Modeling Contractile Mechanisms: Huxley 1957 Model With Slides Courtesy Stuart Campbell, U Kentucky and J. Jeremy Rice IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY

Need to consider both how many XBs are recruited and what is the extension of attached XBs * This requires a modeling approach that simultaneously considers spatial and temporal aspects of XB cycling Thin filament - actin Thick filament - myosin

Alternative theory proposed opposite charges exist in thick and thin filament Contraction from electrostatic attraction Hard to reconcile with maximum velocity being independent of sarcomere length

Derivation of Huxley '57 Model

Review key concepts from basic probability: 1.Probability Density Functions Example: Consider an electron of a hydrogen atom. One could draw an approximate probability density function (p.d.f.) for its location in terms of distance from the nucleus = r. Let h(r) be the function below that defines this p.d.f. r proton r=0 r h(r)h(r) electron

The way to interpret a p.d.f. is that a probability can be computed as an area under the curve. For example the probability of the electron being closer than r 0 is computed as: r=0 r h(r)h(r) r0r0 r1r1 Likewise, the probability of the electron being between r 0 and r 1 is computed as:

By definition, the p.d.f. does not directly report the probability of finding an electron at a given fixed distance, r 0 : In fact, the probability for any given distance, r 0, is 0. See that:

A probability is always between 0 and 1 so if we assume the electron is always found at some distance from the nucleus then: More formally, we define the the p.d.f. as a limit: General rule: The summation of p.d.f. over all possible values must be = 1.

We define a conditional probability as probability of an event A given that an event B has occurred. This is written as: This relation may make more intuitive sense when rearranged as: 2. Conditional Probability

3. State Variables to Represent Probabilities of Stochastic Events When considering a random process like a channel opening and closing, each channel looks like this: C O f g For this channel, if we assume f = 1 s -1 and g = 2 s -1, then steady-state probability of the channel being open is computed as:

Of course, this does not mean the channel is 1/3 open because only states "closed" and "open" exist. However, if we were to average a large number of channel responses together, we would get something like this: Sample many runs and average to get better estimate of probability. } Run 1 Run 2 Run n /3 time

So instead of tracking a each channel, we can define a state variable based system to capture the behavior of the whole population. When implemented this way, average behavior can calculated without averaging (and associated noise and computational cost). C O f g

Derivation of Huxley ’57 Model Assumptions: 1. Contractile machinery only 2. Plateau region of length-tension 3. Muscle fully activated 4. Constant velocity (parameter in model) 5. Crossbridges (XBs) always completes full cycle to detach and uses 1 ATP in the process 6.Single myosin near every A site and interaction between this pair is independent of all other pairs of A sites and myosins

Setup for Huxley '57 model X We consider model on right to be equivalent to model on left. Model is built around actin binding sites called A sites on thin filament. Myosin heads from thick filament can bind to one and only one nearby A site. A site equilibrium myosin position Thick filament Thin filament A site X

EM micrographs shows evidence of crossbridges but no real detail

Setup for Huxley '57 model X 1 < 0 Forces are only considered in left-right directions parallel to thick and thin filaments. When an A site is bound to myosin, force is generated with respect to the distortion of myosin from its equilibrium position. When A site is bound exactly at the equilibrium position (i.e. X 3 ), no force is generated. equilibrium myosin positions X 2 > 0 X 3 = 0 A site 1 A site 2A site 3

Setup for Huxley '57 model X 1 < 0 When an A site is bound to myosin, force is generated with respect to the distortion of myosin from its equilibrium position. Linkage is assumed to be a simple spring so that T = kX. For each A site with myosin bound, T = kX. Therefore, T 1 < T 3 =0 < T 2. A site 1 equilibrium myosin positions X 2 > 0 X 3 = 0 A site 2A site 3 k k k

Setup for Huxley '57 model Distance between A sites = l X1X1 X2X2 X3X3 X4X4 Model shows distance between A sites and equilibrium myosin positions. A whole population of A sites is assumed to sample equally all X values because A sites and equilibrium myosin positions are unequally spaced. (p.d.f is constant) A sites equilibrium myosin positions

