Bachelor Wiskunde

Bachelorproject

A reduced model for the active contractionof cardiac sarcomeres

by

Lonit Peeters

July 1, 2019

Supervisor: Dr. R. Planque

Second examiner: Prof. Dr. R. W. J. Meester

Department of Mathematics

Faculty of Sciences

Abstract

We describe in detail the reduced ODE model for the contraction of cardiac sarcomeres,developed by Regazzoni et al. (2018). This is a spatially explicit model that incorporatescooperative interactions between neighbouring myosin heads. The model was based onthe model described by Rice et al. (2003) with the modification of Washio et al. (2012).In order to reduce the number of degrees of freedom of this model, Regazzoni et al.(2018) assumed conditional independence of specific sets of events. This lead to anenormous reduction in degrees of freedom from approximately 5 · 1021 to 2176, and thusof an enormous reduction of computational time. Numerical tests show that the reducedmodel can adequately predict important experimentally observed behaviour of muscles,such as the steady state force-length and the force-calcium relationships. However, thesteady state force-velocity relationship can unfortunately not be derived from the modelat all. On the other hand, also the results of dynamical tests are similar to experimentaldata.

Title: A reduced model for the active contraction of cardiac sarcomeresAuthor: Lonit Peeters, l.m.peeters@student.vu.nl, 2579568Supervisor: Dr. R. PlanqueSecond examiner: Prof. Dr. R. W. J. MeesterDate: July 1, 2019

Department of MathematicsVU University Amsterdamde Boelelaan 1081, 1081 HV Amsterdamhttp://www.math.vu.nl/

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Contents

1. Introduction 41.1. Contraction of the heart muscle . . . . . . . . . . . . . . . . . . . . . . . 41.2. Important features of muscle contraction . . . . . . . . . . . . . . . . . . 61.3. History of muscle models . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4. Contribution of Regazzoni et al. (2018) . . . . . . . . . . . . . . . . . . . 9

2. Full model 112.1. Model overview and assumptions . . . . . . . . . . . . . . . . . . . . . . 112.2. Markov model for a single myosin head . . . . . . . . . . . . . . . . . . . 13

2.2.1. Formulas for the transition rates . . . . . . . . . . . . . . . . . . . 142.3. Forward Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . . . . 18

3. Reduced model 223.1. The basis of the reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2. Rewriting of the conditional probabilities . . . . . . . . . . . . . . . . . . 233.3. Derivation of the system of ODEs . . . . . . . . . . . . . . . . . . . . . . 27

4. Numerical results 324.1. Steady state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1. Force-length relationship . . . . . . . . . . . . . . . . . . . . . . . 344.1.2. Force-calcium relationship . . . . . . . . . . . . . . . . . . . . . . 344.1.3. Force-velocity relationship . . . . . . . . . . . . . . . . . . . . . . 36

4.2. Dynamical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.1. Twitch contraction . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.2. Rate of force development . . . . . . . . . . . . . . . . . . . . . . 39

5. Discussion 415.1. Possible improvements of the model . . . . . . . . . . . . . . . . . . . . . 425.2. What else Regazzoni et al. (2018) have done . . . . . . . . . . . . . . . . 44

A. Insight in the matrix A 49

B. Matlab code 52B.1. Force-length and force-calcium relationship . . . . . . . . . . . . . . . . . 52B.2. Force-velocity relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 58B.3. Twitch contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58B.4. Rate of force redevelopment . . . . . . . . . . . . . . . . . . . . . . . . . 59

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1. Introduction

The heart is one of the most important organs in the human body. The first attemptsto learn more about the heart and how it works were by examining the muscle itself,for example by dissection. These experiments led to a quite thorough knowledge of thestructures that are present in the heart and of their functions. However, this kind ofresearch can not teach us anything about the interaction of all these individual structuresand processes. Therefore, a lot of the more recent research focuses on modelling theprocesses that, for example, take place during contraction. When creating a model, youneed to make several assumptions. These assumptions can then be tested by comparingthe prediction of the model with what happens in reality. In case of a (heart) musclemodel you might, for instance, want to check whether the predicted force at a certainmuscle length corresponds with the force that can actually be produced at that musclelength.

In this thesis the model that was developed by Regazzoni et al. (2018) will be discussedin detail. Our goals were to thoroughly understand the model, and maybe expand themodel or check whether it works properly. When reading the article of Regazzoni et al.(2018), we also found some minor typos/mistakes, which we want to mention in thisthesis so that others can use the thesis as an erratum when reading the article.

Figure 1.1.: Inner structure of cardiomyocytes. Reprinted from Regazzoni et al. (2018).

1.1. Contraction of the heart muscle

Before going more in depth into the models, we first need to know which elements andprocesses are important in the contraction of the heart muscle. Each cardiac muscle cell(cardiomyocyte) consists of many myofibrils, which in turn consist of many sarcomeresin series. The sarcomeres are the actual contractile elements of each muscle, and aremade of the proteins actin and myosin, together with some other supporting proteins.In a sarcomere, two actin filaments always lie in line with each other, the far end of eachactin filament is anchored in the so-called Z-disk. The myosin filament lies opposite thetwo actin filaments as displayed in Figure 1.1. On the myosin filaments are the myosin

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heads (MHs). They are distributed over the whole filament, except for the middle part.Regazzoni et al. (2018) call this bare zone on the myosin filament the H-zone, which isincorrect1. However, for convenience we will just call it H-zone.

When a muscle (for example the heart) needs to contract, the muscle cell depolarisesand calcium is released into the cell. The calcium molecules bind to the troponin Cproteins that lie on the actin filament, which induces the tropomyosin proteins to changeform from its non-permissive state to its permissive state. When tropomyosin changesto its permissive state the binding sites for the myosin heads become exposed. The MHscan then bind to actin and form so-called crossbridges. The combination of a troponin Cprotein and a tropomyosin protein is called a regulatory unit (RU), since together theyregulate the formation of these crossbridges.

Figure 1.2.: Schematic illustration of a power stroke of myosin heads which generates force. Adaptedfrom Bhagavan (2002).

Figure 1.3.: Sliding-filament model of muscle contraction. The actin filaments slide past the myosinfilaments toward the middle of the sarcomere. The result is shortening of the sarcomerewithout any change in filament length. Adapted from Cooper (2000).

1The H-zone is the zone in between the two actin filaments.

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After binding, the MHs change position in such a way that they pull the actin filamentin the direction of the other actin filament. This is the so-called power stroke of the MHs(Figure 1.2). This means that the two actin filaments are pulled towards each other, andthe sarcomere shortens (Figure 1.3). The shortening of sarcomeres induces the wholemuscle to shorten: muscle contraction. This is called the “sliding filament theory”, andwas proposed independently by both Huxley and Niedergerke (1954) and Huxley andHanson (1954).

1.2. Important features of muscle contraction

The more crossbridges are formed, the more force is produced by the muscle (MusclePhysiology Laboratory, 2008). From the previous section it follows that the calciumconcentration is an important factor that determines how much force can be produced,namely: more calcium means that more binding sites become exposed, so that morecrossbridges can be formed. Here, we see a difference between cardiac and skeletalmuscles. An important feature of the cardiac muscle is the relatively high (with respectto skeletal muscles) sensitivity to the calcium concentration: in the cardiac muscle lesscalcium is needed for the same amount of force. This can be explained by the so-calledcooperative interactions between neighbouring regulatory units inside the sarcomeres.How this works exactly is still unclear, but the three theories that have been proposedall state in some way that the formation of a crossbridge facilitates the formation ofcrossbridges in the neighbouring binding sites. The force-calcium relationship, and thedifference of this relationship between cardiac en skeletal muscle, can be seen in Figure1.4. We remark that here the calcium concentration is expressed on a logarithmic scale,since pCa2+ := − log([Ca2+]), where [Ca2+] is the calcium concentration. The graph isthus semilogarithmic.

Figure 1.4: Force-calcium relationship of a sar-comere of a skeletal muscle. Forsarcomeres of a cardiac muscle, therelationship displaces towards theleft. This indicates that less cal-cium is needed to produce the sameamount of force: a higher sensitiv-ity to calcium. Note that this is asemilogarithmic graph since pCa2+

:= − log([Ca2+]), where [Ca2+] isthe calcium concentration. The au-thors do not state explicitly whatthe force was normalised to, but itseems to be normalised to the max-imum amount of force measured forthe highest calcium concentration.Reprinted from Rassier (2000).

