In this post, I discuss different quantities that can be extracted from in vitro motility assay velocity distributions. I explain how the average velocity, the continuous sliding velocity, and the motile fraction relate to each other. There are two examples how these quantities change for different experimental effects in the motility assay. I close discussing different problems that can arise in the analysis of velocity distributions, along with possible causes and solutions.

At the core of muscle contraction is the interaction between actin and myosin. Filaments of mechanically active myosin proteins move filaments of passive proteins, providing a molecular explanation of how muscle tissue can contract. The in vitro motility assay has been used for decades to recreate these interactions “on glass” (the rough meaning of in vitro) instead of the messy environment inside living muscle cells. This way, many unknown influences on the actin-myosin interaction can be excluded, while the conditions for the interaction can be controlled more precisely. In the video below, one can see fluorescently labeled actin filaments. Their sliding motion is driven by skeletal muscle myosin motors, which adhere to a glass coverslip. So, one can see the actin filaments, but not the molecular motors that move them.

As seen in the example video, actin filaments move unidirectionally, with varying velocities and occasional stopping events. These motion patterns of actin are used to make conclusions about the interaction of myosin proteins with the actin filaments. There are different ideas of how to characterize the actin motion, though. A good way to understand them is to consider the typical distribution of actin sliding velocities – as shown in the next schematic drawing.

This sketch shows a distribution of actin sliding velocities. These velocities are typically created by tracking individual actin filaments and measuring their displacement between two consecutive video frames (instantaneous or “frame-to-frame” velocities). Three crucial quantities can be calculated from the velocity distributions:

1. That average actin sliding velocity ($\nu$) is simply the arithmetic mean of all velocities measured.
2. The sliding velocity of actin during phases of continuous forward sliding ($\nu_{max}$). $\nu_{max}$ can be determined from the position of the second peak of the distribution (the one further away from 0 velocity). $\nu_{max}$ supposedly corresponds to the velocity of relative actin-myosin filament sliding in muscle with no external load ($V_{max}$).
3. The fraction of velocities that can be associated with active forward sliding of actin ($f_{mot}$, the motile fraction).

Assuming that the separation between forward sliding of actin and actin arrest is sufficiently clear (indicated by a deep minimum between the two peaks in the velocity distribution), the mathematical relationship between the three quantities is approximately

$\nu \approx \nu_{max}\cdot f_{mot}$ [1].

In other words, the average sliding velocity of actin can be determined either (1) directly from the mean of the distribution, or (2) by determining the velocity while actin is sliding, and multiplying this velocity with the fraction of time that actin spends actively sliding.

In the motility assay field, different research groups and research objectives lead to different choices in terms of which quantities to use. From my own experience, this should be based on the type of effect or molecular mechanism to be studied. $\nu_{max}$ has seen widespread use when tissue level mechanics of muscle should be compared with molecular mechanics; the direct comparison with $V_{max}$ of the muscle under no load is straight-forward, and potential problems with inferior/difficult/novel protein purification and assay preparation that lead to increased stopping frequency of actin are reduced markedly. Especially for myosin that is difficult to purify, this approach was often used in my time in the Lauzon lab. Another case would be alterations to the basic actin-myosin kinetics – that potentially lead to increased or reduced actin sliding frequency [1,2]; the motile fraction must necessarily be considered to detect this change in the experiment. This is often relevant where actin-regulatory proteins alter the myosin accessibility to and affinity for actin [3].

In the different cases, the changes between experimental conditions reflect differently in the relationship $\nu \approx \nu_{max}\cdot f_{mot}$. The two extreme cases are shown in the above sketches. In the top sketch, $\nu_{max}$ is reduced, but actin slides equally as often irrespective of experimental condition ($f_{mot}\approx\mathrm{const.}$). Correspondingly, $\nu\propto \nu_{max}$, or $\nu/\nu_{max} \approx f_{mot} \approx \mathrm{const.}$. This can be easily checked from the measurements from different experimental conditions. In the bottom sketch, the active sliding velocity of actin is not altered, instead actin is stopping more frequently; $f_{mot}$ is reduced, but not $\nu_{max}$. This also means that $\nu\propto f_{mot}$ and $\nu/f_{mot}\approx \nu_{max}\approx \mathrm{const.}$ Again, this can be checked relatively easy from the experimentally measured filament velocities.

