effect is consistent across the underlying patient subgroups—which may be viewed as some form of sensitivity analysis. In general, for subgroup analyses one is interested in the results for each subgroup, whereas in subgroup “sensitivity” analyses, one is interested in the similarity of results across subgroups (ie. robustness across subgroups). Typically subgroup analyses require specification of the subgroup hypothesis and rationale, and performed through inclusion of an interaction term (i.e. of the subgroup variable x main exposure variable) in the regression model. They may also require adjustment for alpha—the overall level of significance. Furthermore, most studies are not usually powered for subgroup analyses.

## Discussion

Tools commonly used to analyze dose–response data (such as Prism) are not yet capable of computing GR metrics, which is the best method available for eliminating biases in measuring perturbagen dose–response in proliferating cells. Use of GR metrics makes it possible to reliably compare data on drug potency and efficacy across cell lines having different underlying rates of division, assayed for different lengths of time, or growing at different rates due to changes in culture conditions. Given properly processed data, the online and offline tools described here calculate GR values, fit these values to a sigmoidal curve, evaluate the significance of the sigmoidal fit using an F-test, and yield GR metrics. To avoid contaminating dose–response datasets with low reliability values extrapolated from poor fits, non-significant curve fits are replaced by a flat line, and response metrics are set to default values. After calculating the sensitivity metrics, users can quickly and simply visualize results, perform basic analyses, and produce publication-ready figures. Offline R-based GRcalculator tools are designed for computationally sophisticated users and those with proprietary data. The choice of R [7] for online and offline GR calculations facilitates re-use of existing tools for fitting dose–response curves [15] and has enabled creation of a GRmetrics Bioconductor [16] package to facilitate integration of GR metrics within R analytical workflows. For example, combining GRmetrics with the PharmacoGx [17] Bioconductor package facilitates the use of GR metrics in pharmacogenomics analyses.

Reproducibility has become a major concern in contemporary biomedical research and the use of GR metrics increases reproducibility by correcting for factors that are often poorly controlled in large-scale studies involving many cell lines. These factors include plating density and number of cell divisions [3]. Standardization of assay methodology [4] and of computational tools and pipelines for converting raw data into final results [5] are essential for making data acquisition and analysis consistent across experiments the GRcalculator meets these requirement and helps to avoid data processing artefacts. GRcalculator also serves as a repository for large-scale dose–response datasets that have been analyzed using the GR approach, thereby providing a reliable and reusable set of information for the community. The number of such datasets is currently small (primarily due to limitations in existing experimental data), but future dose–response data collected by the NIH LINCS Program will be released in GRcalculator and we anticipate that this will also be true of other efforts focused on characterizing the responses of cells to perturbation. We anticipate further development of the GR method and of other ways of calculating drug response over time [2, 18] and will therefore update the GRcalculator website as needed.

## Dose-Response Relationships

Regardless of how a drug effect occurs—through binding or chemical interaction—the concentration of the drug at the site of action controls the effect. However, response to concentration may be complex and is often nonlinear. The relationship between the drug dose, regardless of route used, and the drug concentration at the cellular level is even more complex (see Pharmacokinetics).

Dose-response data are typically graphed with the dose or dose function (eg, log10 dose) on the x-axis and the measured effect (response) on the y-axis. Because a drug effect is a function of dose and time, such a graph depicts the dose-response relationship independent of time. Measured effects are frequently recorded as maxima at time of peak effect or under steady-state conditions (eg, during continuous IV infusion). Drug effects may be quantified at the level of molecule, cell, tissue, organ, organ system, or organism.

A hypothetical dose-response curve has features that vary (see figure Hypothetical dose-response curve):

### What is the stability of Hygromycin B in solution?

The antibiotic is stable for at least 2 years at 4°C. It is stable for about one month at 37°C.

### What color is the solution?

The color can vary from light-yellow to dark-brown or caramel. The concentrated solution tends to have a darker appearance.