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Another important factor is the total length of the sarcomere. As we can see in Figure1.3, the sarcomere length determines the amount of overlap between the MF and theAFs, and with that also how many MHs face an actin filament. When the sarcomerelength is big, some MHs do not face an actin filament and can therefore not form a crossbridge. Furthermore, in case the sarcomere is very short, the actin filaments slide overeach other. When that happens, some myosin heads at the left half of the myosin (theones in the green circle in Figure 1.5) face the right actin filament. When they forma crossbridge with this AF, their power stroke will pull the actin filament to the right.However, this is lengthening the sarcomere instead of shortening and thus produces acounterforce: a force in the wrong direction. This counterforce is cancelled by the forceproduced by the MHs in the red circle in Figure 1.5. This means that the myosin headsthat face two AFs, do not produce a net force. The resulting force-length relationshipcan be seen in Figure 1.6, together with a sketch of the overlap of the filaments. We seeindeed that for increasing sarcomere length, the force first increases and then decreasesagain.

Figure 1.5.: Schematic representation of a very short sarcomere length. At this length, the actin fila-ments slide over each other so that the MHs in the green circle produce a counterforce thatis canceled by the force produced by the MHs in the red circle. Further explanation in text.Adapted from Gordon et al. (1966).

Figure 1.6: Force-length relationshipof a sarcomere togetherwith sketches of the over-lap of the filaments forthe different sarcomerelengths. Notice that forshort sarcomere lengths,the actin filaments over-lap and less force can begenerated. For long sar-comere lengths, part ofthe myosin filament doesnot overlap with an actinfilament and less forcecan be generated. Fur-ther explanation in text.Reprinted from Stanfieldand Germann (2016).

Finally, another force relationship has been found in muscles, namely the force-velocityrelationship. Here velocity denotes the shortening velocity of the sarcomere. This rela-tionship is comparable for both skeletal and cardiac muscle (Sonnenblick, 1962). How-ever, skeletal muscle can lengthen while contracting, for example when walking down

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the stairs. The muscles at the front part of the thigh then produce force to keep youfrom tumbling downstairs, but are lengthening at the same time. Cardiac muscle, onthe other hand, can only shorten while contracting (Hessel et al., 2017). Therefore, theforce-velocity relationship for cardiac muscle is only defined for shortening velocities.The general force-velocity relationship for skeletal muscle can be seen in Figure 1.7. Wesee that the highest amount of force can be generated when the sarcomere lengthenswhile contracting. For contractions in which the sarcomere shortens, most force can begenerated at low shortening velocities and less force can be generated at high shorteningvelocities. This can be explained by the fact that it takes some time for a myosin headto form a crossbridge. At higher shortening velocities the filaments slide past each otherfaster, and thus the binding sites slide past the myosin heads faster. Therefore, there isless time to form crossbridges, which is why less crossbridges can be formed and henceless force can be generated. On the other hand, when the shortening velocity is small,there is much time for the myosin heads to form a crossbridge. This leads to morecrossbridges being formed and thus to more force produced by the sarcomere (MusclePhysiology Laboratory, 2008).

Figure 1.7.: Schematic representation of the force-velocity relationship of muscle during shortening andlengthening contractions. Notice that most force can be produced during contractions inwhich the sarcomere lengthens; when the sarcomere shortens while contracting, less forcecan be produced when the shortening velocity is higher. Reprinted from Crewther et al.(2005).

1.3. History of muscle models

Early muscle models were based on the phenomenological description of the force-velocityrelationship (Hill) and on the sliding filaments theory (Huxley). However, these modelsdo not incorporate the calcium-based regulation of the activation of the muscle (thefact that more calcium leads to more crossbridge formation, see previous section). Morerecent models do incorporate this using the description of troponin-tropomyosin regula-tory units (RUs). To be able to also incorporate the cooperative interactions betweenthe RUs that are described in section 1.2, a spatially explicit description of the RUsalong the filaments is needed.

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Rice et al. (2003) created such a spatially explicit model. Each filament containedN = 26 RUs, where each was described using a continuous time Markov chain (CTMC)with four states. The state of each regulatory unit was determined by the permissivityof the RU and on whether or not calcium was bounded to the RU. The calcium-basedregulation was incorporated by making the transition rates between the calcium-boundand calcium-unbound state of the RUs depend on the calcium concentration. In thesame way the cooperative interactions were incorporated by letting the transition ratesbetween the permissive and non-permissive states of the RUs depend on the states of thenearest-neighbouring RUs. An additional advantage of such spatially explicit models,is that the force-length relationship of the sarcomere can be incorporated in a moreaccurate way than in non-spatial models, since the overlap of the MF and AFs can beexplicitly incorporated.

The solution of such a CTMC model can be found by solving the associated ForwardKolmogorov Equation (FKE), which describes the time evolution of the probabilities thatare associated to states. The FKE here is a system of ordinary differential equations(ODEs) with the number of variables equal to the number of states in the CTMC. Solvingthe FKE gives the probability of each state as a function of time, so also for the states inwhich force is produced. From this, the total generated force can be derived. However,a major drawback of spatially explicit models is their huge computational complexity:a model with N RUs, each described by an S-state CTMC, has in total SN degrees offreedom. This large number of degrees of freedom makes it impossible to solve the FKEnumerically.

Rice et al. (2003) tried to overcome this drawback by reducing the model to a so-calledIsing problem. However, this approach is restricted only to the steady state, meaningthat they could only solve for steady state solutions. Therefore, Washio et al. (2012)came up with a new model that was based on the model developed by Rice et al. (2003).For example, they modified the transition rates in such a way, that they depend onthe amount of overlap between myosin and actin. Furthermore, instead of looking atthe FKE for each RU (4N ODEs in total), the authors investigated the joint probabil-ities of triplets of consecutive RUs. These joint probabilities were approximated witha function of the individual probabilities of the RUs. This reduces the computationalcomplexity significantly, but they did it in such a way that they needed to determinesome coefficients using fitting on results. Since they determined the coefficients for somespecific conditions, it is unclear whether these coefficients are also applicable for otherconditions. Furthermore, each time the coefficients need to be determined, this requiresa long “off-line phase” in which the coefficients are estimated and the computations ofthe model itself are paused. This takes a lot of time and thus slows down the method.

1.4. Contribution of Regazzoni et al. (2018)

Regazzoni et al. (2018) have tried to come up with a new way to reduce the computa-tional complexity. They started with the model of Rice et al. (2003) and the modificationof Washio et al. (2012). However, instead of using a function of individual probabili-

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ties for the joint probabilities of the triplets, Regazzoni et al. (2018) use a conditionalindependence hypothesis to rewrite the joint probabilities. This leads to a new ODEmodel for the time evolution of the probabilities with a significant reduction in degreesof freedom.

In my thesis, I will first discuss the full model of Rice et al. (2003) with the modificationof Washio et al. (2012). After that, I will show how we can look at the triplets of RUs andshow how the conditional independence hypothesis is used to reduce number of degreesof freedom of the model, which leads to the reduced model. Finally, I will discusssome numerical results obtained from the model and compare them with experimentaldata. We will then see that the force-length and the force-calcium relationship are bothvery well predicted by the model. The force-velocity relationship can, however, not bedetermined using this model. Finally, we will also test two dynamic situations, and wewill see that these predictions are also quite accurate.

AF Actin filamentCTMC Continuous time Markov chainFKE Forward Kolmogorov equationMF Myosin filamentMH Myosin headODE Ordinary differential equationRU Troponin-tropomyosin regulatory unit

Table 1.1.: List of abbreviations.

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2. Full model

In this chapter, the full sarcomere model of Rice et al. (2003) with the modification ofWashio et al. (2012) is presented. First, the model itself is described. After that, theFKE is derived.

