Where does the inverse seconds unit come from in the association constant?

Where does the inverse seconds unit come from in the association constant?

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I'm working to determine Kd(s) kinetically by generating association and disassociation curves. Kon (association constant, or on-rate) is in inverse seconds multiplied by inverse molar. I get that the molar comes from concentration, but where do the seconds come from?

First order association reaction:

$$ce{A + B <=>[k_f][k_r] AB}$$

Rate of association (units: concentration per time) = $k_f imes[A] imes[B]$
Since [A] and [B] are concentrations and LHS is conc. per time the units of $k_f$ has to be such that it gives conc per time when multiplied by conc2 ([A] ×[B])
Which is conc-1 × time-1

Rate of dissociation (conc. per time) = $k_r imes[AB]$
Therefore units of $k_r$ would be time-1.

Dissociation constant is actually $dfrac{k_r}{k_f}$; Its units therefore would be conc.

Live-cell FLIM-FRET using a commercially available system

Colleen M. Castellani , . Thomas J. Maresca , in Methods in Cell Biology , 2020


Förster resonance energy transfer (FRET)-based sensors have been powerful tools in cell biologists' toolkit for decades. Informed by fundamental understanding of fluorescent proteins, protein-protein interactions, and the structural biology of reporter components, researchers have been able to employ creative design approaches to build sensors that are uniquely capable of probing a wide range of phenomena in living cells including visualization of localized calcium signaling, sub-cellular activity gradients, and tension generation to name but a few. While FRET sensors have significantly impacted many fields, one must also be cognizant of the limitations to conventional, intensity-based FRET measurements stemming from variation in probe concentration, sensitivity to photobleaching, and bleed-through between the FRET fluorophores. Fluorescence lifetime imaging microscopy (FLIM) largely overcomes the limitations of intensity-based FRET measurements. In general terms, FLIM measures the time, which for the reporters described in this chapter is nanoseconds (ns), between photon absorption and emission by a fluorophore. When FLIM is applied to FRET sensors (FLIM-FRET), measurement of the donor fluorophore lifetime provides valuable information such as FRET efficiency and the percentage of reporters engaged in FRET. This chapter introduces fundamental principles of FLIM-FRET toward informing the practical application of the technique and, using two established FRET reporters as proofs of concept, outlines how to use a commercially available FLIM system.

"Linearized" Arrhenius Equation

Then, a plot of (ln k) vs. (1/T) and all variables can be found.

This form of the Arrhenius equation makes it easy to determine the slope and y-intercept from an Arrhenius plot. It is also convenient to note that the above equation shows the connection between temperature and rate constant. As the temperature increases, the rate constant decreases according to the plot. From this connection we can infer that the rate constant is inversely proportional to temperature.

Integrated Form

The integrated form of the Arrhenius equation is also useful (Equation ef). This variation of the Arrhenius equation involves the use of two Arrhenius plots constructed on the same graph to determine the activation energy. The above equation, shows temperature's effect on multiple rate constants. This allows easy inference of the rate constants' sensitivity to activation energy and temperature changes. If the activation energy is high for a given temperature range, then the rate constant is highly sensitive changes in temperature have a significant effect on the rate constant. If the activation energy is low for a given temperature range, then the rate constant is less sensitive, and changes in temperature have little effect on the rate constant. This phenomenon is graphically illustrated in the example below:

Experimental Data
1/Temp 0.0085 0.0075 0.0065 0.0055 0.0045 0.0035
lnk (large Ea) 3 2.5 2 1.5 1 0.5
lnk (small Ea) 1.8 1.7 1.6 1.5 1.4 1.3

The graph above shows that the plot with the steeper slope has a higher activation energy and the plot with the flatter slope has a smaller activation energy. This means that over the same temperature range, a reaction with a higher activation energy changes more rapidly than a reaction with a lower activation energy.