### What is the working concentration for selection?

The working concentration for the purpose of selection varies with cell type, media, growth conditions and cell metabolic rate. Recommended concentration for the selection of resistant cells is 25-1000 ug/mL. Commonly used concentrations for selection are 200 ug/mL for mammalian cells, 20-200 ug/mL for plant cells & bacteria cells and 200-1000 ug/mL for fungi. Your optimum concentration should be tested experimentally.

### How can non-transfected cells escape antibiotic selection?

Cells can escape selection if the antibiotic concentration is too low or if the cell density on the plate is too high. Additionally, cells that rapidly proliferate are killed faster than those that proliferate slowly. Control cells should die within 5-7 days after the addition of the antibiotic, allowing colonies of resistant cells to form by 10-14 days.

### How do I determine the toxic concentration?

Hygromycin B is added to the culture medium at a concentration that varies with the cell type transfected. A titration experiment for each cell type may therefore be performed to determine the amount of Hygromycin B needed to kill non-transfected cells. The working concentration for mammalian cell selection is normally between 50 ug/mL and 1 mg/mL, Plant cells: 20-200 ug/mL, Bacteria: 20-200 ug/mL and Fungi: 200 ug-1 mg/mL. Your appropriate concentration should be tested experimentally.

### How do I perform a dose response curve?

To determine the minimum concentration of antibiotic required to kill your non-transfected host cell line:

• Test ranges of concentrations (5-6) to ensure that you determine the minimum concentration necessary for your cell line.
• Seed cells at approximately 20-25% confluency on the appropriate number of plates for each time plate and allow cells to adhere overnight. For cells that require higher densities for viability, increase the number of cells seeded.
• The next day, substitute culture medium with medium containing varying concentrations of the antibiotic.
• Replenish the selective medium every 3-4 days.
• Count the number of viable cells at regular intervals to determine the appropriate concentration of antibiotic that prevents the growth of non-transfected cells. Select the concentration that kills the majority of the cells in the desired number of days (usually 7-10 days).

### How do I maintain Hygromycin-resistant phenotype of transfected cell lines?

To maintain Hygromycin-resistant phenotype of transfected cell lines and for the elimination of revertants, cells may be regularly cultured in medium containing Hygromycin B at the same concentration used for the initial selection.

### When is replacement of media required?

Replacement of the culture media containing Hygromycin B is needed only if nutritional components are consumed by the cells cultured. Acidification of the culture medium is normally a sign of consumption. Utilizing phenol red or media containing phenol red will aid in the detection of acidification. In this case, the media will turn yellow.

### Is Hygromycin B sensitive to acids?

It is sensitive to high concentrations of acids, however a brief a exposure to dilute acids does not affect its stability.

### Can we increase the sensitivity of our cells to the antibiotic?

The sensitivity to Hygromycin B can be increased by increasing the pH of the medium. Sensitivity appears to be greater at lower salt concentrations.

### What enzyme inactivates Hygromycin B?

Hygromycin phosphotransferase (hpt) inactivates the antibiotic through phosphorylation. The hygromycin phosphotransferase gene (hpt, hph or aphIV) codes for hygromycin phosphotransferase, and is utilized as a selectable marker gene for both plant and animal systems.

### What is Hygromycin B’s mode of action?

The compound binds to the 30S ribosomal subunit and affects the fidelity of translation.

### How can I roughly assess the compound’s purity?

The color of a Hygromycin solution is a good indicator: the clearer it is, the purer. Therefore, caramels to brown colored solutions are of inferior grade. The color difference may not be very dramatic with a freshly-made low concentration hygromycin solution.

### What is the spectrum of inhibition?

An aminoglycoside antibiotic that inhibits protein synthesis in bacteria, fungi, and higher eukaryotes. Spectrum also stated as Gram (+), Gram (-) bacilli aerobes and facultative anaerobes.

### Is Hygromycin B hazardous?

Yes. It is considered a toxic and hazardous material. Care should be taken to avoid contact with skin and eyes. Please consult and review Material Safety Data Sheet (MSDS) prior to handling.