2.1. Model overview and assumptions

The goal of the model is to determine the force that is produced by the sarcomere. Itis assumed that the active force that is generated by the sarcomere (FA(t)) is directlyproportional to the fraction of myosin heads (MHs) that is in permissive state (i.e., thepermissivity, P (t)):

FA(t) = αP (t),

where α is the maximum exerted force. This thus means that when all MHs are permis-sive, the maximum amount of force is generated; when no MH is permissive, no force isgenerated at all. Because of this direct proportionality, it is sufficient to have a modelwith which we can compute the permissivity.

Since the sarcomere is symmetric (for example, Figure 1.1), we only need to considerone half of the sarcomere. Each myosin head on this half sarcomere is described usinga CTMC with four states. The state an MH is in, is determined by whether or notcalcium is bound (0 for not bound, 1 for bound) and by the permissivity of the RU (Nfor non-permissive, P for permissive). The set of the four states the MHs can be in, isthus S = {0N , 1N , 0P , 1P}.

With this model, we can compute for each MH the probability that it is in one of thepermissive states (0P or 1P). From this the permissivity can be computed as follows:

P (t) =1

N

N∑i=1

P(X it ∈ {0P , 1P}).

In Figure 2.1, the representation of the (half) sarcomere as used in the model can beseen. The model considers a single (half) myosin filament (MF) and two actin filaments(MFs). The variables displayed in the figure will be further explained in section 2.2.1.

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Figure 2.1.: Sketch of the sarcomere model described in section 2.2. The myosin filament (MF) isrepresented in red and two actin filaments (AFs) are represented in blue (mid). A referencesystem is placed with the origin at the right-hand side of the H-zone (HZ). The geometricalfunctions χLA and χRA, indicating respectively, the region at the right of the left AF, andthe region covered by the right AF, are shown at the top of the figure. The length of theH-zone (LH), of the myosin filament (LM), of a actin filament (LA), and of the sarcomere(SL) are also depicted (bottom). Reprinted from Regazzoni et al. (2018).

Figure 2.2.: The transients for the calcium concentration c (left) and the sarcomere length SL (right).

The sarcomere length (SL) and calcium concentration (c) are inputs of the model.They are assumed to depend on time according to the following c and SL transients (seealso Figure 2.2):

c(t) =

c0 t < tc0

c0 + cmax−c0β

[e− t−t

c0

τc1 − e−t−tc0τc2

]t ≥ tc0

,

SL(t) = SL0

[1 + γmaxf

(max

(0, 1− e

− t−tSL0

τSL0

)−max

(0, 1− e

− t−tSL1

τSL1

))],

where

β =

(τ1

τ2

)−( τ1τ2−1)−1

−(τ1

τ2

)−(1− τ2τ1

)−1

.

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These transients are defined by fitting on experimental data, in order to reproducetransients that are similar to the ones occurring during a heart beat. The values thatare used for the constants can be found in Table 2.1. It is important to remark herethat, since the SL transient is an input of the model, the sarcomere length thus doesnot follow from the model. Therefore, predictions about (the change of) the length ofthe sarcomere are difficult to obtain from the model.

Finally, it is assumed that the outer MHs (i = 1 and i = N) always have a non-permissive neighbour with no calcium bound (0N ) at the side where they do not have anactual neighbour. This is important since we will see (section 2.2.1) that the transitionrates depend on the states of the neighbours of an MH, in order to account for thecooperative interactions between neighbouring MHs.

Variable Value UnitsDynamics of cc0 0.1 µMcmax 1.1 µMtc0 0.1 sτ c1 0.02 sτ c2 0.11 s

Dynamics of SLSL0 2.2 µmγmaxf -0.07 -tSL0 0.15 stSL1 0.55 sτSL0 0.05 sτSL1 0.02 s

Table 2.1.: Constants associated with the dynamics of c and SL; values of the constants for SL set toreproduce a realistic SL transient. All values adopted from Washio et al. (2012);

In the rest of this chapter, we will first discuss the Markov model for each of the MHs.Then we will again look at the full sarcomere, to find the FKE that describes the fulldynamics of a sarcomere.

2.2. Markov model for a single myosin head

As mentioned in the previous section, each myosin head can be in one of four states. Thetransition rates between these four states are summarised in Figure 2.3, and depend onthe free calcium concentration c, where high concentrations facilitate the transition fromnot bound (0) to bound (1); on the number of actin filaments opposite the MH (throughthe index i and the variable SL); and on the number of neighbours that are permissive,n ∈ {0, 1, 2}, where a large value of n facilitates the transition from non-permissive (N )to permissive (P). This last dependence is thus the one that causes the cooperativeinteractions. We will denote the number of neighbours in permissive state by n(ξ, η),

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when ξ ∈ S and η ∈ S are the states of the direct neighbours. Now we can write thetransition rate of the ith MH from some state β ∈ S to the state α ∈ S \ {β}, withneighbours in state ξ ∈ S and η ∈ S, respectively, as

Aαβ(c, SL, i, n(ξ, η)).

For example, if α = 0N and β = 0P , then Aαβ(c, SL, i, n(ξ, η)) = γnKnp0(SL, i), as canbe seen in Figure 2.3. The meaning and interpretation of the variables and constantsused in the transition rates, will be discussed in section 2.2.1.

When we determine the transition rate for each possible combination for α and β, andwe give the states 0N , 1N , 1P , and 0P the indices 1, 2, 3, and 4, respectively, then wecan write the transition rates in a matrix A = (Aαβ), where 1 ≤ α, β ≤ 4:

A(c, SL, i, n) =

0 Koff 0 γ−nKpn0

Kon(SL, i)c 0 γ−nKpn1 00 γnKnp1(SL, i) 0 K ′on(SL, i)c

γnKnp0(SL, i) 0 K ′off 0

(2.1)

Here, the zeros indicate that for example a transition from 0P to 1N , or staying in thesame state, is not possible.

Figure 2.3.: The cooperative four states Markov model of a single myosin head (MH). The terms de-pending on the intracellular calcium concentration c are coloured in red; terms dependingon the state of the neighbouring MHs (i.e. depending in n) are coloured in blue; termsdepending on the position of the MH and the current sarcomere elongation are coloured ingreen. Reprinted from Regazzoni et al. (2018).

2.2.1. Formulas for the transition rates

In this subsection, the formulas for the transition rates between the states will be dis-cussed; the values of the constants can be found in Table 2.2.

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Parameter Value UnitsSarcomere geometryLA (length of AF) 1.2 µmLM (length of MF) 1.65 µmLH (length of H-zone) 0.1 µmN (number of MHs) 36 -

Transition rates Ca bindingKon 80 µM−1s−1

Koff 80 s−1

Transition rates permissivityQ0 3 -SLQ 2.2 µmαQ 1.4 µm−1

Kbasic 10 s−1

µ 10 -γ 40 -

SL dependenceaR 0.1 µmaL 0.1 µm

Table 2.2.: Model parameters; values reprinted from Washio et al. (2012).

First, the constant transition rates used by Rice et al. (2003), are given by:

Knp0 = QKbasic/µ,

Knp1 = QKbasic,

Kpn0 = Kbasicγ2,

Kpn1 = Kbasicγ2,

K ′on = Kon,

K ′off = Koff/µ.

The explanation of these constant transition would be very time-consuming rates takeslong and is beyond the scope of this thesis. For a thorough explanation we thereforerefer to the article of Rice et al. (2003).

Since the force-calcium relationship depends on the sarcomere length (see section4.1.2), Washio et al. (2012) changed the constant Q as used by Rice et al. (2003) into afunction that depends on SL. This function is given by:

Q(SL) =

{Q0 SL ≥ SLQ

Q0 − αQ(SLQ − SL) SL < SLQ,

where they used values for Q0, αQ, and SLQ that they obtained the best results with (seeTable 2.2). The value of Q thus decreases linearly when the sarcomere length becomessmaller than SLQ = 2.2 µm.

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Furthermore, Rice et al. (2003) already introduced the terms γn and γ−n in the tran-sition rates from N to P and from P to N , respectively, to incorporate the cooperativeinteractions. However, their transition rates did not depend on the overlap of the fila-ments. As discussed in section 1.2, the amount of overlap is important since it determineshow many myosin heads face an actin filament and can form a crossbridge. For an indi-vidual MH, this means that the transition rate should be adopted when it faces eitherzero or two AFs, since then it can not produce force. How many AFs an myosin headfaces, depends on both the sarcomere length and the position of the MH. For example,there might be a region with no overlap between the MF and the AFs, but when the MHis not in that region, it can still form a crossbridge. Therefore, we need transition ratesthat depend on both the sarcomere length (SL) and the position of the MH (determinedby the number of i).