One problem can be the detection of $\nu_{max}$ from the second mode in the distribution. Often, histograms are used, and $\nu_{max}$ is determined simply from the histogram bar that has the highest count. This approach comes with the typical problems of histogram graphs – for example that often the apparent distributions is highly dependent on the chosen bin size. A more consistent approach is the fitting of Gaussian mixture models to the distribution, and extracting the means of the second Gaussian. This will give a more precise estimate of $\nu_{max}$. I used this procedure in reference [2], and it can also be directly executed from within the ivma3 software I wrote during my PhD. Other procedures are sometimes used, where all filament velocities below a specified threshold are discarded; these are prone to artificial biases and sometimes difficult to reproduce due to the choice of the threshold and rejection parameters – but this approach can also be executed using ivma3.

There is also the question how to determine $f_{mot}$. I have used two different ways. The simple and robust way is to pick a threshold velocity, then calculate the fraction of total filament velocities above that threshold. The resulting fraction is a number between 0 and 1 (otherwise the calculation must be wrong). The other, more complicated but slightly more accurate, method is to fit two Gaussians, so that one corresponds to each mode in the velocity distribution. It is technically more challenging, you will likely have to write a custom MatLab (or other software) script for this procedure. Also, in situations with $f_{mot}$ close to either 0 or 1, the distribution will be almost unimodal and fitting a two Gaussian mixture model will fail. There are also issues with numerical stability of the fitting procedure, so doing repeated randomized fits often becomes necessary.

A third problem can be that the velocity distribution simply isn’t bimodal as shown in the sketch. One cause can be a badly adjusted time resolution of the video analysis. When estimating instantaneous velocities, I usually use the mean square displacement of filament centroids in two consecutive frames. When the frame rate for analysis is set too low, noise will obscure the bimodal distribution of centroid displacements [2]. When the frame rate is chosen too low, again, the bimodal distribution is blurred, because between two frames filaments switch between motion and arrest. These two states are then effectively averaged, and give an intermediate actin velocity between the two modes of the distribution. Practically, we used $\approx 10$ frames per second for skeletal muscle myosin ($\nu_{max}\approx 6\,\mu \mathrm{m/s}$) or $\approx 3$ frames per second for smooth muscle myosin ($\nu_{max} \approx 0.7\,\mu\mathrm{m/s}$).

An interesting statistical method in this context is the Hartigans’ Dip Test [4]. It is a computational statistical method that allows to test the probability that a given data set (one-dimensional and continuous) is a sample from a unimodal distribution. If it is not, it gives strong evidence of multimodality – together with visual inspection of distributions, this is as good a test for bimodality as I have come across so far.

A good alternative to the ivma3 software is FIESTA, which is developed by Felix Ruhnow. It is more geared towards a good user interface, and admittedly there seem to be more tutorials and wiki descriptions than for ivma3. In case you wonder which software to try: in case you want to process huge amounts of motility videos and are not afraid of MatLab code, you might go for ivma3; if you need precision and more of a well-working user interface, you might rather give FIESTA a try.

[1] Hilbert, Balassy, Zitouni, Mackey, Lauzon; Phosphate and ADP Differently Inhibit Coordinated Smooth Muscle Myosin Groups. Biophysical Journal, In press
[2] Hilbert, Bates, Roman, Blumenthal, Zitouni, Sobieszek, Mackey, Lauzon; Molecular Mechanical Differences between Isoforms of Contractile Actin in the Presence of Isoforms of Smooth Muscle Tropomyosin. PLoS Computational Biology 9(10):e1003273, 2013
[3] Marston; Random walks with thin filaments: application of in vitro motility assay to the study of actomyosin regulation. Journal of Muscle Research and Cell Motility 24:149–156, 2003
[4] Hartigan, Hartigan; The Dip Test of Unimodality. The Annals of Statistics 13:70-84, 1985

## 2 thoughts on “Motile fraction and actin velocity in in vitro motility assays”

1. David says:

Nice, concise explanation!
And side note: I like the typo in the last reference: “The Dip Test of Unmorality” 😉

1. My spell checker seems to have clear preferences…thanks for the comment David, and sorry for not having fixed the IVMA3 bug yet.