The Arrhenius plot may become non-linear if steps become rate-limiting at different temperatures. Such an example can be found with Fox and co-workers in 1972 with beta-glycoside transport in E. coli. The differences in the transition temperatures are due to fatty acid composition in cell membranes. The transition state difference is a result of the sharp change of fluidity of the membrane. Another example includes a sudden drop at low 1/T (high temperatures), a result of protein denaturation.

Software Activity

The types of variables you are analyzing directly relate to the available descriptive and inferential statistical methods.

  • assess how you will measure the effect of interest and
  • know how this determines the statistical methods you can use.

As we proceed in this course, we will continually emphasize the types of variables that are appropriate for each method we discuss.

From the documentation, it seems like the :inverse_of option is a method for avoiding SQL queries, not generating them. It's a hint to ActiveRecord to use already loaded data instead of fetching it again through a relationship.

In this case, calling dungeon.traps.first.dungeon should return the original dungeon object instead of loading a new one as would be the case by default.

I think :inverse_of is most useful when you are working with associations that have not yet been persisted. E.g.:

Without the :inverse_of arguments, t.project would return nil , because it triggers an sql query and the data isn't stored yet. With the :inverse_of arguments, the data is retrieved from memory.

After this pr ( inverse_of is not required in most cases.

Active Record supports automatic identification for most associations with standard names. However, Active Record will not automatically identify bi-directional associations that contain a scope or any of the following options:

In the above example, a reference to the same object is stored in the variable a and in the attribute writer .

Just an update for everyone - we just used inverse_of with one of our apps with a has_many :through association

It basically makes the "origin" object available to the "child" object

So if you're using the Rails' example:

Using :inverse_of will allow you to access the data object that it's the inverse of, without performing any further SQL queries

When we have 2 models with has_many and belongs_to relationship, it's always better to use inverse_of which inform ActiveRecod that they belongs to the same side of the association. So if a query from one side is triggered, it will cache and serve from cache if it get triggered from the opposite direction. Which improves in performance. From Rails 4.1, inverse_of will be set automatically, if we uses foreign_key or changes in class name we need to set explicitly.

Best article for details and example.

Take a look at this article!!

If you have a has_many_through relation between two models, User and Role, and want to validate the connecting model Assignment against non existing or invalid entries with validates_presence of :user_id, :role_id , it is useful. You can still generate a User @user with his association @user.role(params[:role_id]) so that saving the user would not result in a failing validation of the Assignment model.

Please take a look 2 two useful resources

And remember some limitations of inverse_of :

does not work with :through associations.

does not work with :polymorphic associations.

for belongs_to associations has_many inverse associations are ignored.

5 Answers 5

take a signal, a time-varying voltage v(t)

units are V, values are real.

throw it into an FFT -- ok, you get back a sequence of complex numbers

units are still V, values are complex ( not V/Hz - the FFT a DC signal becomes a point at the DC level, not an dirac delta function zooming off to infinity )

units are still V, values are real - magnitude of signal components

units are now V 2 , values are real - square of magnitudes of signal components

shall I call these spectral coefficients?

It's closer to an power density rather than usual use of spectral coefficient. If your sink is a perfect resistor, it will be power, but if your sink is frequency dependent it's "the square of the magnitude of the FFT of the input voltage".

AT THIS POINT, you have a frequency spectrum g(w): frequency on the x axis, and. WHAT PHYSICAL UNITS on the y axis?

The other reason the units matter is that the spectral coefficients can be tiny and enormous, so I want to use a dB scale to represent them. But to do that, I have to make a choice: do I use the 20log10 dB conversion (corresponding to a field measurement, like voltage)? Or do I use the 10log10 dB conversion (corresponding to an energy measurement, like power)?

You've already squared the voltage values, giving equivalent power into a perfect 1 Ohm resistor, so use 10log10.

log(x 2 ) is 2 log(x), so 20log10 |fft(v)| = 10log10 ( |fft(v)| 2 ), so alternatively if you did not square the values you could use 20log10.

The y axis is complex (as opposed to real). The magnitude is the amplitude of the original signal in whatever units your original samples were in. The angle is the phase of that frequency component.