Washio et al. (2012) introduced this dependence by multiplying the constant transitionrates as used by Rice et al. (2003) by the geometrical factors χRA(SL, i) and χLA(SL, i):

Knp0(SL, i) = χLA(SL, i)χRA(SL, i)Knp0,

Knp1(SL, i) = χLA(SL, i)χRA(SL, i)Knp1,

Kon(SL, i) = χRA(SL, i)Kon,

K ′on(SL, i) = χRA(SL, i)K ′on.

The geometrical factors are functions of the position of the myosin head they are com-puted for, and of the amount of overlap between the MF and AFs. The position ofthe MH can be determined using the length of the MF (LM), the length of the H-zone(LH), and the number of MHs on the MF (N), see also Figure 2.1:

xi =LM − LH

2Ni.

The amount of overlap is determined by the positions of the free ends (i.e. the ones notconnected to the Z-disk) of the AFs. These positions can be determined using the lengthof the AFs (LA), the length of the H-zone and the position of the Z-disk connected tothe right AF (xAZ), see also Figure 2.1:

xLA = LA− xAZ − LH,xRA = xAZ − LA,

where xAZ can be computed as

xAZ =SL− LH

2.

For the overlap with the right AF there are three kinds of regions possible on the right(half) MF (see also Figure 2.4), namely a region at the left where there is no overlapwith the right AF (xi ≤ xRA, Figure 2.4 (a1)), a region where there is overlap with theright AF (xRA ≤ xi ≤ xAZ), and a region at the right where there is no overlap withthe right AF (xi ≥ xAZ , Figure 2.4 (a3)). In the region with overlap, the rate constant

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Figure 2.4: Three different kinds of over-lap between the myosin fila-ment (MF) and the two actinfilaments (AFs): in (a1) thereis a region in the middle of thesarcomere where the two AFsdo not overlap; in (a2) the twoAFs and the MF overlap every-where; in (a3) the boundaryregions of the MF do not haveoverlap with an AF. Reprintedfrom Washio et al. (2012).

for the forming formation of crossbridges is not modified. In the non-overlap regions,the rate constant is modified such that it decreases when the MH is further away fromthe AF. Here, a step function is used that is smoothed at the end, such that χRA iscontinuous with respect to SL:

χRA(SL, i) =

exp

(− (xRA−xi)2

a2R

)xi ≤ xRA

1 xRA < xi < XAZ

exp(− (xi−xAZ)2

a2R

)xi ≥ xAZ .

We see that the function is also differentiable with respect to xi, and thus with respectto SL, since the limits of the derivatives from left and right are equal for both transitionpoints (all zero).

The parameter aR determines how fast the MHs at the boundary of a region withoverlap lose their capability to form a crossbridge when they move away from the ends ofthe filaments. The value was chosen such that it agrees with the force-length relationship(see Table 2.2).

Since the model includes only half a sarcomere, for the overlap with the left AF, thereare two kinds of regions possible (see also Figure 2.4), namely a region at the left wherethere is overlap with both the left and the right AF (xi ≤ xLA), and a region at the rightwhere there is overlap with just the right AF. Here it is assumed that the formation ofcrossbridges in the region with double overlap is inhibited. The modification is againsuch that the rate constant decreases when the MH is further away from the regionwith single overlap. In the region with single overlap the rate constant is not modified.Furthermore, the step function used is smoothed again such that χLA is continuous anddifferentiablewith respect to SL1:

χLA(SL, i) =

{exp

(− (xLA−xi)2

a2L

)xi ≤ xLA

1 xi > xLA.

The parameter aL has the same interpretation as aR in χRA(SL, i).

1Here, I found a typo: Regazzoni et al. (2018) wrote aR where they should have written aL.

17

2.3. Forward Kolmogorov equation

Now that we have worked out the Markov model for the individual MHs, we can againlook at the the sarcomere and its dynamics. The model consists of a CTMC on the statespace SN , where S is the set of possible states of each MH and N is the number of MHs.Since the transition rates depend on time (since they depend on the time-dependent c(t)and SL(t)), the CTMC is not time-homogeneous. We define X i

t ∈ S to be the randomprocess that is associated with the ith MH. Similar to Regazzoni et al. (2018), in therest of this thesis we will use the following notation to denote events at time t:

(α1, α2, . . . , αN)t := {X1t = α1, . . . , X

Nt = αN},

(α,i

β)t := {X i−1t = α,X i

t = β},

(α,i

β, δ)t := {X i−1t = α,X i

t = β,X i+1t = δ},

(α, β,i

δ, η)t := {X i−2t = α,X i−1

t = β,X it = δ,X i+1

t = η}.

The superscript i is thus used to denote that the ith MH is in the associated state.First, we need an interpretation of the transition rates in the matrix in equation

(2.1). These transition rates can be interpreted as follows: multiplying the rates by aninfinitesimal time step ∆t gives a first order (in ∆t) approximation of the probabilitythat a single MH changes state in the time interval [t, t+ ∆t]. For each η 6= αi, we canthus write:

P((α1, . . . , αi−1, η, αi+1, . . . , αN)t+∆t|(α1, . . . , αN)t

)= ∆t · Aηαi(c(t), SL(t), i, n(αi−1, αi+1)) +O(∆t2).

(2.2)

Transitions of two or more MHs in the time interval [t, t+ ∆t] are of second order in ∆t.For each η 6= αi and ξ 6= αj, we can thus write:

P((α1, . . . , η, . . . , ξ, . . . , αN)t+∆t|(α1, . . . , αi, . . . , αj, . . . , αN)t

)= O(∆t2). (2.3)

To find the FKE for this model, we first want to find an expression for the probabilityof a combination of states at time t+ ∆t, so we want to find an expression for

P((α1, . . . , αN)t+∆t

).

Using the law of total probability, we can write this as

P((α1, . . . , αN)t+∆t

)=

N∑i=1

∑ξ∈S

[P((α1, . . . , αN)t+∆t|(α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2)

18

= P((α1, . . . , αN)t+∆t|(α1, . . . , αN)t

)P((α1, . . . , αN)t

)(2.4)

+N∑i=1

∑ξ∈S\{αi}

[P((α1, . . . , αN)t+∆t|(α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2),

where we use that the probability of two transitions is of order ∆t2 (equation (2.3)).We split the sum into two parts (no transition and one transition), since we need totreat them differently. The part with one transition can directly be written in termsof transition rates using equation (2.2). The part with no transition first needs to berewritten in terms of probabilities of one transition. This can be done using the fact thatthe probability of no transition is one minus the probability of one or more transitions;here again the probabilities of more than one transition are contained in the termO(∆t2):

P((α1, . . . , αN)t+∆t|(α1, . . . , αN)t

)= 1−

N∑i=1

∑ξ∈S\{αi}

P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t+∆t|(α1, . . . , αN)t

)+O(∆t2).

Now we can substitute this expression into equation (2.4), which gives:

P((α1, . . . , αN)t+∆t

)= P

((α1, . . . , αN)t

)·

1−N∑i=1

∑ξ∈S\{αi}

P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t+∆t|(α1, . . . , αN)t

)+O(∆t2)

+

N∑i=1

∑ξ∈S\{αi}

[P((α1, . . . , αN)t+∆t|(α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2)

= P((α1, . . . , αN)t

)− P

((α1, . . . , αN)t

) N∑i=1

∑ξ∈S\{αi}

P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t+∆t|(α1, . . . , αN)t

)+

N∑i=1

∑ξ∈S\{αi}

[P((α1, . . . , αN)t+∆t|(α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2).