Position of myosin heads on thick filament Pairs of heads emanate 180 degree apart in radial direction at each step Radial direction of heads rotate ~60 degrees at next step in axial direction (distance = ~14.3 nm) a "pseudo-repeat" happens on the 3th steps as heads will be emanate in same radial direction (distance = ~43 nm) axial direction radial direction

Thin filament is a two-stranded helix of actin monomers From "Pseudo-repeat" = 13 units 5.54 nm 2.77 nm "Pseudo-repeat" 37 nm True repeat = 26 units

Setup for Huxley '57 model Distance between A sites = l (small L) X1X1 X2X2 X3X3 X4X4 Model assumes distance l between A sites is large and interactions are with only one nearby myosin. Hence, each myosin can interact with only one A site at a time. Therefore, the cases shown above (for X 2 and X 3 ) cannot happen. A sites equilibrium myosin positions

Setup for Huxley '57 model X1X1 The thick and thin filaments slide past each other at a constant velocity V. We assume that the motion results from combined action of many force generators acting across the whole muscle, so the sliding velocity is not affected by the local attachment or detachment events. Note: velocity is a parameter in the model. equilibrium myosin positions X2X2 X3X3 Sliding in V > 0 direction (if thick filament fixed) Sliding in V < 0 direction (if thick filament fixed)

Setup for Huxley '57 model X1X1 As thick and thin filaments slide past each other at a constant velocity V, the relative position of A sites compared to equilibrium myosin positions changes. Therefore, when V>0, X 1, X 2 and X 3 all get smaller (less positive or more negative) with time. equilibrium myosin positions X2X2 X3X3 Sliding in V > 0 direction (if thick filament fixed) Sliding in V < 0 direction (if thick filament fixed)

g2g2 g f h x g1g1 f1f1 XB attach only in this range XB can detach at any distortion Attachment rates as function of X

X < 0 Attachment rate between A site and myosin are a function of the relative distance between the A site and equilibrium myosin position. In the model, f(X) is the function defined to control attachment. Function f(X) increases linearly from X=0 to X=h. X = h/2 X = 0 f(X) g f h x

g f h x Detachment rates as function of X X 1 < 0 In the model, g(X) is the function defined to control detachment as a function of the relative distance between the A site and equilibrium myosin position. In the positive range, g(X) increases linearly from X=0. In the negative range, g(X) is large so that the negative distortion XBs ("draggers") detach quickly. X > h X = 0 g(X)

Let n(x,t) be a conditional probability describing the likelihood that an XB is attached given that the A site is at displacement x from the nearest XB equilibrium position. To be more rigorous: Conditional probability vertical bar means “given that”

Note that a more intuitive function describes when an A site is attached and the A site is between x and x +  x We can use rule from basic probability

Make a substitution for probability inside limit Substitute limit above with product of limits below Probability Density Function Conditional Probability

is a probability density function describing the positions of A sites with respect to equilibrium XB positions Model assumes is constant over all possible x values between - l /2 and l /2 as shown below: l/2 -l/2 1/l x

Continue with derivation For steady-state response: Apply chain rule: We know that: Rearrange to get:

Unattached A site fraction Attached A site fraction

Combine above results: Can find solution for specific cases if we define f(x) and g(x) with units of s -1 g2g2 g f h x g1g1 f1f1

Apply constraint that if then solution is continuous at x=0 and x=h. One can write: Now solve for three region assuming V > 0: Region 1 -

Integrate to get: Region 2 -

Rearrange to get:

Integrate to get:

Region 3 - If we assume shortening then no crossbridges can be attached at x>h. This is equivalent to n(x,t) =0 for x>h. Now determine the constants using the continuity conditions:

Substitute constant back into original equation: Find C 1 using the other continuity condition:

Substitute constant back into original equation: Define where S is a full sarcomere length (~ 2  m), S /2 is a half sarcomere length, and V is normalized velocity in half sarcomere lengths per second. Make a change of variables: and

The three regions can now be defined as: Must define rates for XB cycling. Can use the ratios below: The following plots on following slides are the result.