Here's what I've been able to come up with so far:

The y-axis seems likely to be in units of [Energy / Hz] !?

Here's how I'm deriving this (feedback welcomed!):

the signal v(t) is in volts

so after taking the Fourier integral: integral e^iwt v(t) dt , we should have units of [volts*seconds], or [volts/Hz] (e^iwt is unitless)

taking the magnitude squared should then give units of [volts^2 * s^2], or [v^2 * s/Hz]

we know Power is proportional to volts ^2, so this gets us to [power * s / Hz]

but Power is the time-rate of change in energy, i.e. power = energy/s, so we can also write Energy = power * s

this leaves us with the candidate conclusion [Energy/Hz]. (Joules/Hz ?!)

. which suggests the meaning "Energy content per Hz", and suggests as a use integrating frequency bands and seeing the energy content. which would be very nice if it were true.

Continuing. assuming the above is correct, then we are dealing with an Energy measurement, so this would suggest using 10log10 conversion to get into dB scale, instead of 20log10.

V^2 / R with R taken as the 50 ohm constant load. &ndash Assad Ebrahim Oct 6 '09 at 9:52

The power into a resistor is v^2/R watts. The power of a signal x(t) is an abstraction of the power into a 1 Ohm resistor. Therefore, the power of a signal x(t) is x^2 (also called instantaneous power), regardless of the physical units of x(t) .

For example, if x(t) is temperature, and the units of x(t) are degrees C , then the units for the power x^2 of x(t) are C^2 , certainly not watts.

If you take the Fourier transform of x(t) to get X(jw) , then the units of X(jw) are C*sec or C/Hz (according to the Fourier transform integral). If you use (abs(X(jw)))^2 , then the units are C^2*sec^2=C^2*sec/Hz . Since power units are C^2 , and energy units are C^2*sec , then abs(X(jw)))^2 gives the energy spectral density, say E/Hz . This is consistent with Parseval's theorem, where the energy of x(t) is given by (1/2*pi) times the integral of abs(X(jw)))^2 with respect to w , i.e., (1/2*pi)*int(abs(X(jw)))^2*dw) > (1/2*pi)*(C^2*sec^2)*2*pi*Hz > (1/2*pi)*(C^2*sec/Hz)*2*pi*Hz > E .

Conversion to a dB (log scale) scale does not change the units.

If you take the FFT of samples of x(t) , written as x(n) , to get X(k) , then the result X(k) is an estimate of the Fourier series coefficients of a periodic function, where one period over T0 seconds is the segment of x(t) that was sampled. If the units of x(t) are degrees C , then the units of X(k) are also degrees C . The units of abs(X(k))^2 are C^2 , which are the units of power. Thus, a plot of abs(X(k))^2 versus frequency shows the power spectrum (not power spectral density) of x(n) , which is an estimate the power of a set of frequency components of x(t) at the frequencies k/T0 Hz .


During a traumatic experience, it’s common to feel that time has slowed down or even stood still. Although skydiving might be counted as deliberate exposure to trauma, your brain still processes the stress the same way, causing your perception of time to shift.

In 2007, researchers published an article in the journal Behaviour Research and Therapy on time perception for first-time skydivers and found that people who reported being more frightened also reported the fall lasting longer than it actually was. People who were excited, however, had a “time flies when you’re having fun” experience and thought the dive went by faster than it really did. Objectively speaking, that free fall normally lasts for about a minute.

Where does the inverse seconds unit come from in the association constant? - Biology


This page looks at the way that rate constants vary with temperature and activation energy as shown by the Arrhenius equation.

Note: If you aren't sure what a rate constant is, you should read the page about orders of reaction before you go on. This present page is at the hard end of the rates of reaction work on this site. If you aren't reasonably confident about the basic rates of reaction work, explore the rates of reaction menu first.

The Arrhenius equation

Rate constants and rate equations

You will remember that the rate equation for a reaction between two substances A and B looks like this:

Note: If you don't remember this, you must read the page about orders of reaction before you go on. Use the BACK button on your browser to return to this page.