19

We have now written the probability of the event at time t+∆t in terms of probabilities ofan event at time t and probabilities of one transition. The probabilities of one transitioncan now be substituted by the expression in equation (2.2) to get

P((α1, . . . , αN)t+∆t

)= P

((α1, . . . , αN)t

)− P

((α1, . . . , αN)t

) N∑i=1

∑ξ∈S\{αi}

[∆t · Aξαi(c(t), SL(t), i, n(αi−1, αi+1))]

+N∑i=1

∑ξ∈S\{αi}

[∆t · Aαiξ(c(t), SL(t), i, n(αi−1, αi+1))

·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2)

= P((α1, . . . , αN)t

)−∆t · P

((α1, . . . , αN)t

) N∑i=1

∑ξ∈S\{αi}

Aξαi(c(t), SL(t), i, n(αi−1, αi+1))

+ ∆t ·N∑i=1

∑ξ∈S\{αi}

[Aαiξ(c(t), SL(t), i, n(αi−1, αi+1))

·P((α1, . . . , αi−1, ξ, αi+1, . . . , αN)t

)]+O(∆t2).

(2.5)

When we now write p(t) for the vector with length 4N containing each possibleP ((α1, . . . , αN)t), we can write

p(t+ ∆t) = p(t) + ∆tAp(t) +O(∆t2), (2.6)

where A is a 4N × 4N matrix containing transition rates. For N = 1 this matrix is (seeAppendix A):

A =−Aβα − Aγα − Aδα Aαβ Aαγ Aαδ

Aβα −Aαβ − Aγβ − Aδβ Aβγ AβδAγα Aγβ −Aαγ − Aβγ − Aδγ AγδAδα Aδβ Aδγ −Aαδ − Aβδ − Aγδ

.

In general, the matrix A contains many zeros, for example because the transitions 0N ↔1P and 1N ↔ 0P are not possible. Furthermore, it contains zeros for transitions suchas γj → βj when the MH at position j is not in state γj at time t. For more insightin how the matrix A is formed and what it looks like, we have provided a step-by-stepderivation for N = 1 and N = 2 in Appendix A.

20

Now we go back to equation (2.6). Since we want to derive a system of ODEs fromthis, we first rewrite it to

p(t+ ∆t)− p(t)

O(∆t)=

∆tAp(t) +O(∆t2)

O(∆t)= Ap(t) +O(∆t). (2.7)

Now we let ∆t→ 0, which gives the following system of ODEs (the FKE):

p = Ap(t). (2.8)

21

3. Reduced model

In the previous chapter, we described the full model as of Rice et al. (2003) with themodification of Washio et al. (2012). Using equation (2.8) we could compute the exactevolution of the probabilities of the 4N states. However, we have a model with N =36 MHs, which means that there are 436 ≈ 5 · 2021 degrees of freedom in the model.Therefore, storing only the vector p(t) would already need extremely many bytes. Thismeans that computing solutions of equation (2.8) is computational impossible. Hence, weneed to reduce the number of degrees of freedom a lot before we can compute solutions.In this chapter we will first discuss the idea of the reduction. After that, some rewritingof probabilities will be done in order to be able to substitute the transition rates intothe equations. Finally, these rewritten probabilities will be combined such that a newsystem of ODEs will be found.

3.1. The basis of the reduction

The first step in the reduction of the model, is the realisation that the transition rates ofthe ith MH are determined entirely by the states of the triplet of MHs that is centeredin the ith head. That is why Regazzoni et al. (2018) came up with the idea that itis reasonable to assume that the set of the joint probabilities of such triplets providesa good description of the state of the whole system. Therefore, from now on we willconsider events of the kind

(α,i

β, δ)t,

with i = 2, . . . , N − 1 and α, β, δ ∈ S. Here, we need to remark that these triplets arecoupled, and can thus not be considered separately. Just as in the full model, we willnow look at the time evolution of the probabilities of such events. Again, we want torewrite the probability of the combination of states of a triplet at time t+ ∆t in termsof transition rates and probabilities of the combination of states of the triplet at time t.Using the law of total probability, we can now write

P((α,i

β, δ)t+∆t)

= P((α,i

β, δ)t+∆t|(α,i

β, δ)t) · P((α,i

β, δ)t)

+∑

η∈S\{α}

[P((α,

i

β, δ)t+∆t|(η,i

β, δ)t) · P((η,i

β, δ)t)

]

22

+∑

η∈S\{β}

[P((α,

i

β, δ)t+∆t|(α, iη, δ)t) · P((α,iη, δ)t)

](3.1)

+∑

η∈S\{δ}

[P((α,

i

β, δ)t+∆t|(α,i

β, η)t) · P((α,i

β, η)t)

]+O(∆t2).

Here, the first term is the probability of no transition, the second term is the probabilityof a transition in the left MH of the triplet, the third term is the probability of a transitionin the central MH of the triplet, and the fourth term is the probability of a transitionin the right MH of the triplet. Again, the probability of more than one transition iscontained in the term O(∆t2).

3.2. Rewriting of the conditional probabilities

Each of these terms now needs to be rewritten in terms of the transition rates. Bydefinition of the transition rate we know that the probability of a transition in thecentral MH of the triplet is given by

P((α,i

β, δ)t+∆t|(α, iη, δ)t) = ∆t · Aβη(c(t), SL(t), i, n(α, δ)) +O(∆t2),

where i = 2, . . . , N − 1 and η 6= β. Also the probability of transition of the left MHin the first triplet (i = 2) and the probability of transition of the right MH in the lasttriplet (i = N − 1), can directly be written in terms of the transition rates, using theassumption that the outer MHs of the sarcomere always have a neighbour in state 0Nwhere they do not have an actual neighbour (section 2.1):

P((1α, β, δ)t+∆t|(1

η, β, δ)t) = ∆t · Aαη(c(t), SL(t), 1, n(0N , β)) +O(∆t2),

P((α, β,N

δ )t+∆t|(α, β,Nη)t) = ∆t · Aδη(c(t), SL(t), N, n(β, 0N )) +O(∆t2).

However, the probabilities of a transition in the left or right MH of the middle triplets,in the left MH of the last triplet, and in the right MH of the first triplet, can notbe written in terms of the transition rates that easily. This is because the transitionrates depend on states of the two the neighbours of the changing myosin head, so theprobabilities should be in term of the changing MH and its two neighbours, but this isnot the case. Therefore, we first need to rewrite the probabilities such that they are interms of triplets of which the central MH changes. To do so, we make use of the law oftotal probability and of the definition of a conditional probability:

P(A|B) =P(A ∩B)

P(B)⇐⇒ P(A|B)P(B) = P(A ∩B),

where A and B are events.

23

We will first work out the details for the probability of a transition in the left MHof a middle or last triplet (i = 3, . . . , N − 1); for the probability of a transition in theright MH of a middle or first triplet (i = 2, . . . , N − 2) it works exactly the same (againη 6= β):

P((α,i

β, δ)t+∆t|(η,i

β, δ)t)

=P((α,

i

β, δ)t+∆t ∩ (η,i

β, δ)t)

P((η,i

β, δ)t)

∗=

∑ξ∈S P((ξ, α,

i

β, δ)t+∆t ∩ (ξ, η,i

β, δ)t)

P((η,i

β, δ)t)

+O(∆t2)

=

∑ξ∈S

[P((ξ, α,

i

β, δ)t+∆t|(ξ, η,i

β, δ)t) · P((ξ, η,i

β, δ)t)

]P((η,

i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) · P((ξ, η,i

β, δ)t)

P((η,i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) · P((ξ, η,i

β)t ∩ (i+1

δ )t)

P((η,i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) · P((i+1

δ )t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) · P((i+1

δ )t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2).

(3.2)

Where in the step marked with a ∗ the term O(∆t2) appears in order to account forprobabilities of more than one transition.

Now we remark that P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) is the probability that the (i− 1)thMH changes from η to α. This means that, when δ would also change, the probabilitywould become of second order. Therefore, we can drop the δ at time t + ∆t to furthersimplify the expression. In symbols this is:

P((ξ, α,i

β)t+∆t|(ξ, η,i

β, δ)t)

= P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) +∑γ∈S

P((ξ, α,i

β, γ)t+∆t|(ξ, η,i

β, δ)t)

= P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) +O(∆t2).

24

So now we can write:

P((ξ, α,i

β, δ)t+∆t|(ξ, η,i

β, δ)t) = P((ξ, α,i

β)t+∆t|(ξ, η,i

β, δ)t) +O(∆t2).

Now substituting this in the last expression of equation (3.2), we get:

P((α,i

β, δ)t+∆t|(η,i

β, δ)t)

=∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β, δ)t) · P((i+1

δ )t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2).