Results for Huxley '57 model In isometric conditions (V=0), for given distances between X=0 and X=h, there is a high probability of attachment of the A sites to the myosin. Attached XBs with positive distortion are "pullers". X < 0X = h - X = h/2 (a) (c) (b) (a) (b) (c) n(h/2) = 0.8 n(h - ) = 0.8

Results for Huxley '57 model. X < 0X = -h/2 X = h/2 (a) (c) (b) (a) (b) (c) As velocity increases (V>0), probability of attachment of the A site to the myosin can be above zero for X<0 because XBs may attach between X=0 and X=h and be dragged to negative distortions. n(h/2) ~ 0.4 n(-h/2) ~ 0

Results for Huxley '57 model. X < 0X = -h/2 X = h/2 (a) (c) (b) (a) (b) (c) As velocity increases (V>0), the average distortion of the attached XBs decreases (less positive or more negative). At V max the "pullers" (X>0, i.e. (a)) and "dragger" (X<0, i.e. (b)) cancel so that net tension T = 0. Detached XBs don't contribute. n(-h/2) ~ 0.1 n(h/2) ~ 0.15

Now we want to generate a Force-Velocity relation. Velocity is an input parameter, and force is computed. Must define force per single XB as Where x is the distortion (= distance from A site to equilibrium position of attached crossbridge) Now we want to compute total tension as a the sum of the the contributions of all the XBs. To do this, we need to compute an “expected value” that is analogous to average value. First define an expected value as: expected value of u value of u probability density function of finding u

In our case, we want expected value of tension for all A sites, both attached and detached. We can write expected value as: Force of attached A site Attached A sites True p.d.f. of all A site being at distance = x Detached A sites Force of detached A site

A p.d.f. describing likelihood of A sites being distance x from the nearest equilibrium XB position Model assumes is constant over all possible x values between - l /2 and l /2 as shown below: l/2 -l/2 1/l x Conditional probability of an A site having an attached XB given its distance is x from the nearest equilibrium XB position Recall these features of the model:

Multiplying the terms produces: Substitute for n(x) and integrate to get: Then substituting into expected value of force:

For isometric force, set v =0 to get: Now normalize force by isometric force to get: These parameters give best fit to experimental data at right

Successes of model - - Good basic framework for cycling XB distribution - Reproduces Force – Velocity relationship - Reproduces energy use vs. tension relationship - Superb first attempt given the knowledge of the system at the time Problems - - XB cycle is simplistic - Restrictive set of conditions - isotonic, constant velocity - full activation - Cycling rate increases with lengthening causing increased ATP usage in disagreement with experimental results

Excel-based simulation package for Huxley '57 (developed by D. Yue and J. Rice)

Huxley '57 assumes continuous time and space - first must make discrete for Excel simulation Solve model on a grid of points as shown below N(-15)N(0)N(15)N(8)N(-8) xx tt t(0) t(15) t(5)

Attachment and detachment rates g f h x Rates from Huxley '57 model As implemented in Excel model

How to run model Model parameters Input velocity ( ) Force is computed Normalized velocity and force are computed

This model uses a discrete time and space approximation to continuous values equations from Huxley '57 We know that:

From previous page: Then use Euler integration to evolve forward in time at every location Substitute to get:

From class derivation: Substitute to get: Now we just need to compute approximation for using a difference equation

One minor complication is approximation for the spatial derivative - must use different forms for positive and negative velocities Positive velocity: Negative velocity:

Discrete time step =  t Corresponds to (1-n(x,t))*f(x) - n(x,y)*g(x) Corresponds to V*dn(x,t)/dx Slightly rearrange previous results to better correspond to Excel formula Corresponds to n(x,t) (V>0 case)

Model results for V = 0 Shows h

Model results for V = ~.25V max Shows h

Calculation of n(x) is n(x,t) after waiting for "steady-state"

Combine contributions of kx and h > l/2 x -l/2 l/2 x -l/2 1/l

Multiply to help compute X

Compute by summing over all distortions This step corresponds to this step in the continuous time derivation:

Exercises 1. Generate a classical force-velocity curve using the discretized Huxley ‘57 solver in Excel. Plot the result. 2. Choose one of the model parameters to alter, and predict its effects on the force- velocity curve. 3. Use the model implementation to test your hypothesis.