The rate equation shows the effect of changing the concentrations of the reactants on the rate of the reaction. What about all the other things (like temperature and catalysts, for example) which also change rates of reaction? Where do these fit into this equation?

These are all included in the so-called rate constant - which is only actually constant if all you are changing is the concentration of the reactants. If you change the temperature or the catalyst, for example, the rate constant changes.

This is shown mathematically in the Arrhenius equation.

The Arrhenius equation

What the various symbols mean

Starting with the easy ones . . .

To fit into the equation, this has to be meaured in kelvin.

This is a constant which comes from an equation, pV=nRT, which relates the pressure, volume and temperature of a particular number of moles of gas. It turns up in all sorts of unlikely places!

This is the minimum energy needed for the reaction to occur. To fit this into the equation, it has to be expressed in joules per mole - not in kJ mol -1 .

Note: If you aren't sure about activation energy, you should read the introductory page on rates of reaction before you go on. Use the BACK button on your browser to return to this page.

And then the rather trickier ones . . .

This has a value of 2.71828 . . . and is a mathematical number, a bit like pi. You don't need to worry exactly what it means, although if you have to do calculations with the Arrhenius equation, you may have to find it on your calculator. You should find an e x button - probably on the same key as "ln".

The expression, e -(EA / RT)

For reasons that are beyond the scope of any course at this level, this expression counts the fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature. You will find a simple calculation associated with this further down the page.

The frequency factor, A

You may also find this called the pre-exponential factor.

A is a term which includes factors like the frequency of collisions and their orientation. It varies slightly with temperature, although not much. It is often taken as constant across small temperature ranges.

By this time you've probably forgotten what the original Arrhenius equation looked like! Here it is again:

You may also come across it in a different form created by a mathematical operation on the standard one:

"ln" is a form of logarithm. Don't worry about what it means. If you need to use this equation, just find the "ln" button on your calculator.

Using the Arrhenius equation

The effect of a change of temperature

You can use the Arrhenius equation to show the effect of a change of temperature on the rate constant - and therefore on the rate of the reaction. If the rate constant doubles, for example, so also will the rate of the reaction. Look back at the rate equation at the top of this page if you aren't sure why that is.

What happens if you increase the temperature by 10°C from, say, 20°C to 30°C (293 K to 303 K)?

The frequency factor, A, in the equation is approximately constant for such a small temperature change. We need to look at how e -(EA / RT) changes - the fraction of molecules with energies equal to or in excess of the activation energy.

Let's assume an activation energy of 50 kJ mol -1 . In the equation, we have to write that as 50000 J mol -1 . The value of the gas constant, R, is 8.31 J K -1 mol -1 .

At 20°C (293 K) the value of the fraction is:

By raising the temperature just a little bit (to 303 K), this increases:

You can see that the fraction of the molecules able to react has almost doubled by increasing the temperature by 10°C. That causes the rate of reaction to almost double. This is the value in the rule-of-thumb often used in simple rate of reaction work.

Note: This approximation (about the rate of a reaction doubling for a 10 degree rise in temperature) only works for reactions with activation energies of about 50 kJ mol -1 fairly close to room temperature. If you can be bothered, use the equation to find out what happens if you increase the temperature from, say 1000 K to 1010 K. Work out the expression -(EA / RT) and then use the e x button on your calculator to finish the job.

The rate constant goes on increasing as the temperature goes up, but the rate of increase falls off quite rapidly at higher temperatures.

The effect of a catalyst

A catalyst will provide a route for the reaction with a lower activation energy. Suppose in the presence of a catalyst that the activation energy falls to 25 kJ mol -1 . Redoing the calculation at 293 K:

If you compare that with the corresponding value where the activation energy was 50 kJ mol -1 , you will see that there has been a massive increase in the fraction of the molecules which are able to react. There are almost 30000 times more molecules which can react in the presence of the catalyst compared to having no catalyst (using our assumptions about the activation energies).

It's no wonder catalysts speed up reactions!