To simplify the expression even more, we remark that the transition rate of the (i−1)thMH from η to α is independent of the state of the (i + 1)th MH (more than one MHaway). This is due to the definition of the transition rate: it depends on the states ofthe two closest neighbours, not on the states of the further MHs. Therefore, we can alsodrop the δ at time t, which leads to the expression

P((α,i

β, δ)t+∆t|(η,i

β, δ)t)

=∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β)t) · P((i+1

δ )t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2).

At this stage, the most important assumption of Regazzoni et al. (2018) comes intothe picture. Namely, they assume that at each time instant t the states of MHs threepositions away from each other are conditionally independent given the states of the twoMHs in between them. We write this as

X it ⊥ X i+3

t |(X i+1t , X i+2

t ),

for any t > 0 and i = 1, . . . , N − 3. In words this says that knowledge of the (i − 3)thMH does not provide any extra information about the probability distribution of the ithhead beyond the knowledge of the states of the (i− 1)th and (i− 2)th MHs. Accordingto this assumption, we see that

P((i

δ)t|(ξ, η,i−1

β )t) ≈ P((i

δ)t|(η,i−1

β )t).

This means that we can now write

P((α,i

β, δ)t+∆t|(η,i

β, δ)t)

≈∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β)t) · P((i+1

δ )t|(η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2).

25

Now, we can again use the definition of a conditional probability to write

P((α,i

β, δ)t+∆t|(η,i

β, δ)t)

≈∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β)t) · P((i+1

δ )t ∩ (η,i

β)t)

P((η,i

β)t)

· P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β)t) · P((η,i

β, δ)t)

P((η,i

β)t)

· P((ξ, η,i

β)t)

P((η,i

β, δ)t)

+O(∆t2)

=∑ξ∈S

P((ξ, α,i

β)t+∆t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β)t)

+O(∆t2)

=

∑ξ∈S P((ξ, α,

i

β)t+∆t|(ξ, η,i

β)t) · P((ξ, η,i

β)t)

P((η,i

β)t)

+O(∆t2)

=

∑ξ∈S P((ξ,

i−1α , β)t+∆t|(ξ, i−1

η , β)t) · P((ξ,i−1η , β)t)

P((η,i

β)t)

+O(∆t2)

=

∑ξ∈S [∆t · Aαη(c(t), SL(t), i− 1, n(ξ, β)) +O(∆t2)] · P((ξ,

i−1η , β)t)

P((η,i

β)t)

+O(∆t2)

=

∑ξ∈S Aαη(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,

i−1η , β)t)

P((η,i

β)t)

·∆t+O(∆t2)

=

∑ξ∈S Aαη(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,

i−1η , β)t)∑

ξ∈S P((ξ,i−1η , β)t)

·∆t+O(∆t2).

Here we have thus rewritten the conditional probability such that we can express it interms of the transition rates.

As stated earlier, we can derive an expression for the probability of a transition in theright MH of a middle or first triplet in exactly the same way. If we do so, we find thatfor i = 2, . . . , N − 2 and η 6= δ the probability is well approximated by

P((α,i

β, δ)t+∆t|(α,i

β, η)t)

≈∑

ξ∈S Aδη(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1η , ξ)t)∑

ξ∈S P((β,i+1η , ξ)t)

·∆t+O(∆t2).

Now we also need to find an expression for the probability of no transition. As we didearlier, we will use the fact that the probability of no transition is equal to one minus

26

the probability of at least one transition. When we do so, we find for i = 2, . . . , N − 1that

P((α,i

β, δ)t+∆t|(α,i

β, δ)t)

= 1−∑

η∈S\{α}

P((η,i

β, δ)t+∆t|(α,i

β, δ)t)

−∑

η∈S\{β}

P((α,iη, δ)t+∆t|(α,

i

β, δ)t)

−∑

η∈S\{δ}

P((α,i

β, η)t+∆t|(α,i

β, δ)t) +O(∆t2).

(3.3)

We see that we have already found an expression in terms of the transition rates foreach term in this expression.

3.3. Derivation of the system of ODEs

Now that we have rewritten each term in equation (3.1), we can substitute the foundexpressions in equation (3.1). First we substitute the expression from equation (3.3)into equation (3.1), which gives

P((α,i

β, δ)t+∆t)

= P((α,i

β, δ)t) ·

1−∑

η∈S\{α}

P((η,i

β, δ)t+∆t|(α,i

β, δ)t)

−∑

η∈S\{β}

P((α,iη, δ)t+∆t|(α,

i

β, δ)t)−∑

η∈S\{δ}

P((α,i

β, η)t+∆t|(α,i

β, δ)t) +O(∆t2)

+

∑η∈S\{α}

P((α,i

β, δ)t+∆t|(η,i

β, δ)t) · P((η,i

β, δ)t)

+∑

η∈S\{β}

P((α,i

β, δ)t+∆t|(α, iη, δ)t) · P((α,iη, δ)t)

+∑

η∈S\{δ}

P((α,i

β, δ)t+∆t|(α,i

β, η)t) · P((α,i

β, η)t) +O(∆t2)

= P((α,i

β, δ)t)

− P((α,i

β, δ)t)∑

η∈S\{α}

P((η,i

β, δ)t+∆t|(α,i

β, δ)t)

− P((α,i

β, δ)t)∑

η∈S\{β}

P((α,iη, δ)t+∆t|(α,

i

β, δ)t)

27

− P((α,i

β, δ)t)∑

η∈S\{δ}

P((α,i

β, η)t+∆t|(α,i

β, δ)t)

+∑

η∈S\{α}

P((α,i

β, δ)t+∆t|(η,i

β, δ)t) · P((η,i

β, δ)t)

+∑

η∈S\{β}

P((α,i

β, δ)t+∆t|(α, iη, δ)t) · P((α,iη, δ)t)

+∑

η∈S\{δ}

P((α,i

β, δ)t+∆t|(α,i

β, η)t) · P((α,i

β, η)t) +O(∆t2).

Now we can also substitute the expressions of the conditional probabilities in terms ofthe transition rates into this expression. Since these expressions are not completely thesame for the middle triplets and the outer triplets, we will first do this for the middletriplets and after that for the outer triplets. For the middle triplets (i = 3, ..., N − 2),we see that

P((α,i

β, δ)t+∆t)

≈ P((α,i

β, δ)t)

− P((α,i

β, δ)t)∑

η∈S\{α}

∑ξ∈S Aηα(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1α , β)t)∑

ξ∈S P((ξ,i−1α , β)t)

·∆t

− P((α,

i

β, δ)t)∑

η∈S\{β}

[∆t · Aηβ(c(t), SL(t), i, n(α, δ))]

− P((α,i

β, δ)t)∑

η∈S\{δ}

∑ξ∈S Aηδ(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1

δ , ξ)t)∑ξ∈S P((β,

i+1

δ , ξ)t)

·∆t

+

∑η∈S\{α}

∑ξ∈S Aαη(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1η , β)t)∑

ξ∈S P((ξ,i−1η , β)t)

·∆t · P((η,i

β, δ)t)

+

∑η∈S\{β}

[∆t · Aβη(c(t), SL(t), i, n(α, δ)) · P((α,

iη, δ)t)

]

+∑

η∈S\{δ}

∑ξ∈S Aδη(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1η , ξ)t)∑

ξ∈S P((β,i+1η , ξ)t)

·∆t · P((α,i

β, η)t)

+O(∆t2)

= P((α,i

β, δ)t)

+ ∆t∑

η∈S\{α}

∑ξ∈S Aαη(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1η , β)t)∑

ξ∈S P((ξ,i−1η , β)t)

· P((η,i

β, δ)t)

28

−∑

ξ∈S Aηα(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1α , β)t)∑

ξ∈S P((ξ,i−1α , β)t)

· P((α,i

β, δ)t)

+ ∆t

∑η∈S\{β}

[Aβη(c(t), SL(t), i, n(α, δ)) · P((α,

iη, δ)t)

−Aηβ(c(t), SL(t), i, n(α, δ)) · P((α,i

β, δ)t)

]

+ ∆t∑

η∈S\{δ}

∑ξ∈S Aδη(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1η , ξ)t)∑

ξ∈S P((β,i+1η , ξ)t)

· P((α,i

β, η)t)

−∑

ξ∈S Aηδ(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1

δ , ξ)t)∑ξ∈S P((β,

i+1

δ , ξ)t)

· P((α,i

β, δ)t)

.We have now thus rewritten the probability of the combination of states of a triplet attime t+ ∆t in terms of transition rates and probabilities of the combination of states ofthe triplet at time t, which is exactly what we wanted.

Now we still need to do the same for the first and last triplet. When we do this, wefind for the first triplet (i = 2) that

P((1α, β, δ)t+∆t)

≈ P((1α, β, δ)t)

+ ∆t∑

η∈S\{α}

[Aαη(c(t), SL(t), 1, n(0N , β)) · P((η,

i

β, δ)t)

−Aηα(c(t), SL(t), 1, n(0N , β)) · P((α,i

β, δ)t)

]+ ∆t

∑η∈S\{β}

[Aβη(c(t), SL(t), i, n(α, δ)) · P((α,

iη, δ)t)

−Aηβ(c(t), SL(t), i, n(α, δ)) · P((α,i

β, δ)t)

]

+ ∆t∑

η∈S\{δ}

∑ξ∈S Aδη(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1η , ξ)t)∑

ξ∈S P((β,i+1η , ξ)t)

· P((α,i

β, η)t)

−∑

ξ∈S Aηδ(c(t), SL(t), i+ 1, n(β, ξ)) · P((β,i+1

δ , ξ)t)∑ξ∈S P((β,

i+1

δ , ξ)t)

· P((α,i

β, δ)t)

.For the last triplet (i = N − 1), we find that

P((α, β,N

δ )t+∆t)

29

≈ P((α, β,N

δ )t)

+ ∆t∑

η∈S\{α}

∑ξ∈S Aαη(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1η , β)t)∑

ξ∈S P((ξ,i−1η , β)t)

· P((η,i

β, δ)t)

−∑

ξ∈S Aηα(c(t), SL(t), i− 1, n(ξ, β)) · P((ξ,i−1α , β)t)∑

ξ∈S P((ξ,i−1α , β)t)

· P((α,i

β, δ)t)

+ ∆t

∑η∈S\{β}

[Aβη(c(t), SL(t), i, n(α, δ)) · P((α,

iη, δ)t)

−Aηβ(c(t), SL(t), i, n(α, δ)) · P((α,i

β, δ)t)

]+ ∆t

∑η∈S\{δ}

[Aδη(c(t), SL(t), N, n(β, 0N )) · P((α,

i

β, η)t)

−Aηδ(c(t), SL(t), N, n(β, 0N )) · P((α,i

β, δ)t)

].

Now for each case we subtract the probability of the same state at time t and divideby ∆t (just like we did in equation (2.7)) and let ∆t→ 0. Combining the result for themiddle and outer triplets, gives the following nonlinear system of ODEs:

d

dtP((α,

i

β, δ)t)

=∑

η∈S\{α}

[ΦiL(η, β, δ;α; t)− Φi

L(α, β, δ; η; t)]

+∑

η∈S\{β}

[ΦiC(α, η, δ; β; t)− Φi

C(α, β, δ; η; t)]

+∑

η∈S\{δ}

[ΦiR(α, β, η; δ; t)− Φi

R(α, β, δ; η; t)],

where

ΦiC(α, β, δ; η; t) = Aηβ(c(t), SL(t), i, n(α, δ)) · P((α,

i

β, δ)t),

ΦiL(α, β, δ; η; t) =

∑ξ∈S Φi−1

C (ξ,α,β;η;t)∑ξ∈S P((ξ,

i−1α ,β)t)

· P((α,i

β, δ)t) i = 3, ..., N − 1

Aηα(c(t), SL(t), 1, n(0N , β)) · P((α,i

β, δ)t) i = 2,

ΦiR(α, β, δ; η; t) =

∑ξ∈S Φi+1

C (β,δ,ξ;η;t)∑ξ∈S P((β,

i−1δ ,ξ)t)

· P((α,i

β, δ)t) i = 2, ..., N − 2

Aηδ(c(t), SL(t), N, n(β, 0N )) · P((α,i

β, δ)t) i = N − 1.

Here, I found another typo in the article of Regazzoni et al. (2018)): in the special cases

30

(i = 2 and i = N − 1) the transition rates from respectively η to α and η to δ are giveninstead of those from α to η and δ to η.

We see that the reduced model is thus a nonlinear system, while the full model (equa-tion 2.8) is linear. However, the pay-off is that we have reduced the number of degreesof freedom drastically from 4N ≈ 5 · 1021 to (N − 2) · 43 = 2176. Namely, we are nowlooking at N − 2 triplets that can all be in 43 states, where we were first looking at onesarcomere that can be in 4N states. This significant reduction in number of degrees offreedom, makes it now possible to compute solutions.

31

4. Numerical results

Now that we have discussed the reduced model, we can run it to do some numerical tests.The goal of these tests is to validate the model by testing whether the numerical resultsderived from the model are in line with what is observed in muscles experimentally.Therefore, we will compare the results derived from the model with experimental data.

In the numerical tests, a Forward Euler scheme with time steps of 2.5 · 10−5 is usedfor the time discretisation. Furthermore, since the transition rates depend on SL andc, which change slowly over time, the transition rates are updated only every 0.25 ms.By doing so, we do not compute the transition rates more often than actually needed.Finally, it is assumed that at time t = 0 all MHs are in the state 0N ; in other words, itis assumed that the sarcomere is fully inactive at time t = 0. In symbols, the followinginitial condition is used:

P((α,i

β, δ)0) =

{1 if α = β = δ = 0N ,0 else.

As discussed in section 1, three important force-relationships can be found in mus-cles: the force-length relationship, the force-calcium relationship, and the force-velocityrelationship. Therefore, we will test our model for these relationships in the followingsections. These relationships only hold for steady state conditions. However, we are alsointerested in dynamic situations, since in these situations the results really depend onthe calcium and sarcomere length transients that are discussed in section 2.1. Therefore,we will also test the model for two dynamic situations.

In experimental research, the force produced by a muscle is often normalised to themaximum amount of force that can be generated in a certain condition. This is donedifferently by different authors and will therefore be specified for each case separately.We have to remark that the normalised force is not always between 0% and 100% sinceat a different sarcomere length a higher maximum amount of force might be produced.However, we can still compare the shapes of the graphs, since these are not affected bythe way of scaling.

4.1. Steady state analysis

Steady state relationships can be derived from the model by letting the system reacha steady state under different conditions. When steady state is reached, we determinethe permissivity associated with that condition. What these conditions are for eachrelationship, will be explained further in the corresponding sections. In our analysis, we

32

assume that after four seconds, steady state has always been reached. This has beenverified by looking at graphs of the solutions over time.

(a) Steady state force-length relationship as derived from the model for different values of the calciumconcentration c.

(b) Steady state force-length relationship as derived from two rat cardiac sarcomeres for different valuesof the calcium concentration; only available in the physiological range (approximately 1.6−2.2 µm).It is unclear to what the force is normalised, we think it might be normalised to the maximumamount of force that has been measured at SL = 2.00 µm and c = 8.9 µM. Adopted from Kentishet al. (1986).

Figure 4.1.: Steady state force-length relationship for different values of the calcium concentration:comparison of the results of the reduced model (a) and experimental curves (b).

33

4.1.1. Force-length relationship

The force-length relationship holds for steady state and fixed calcium concentration. Toobtain this relationship out of the model, we will thus have to fix c and let the systemreach the steady state. Since we have assumed that the force is directly proportional tothe permissivity, we want to find the permissivity for different values of the sarcomerelength. Therefore, we will also fix SL, and then we check what the permissivity is inthe steady state. When we do this for many values of SL, we find the force-length rela-tionship. It is known that the force-length relationship is slightly different for differentvalues of the calcium concentration: for higher values of c the force for a certain value ofSL is higher (Gordon and Pollack, 1980). Therefore, we repeat the above for differentvalues of the calcium concentration.