Note: If you read this carefully, you should notice that I am not saying that the reaction will be 30000 times faster. There may well be 30000 times more molecules which can react, but it is highly likely that the frequency factor will have changed in the presence of the catalyst. And the rate constant k is just one factor in the rate equation. You won't just have the original reactants present as before. The catalyst is bound to be involved in the slow step of the reaction, and a new rate equation will have to include a term relating to the catalyst.

Nevertheless, the catalysed reaction is still going to be a lot faster than the uncatalysed one because of the huge increase in sufficiently energetic molecules.

Other calculations involving the Arrhenius equation

If you have values for the rate of reaction or for the rate constant at different temperatures, you can use these to work out the activation energy of the reaction. Only one UK A' level Exam Board expects you to be able to do these calculations. They are included in my chemistry calculations book, and I can't repeat the material on this site.

Note: There is no way of making this sufficiently different from what is in the book to avoid being in breach of contract with my publishers if I included it on this site.

If you are interested in my chemistry calculations book you might like to follow this link.

Questions to test your understanding

If this is the first set of questions you have done, please read the introductory page before you start. You will need to use the BACK BUTTON on your browser to come back here afterwards.

Dimensions and units

In the former case (angles as fundamental quantity with their own unit), since the argument of any analytic function - such as exponential and trigonometric functions - HAS TO BE dimensionless (because otherwise one would end up mixing apples with oranges), a conversion factor HAS TO BE introduced to leave only the magnitude of the angle. In the same way chemists have pressures inside logarithms by dividing by a reference pressure, one should be using a unit angle (that some, like Gibbins (1) call $ eta_0 $) that will divide angle quantities wherever they appear. This will make the [$alpha$] dimension disappear and from the argument of any analytic function, avoiding the orange-apple paradox.

In the latter case (angles as ratios, hence dimensionless), the paradox is avoided from the start since angles are pure numbers to begin with. Some author (for example Szirtes (2)) prefer to view dimensionless quantities as quantities with dimension $[1]$, so that the any power of such quantity will still have dimension $[1]^n=[1]$ (as opposed to dimension $[alpha]^n$ , thus saving the day and restoring homogeneity.

In practice, people will not use the conversion factor $eta_0$ but will still consider angles dimensionless while using the unit $rad$ for the following reason illustrated by the Bureau International des Poids et Mesures (BIPM):

In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension. When using this freedom, one may recall the process by which the quantity is defined. [. ] The SI unit of frequency is given as the hertz, implying the unit cycles per second the SI unit of angular velocity is given as the radian per second and the SI unit of activity is designated the becquerel, implying the unit counts per second. Although it would be formally correct to write all three of these units as the reciprocal second, the use of the different names emphasises the different nature of the quantities concerned. Using the unit radian per second for angular velocity, and hertz for frequency, also emphasizes that the numerical value of the angular velocity in radian per second is 2 pi times the numerical value of the corresponding frequency in hertz.

But all trig, exp and log functions need to have dimensionless arguments. Period.

So, if you use dimensionless angles, $omega$ will have dimension $[T]^-1$ and unit $s^<-1>$ while if you use angles of dimension $[alpha]$ and unit $rad$, by dividing by the conversion factor $eta_0$, you will still end up with the same result. The 'ghost' $rad$ you put 'as a reminder' in the $omega$ case is there to remember you that there is a $2 pi$ that derives from the fact that trig functions are periodic of period $2 pi$.

Newton's second law in action

Rockets traveling through space encompass all three of Newton's laws of motion.

If the rocket needs to slow down, speed up, or change direction, a force is used to give it a push, typically coming from the engine. The amount of the force and the location where it is providing the push can change either or both the speed (the magnitude part of acceleration) and direction.

Now that we know how a massive body in an inertial reference frame behaves when it subjected to an outside force, such as how the engines creating the push maneuver the rocket, what happens to the body that is exerting that force? That situation is described by Newton’s Third Law of Motion.

Watch the video: Section 1 6 Part 2 (August 2022).