The results can be seen in Figure 4.1a. The first thing we notice is that the shape ofthe graphs is similar to the shape of the force-length relationship that we saw in section1.2 (Figure 1.6). Indeed, our model predicts that the sarcomere can produce most force(highest permissivity is obtained) when it has a length such that the whole MF facesexactly one AF (2.2 − 2.6 µm). When the sarcomere is shorter than 2.2 µm, the AFsoverlap and the permissivity is lower. When the sarcomere is longer than 2.6 µm, someMHs in the middle part of the MF do not face an AF, and again the permissivity islower.

When we look at the different curves for different values of the calcium concentration,we see that for higher values of c the force that can produced is higher. Furthermore, inthe region 1.5 µm ≤ SL ≤ 2.2 µm, we see that the curves are convex for low values ofc. For higher values of c, the curves first become concave, and at the highest values ofc they become approximately linear.

Now, if we look at the experimental force-length relationship for different values of thecalcium concentration (Figure 4.1b), we see that the same change in convexity occurs.We can thus conclude that the reduced model can accurately predict the experimentallyobserved force-length relationship.

4.1.2. Force-calcium relationship

Just like the force-length relationship, also the force-calcium relationship holds for steadystate. In this case, we fix sarcomere length, and we want to obtain the permissivity fordifferent values of the calcium concentration. Therefore, we will let the system reachsteady state for several different combinations of fixed values of c and SL. Then wedetermine the permissivity in steady state, which gives the force-calcium relationshipfor different values of SL.

The results can be seen in Figure 4.2a. The first thing we notice is that, again, theshape of the graphs is similar to the shape of the force-calcium relationship that wesaw in section 1.2 (Figure 1.4). Indeed, our model predicts an approximately sigmoidal(S-shaped) curve on a semilogarithmic scale. The model thus indeed predicts the highsensitivity of the developed force to the calcium concentration that was discussed insection 1.2.

34

Furthermore, we see that for higher values of the sarcomere length, the permissivityat a certain calcium concentration is higher. This is of course in line with what we sawin the previous section. Finally, we see that for increasing SL values, the curve shiftsto the left. This means that the sarcomere becomes more sensitive to calcium when thesarcomere is longer.

(a) Steady state force-calcium relationship as derived from the model for different values of the sarcomerelength SL.

(b) Steady state force-calcium relationship as derived from 10 rat cardiac sarcomeres (average is dis-played) for different values of the sarcomere length, namely from bottom to top: SL = 1.85, 1.95,2.05, 2.15, and 2.25 µm. The force is normalised to the maximum amount of force that has beenmeasured at SL = 2.05 µm. Adopted from Dobesh et al. (2002).

Figure 4.2.: Steady state force-calcium relationship for different values of the sarcomere length: com-parison of the results of the reduced model (a) and experimental curves (b).

When looking at the experimental data in Figure 4.2b, we see the same happening:for larger values of the sarcomere length, the relative force is higher at the same value

35

of c; furthermore, for larger values of the sarcomere length, the curves shift to the left.Therefore, we can conclude that the reduced model can also predict the force-calciumrelationship quite accurately.

4.1.3. Force-velocity relationship

The main reason for checking the model for the force-velocity relationship, is that Regaz-zoni et al. (2018) state in their discussion that, “since the original model of Washio et al.(2012) is restricted to thin (actin) filament dynamics, it does not incorporate velocity-dependent effects, which are linked to crossbridge dynamics” (see also section 5.1). How-ever, in their article they do not test whether it is indeed the case that the model cannot predict the force-velocity relationship.

In real experiments there are several ways to determine the force-velocity relationshipof a sarcomere. Ideally, we would want to measure steady state force and steady statevelocity during steady state activation. However, it is impossible to reach steady state inboth the force and the velocity due to the interaction between force and other dynamicproperties (Binder et al., 2009). The experimental method that can be imitated withour model, is the so-called isovelocity experimental paradigm. In these experiments,the muscle/sarcomere is activated while its length is fixed, until the generated force isconstant (steady state). Then a constant shortening is applied. This induced a changein the force generated by the muscle. Usually, first a fast transient is observed. Then,the force is relatively constant for some time. Finally, the force will start to decreaseslowly, due to the SL-dependence of the force. The force that is generated in the secondphase, is the force that is associated with the imposed shortening velocity (Binder et al.,2009; personal communication with Dr. F. Regazzoni, June 22, 2019).

To determine the force-velocity relationship with our model, we thus need to fix boththe calcium concentration and the sarcomere length and let the system reach steadystate, say at time t = tss. At that moment, we need to apply a constant velocity v,which we can do by changing the constant sarcomere length to SL(t) = SL0−v ·(t−tss),where SL0 is the initial fixed value of the sarcomere length. Then we need to determinethe permissivity in the second phase, where it gets relatively constant. When we do thisfor different shortening velocities, this will give the force-velocity relationship.

We might, for example, initially fix the calcium concentration and sarcomere lengthat respectively c = 1.0 µM and SL = 2.3 µm, and apply a shortening velocity of v = 1.0µm/s after one second, since then steady state is reached as can be seen in Figure 4.3.Then we find that the permissivity does not change as seen in experimental observations(Figure 4.3). First, there is no fast transient just after the start of the shortening. Moreimportantly, the permissivity does not become approximately constant. This meansthat we can not determine the force that is associated to the shortening velocity. Wetried this for several initial conditions and shortening velocities to make sure this wasnot due to this specific initial condition. However, the same pattern is seen for allinitial conditions we tried. Therefore it is unfortunately not possible to determine aforce-velocity relationship at all using this model.

36

Figure 4.3.: Plots of the sarcomere length (top) and permissivity (bottom). Initially, the calcium con-centration is 1.0 µM and the sarcomere length is 2.3 µm. After four seconds (dotted verticalline) the sarcomere shortens with a shortening velocity of 1.0 µm/s, this is equivalent tothe sarcomere length changing according to SL(t) = 2.3− 1 · (t− 4) after t = 4 s.

4.2. Dynamical analysis

As discussed earlier (see section 2.1) the values of the calcium concentration and thesarcomere length are functions of time that are inputs of the model. To investigatewhether this choice can be justified, we test whether the model is capable of reproducingthe dynamics of the sarcomere. We will do this in two situations. In the first, we will testwhether our reduced model can reproduce a so-called twitch contraction (see section 4.2.1for further explanation). In the second, we will investigate the rate of force developmentafter a sudden detachment of all crossbridges.

4.2.1. Twitch contraction

In this section, we will simulate a so-called twitch contraction. This is a single contractionof the sarcomere resulting from a short, sudden increase and subsequent decrease incalcium concentration. This is thus different from simulating steady states as in theprevious sections, where we kept the calcium concentration at a constant level. A twitchcan be simulated by imposing the calcium transient that we discussed in section 2.1,and a constant sarcomere length. We do this for several values of the sarcomere length.This results in the upper panel of Figure 4.4a. Here we see that for higher sarcomerelengths a higher permissivity is reached. When we normalise each curve to its maximumpermissivity value, we get the curves in the lower panel of Figure 4.4a. Here we see that

37

for higher values of the sarcomere length, the relaxation time of the sarcomere increases.Furthermore, we see that the rate of force development (how fast the amount of forceincreases) is approximately equal for all different values of the sarcomere length.

When looking at the experimental data of Janssen and Hunter (1995) we see the samepatterns. For higher sarcomere lengths a higher force is produced by the sarcomere(upper panel of Figure 4.4b). Furthermore, in the lower panel of Figure 4.4b, we alsosee that the relaxation time increases with increasing sarcomere length. Finally, also inthe experimental data the rate of force development is approximately the same for eachsarcomere length. Therefore, we can conclude that our model can adequately predicta twitch contraction. Since we impose the calcium transient in simulation the twitchcontraction, this conclusion supports the use of this transient.

(a) Numerical force transients in twitch contrac-tions in which the sarcomere length is kept con-stant. The value of cmax is set to be 2.0 µM inorder to imitate the experimental conditions aswell as possible.

(b) Experimental force transients in twitch con-tractions in which the sarcomere length is keptconstant. The extracellular calcium concentra-tion (comparable to cmax) was 2.0 µM. Adoptedfrom Janssen and Hunter (1995).

Figure 4.4.: Numerical and experimental force transients in twitch contractions in which the sarcomerelength is kept constant. In the bottom panels, the force is normalised to the maximumamount of force/permissivity for each sarcomere length separately. In both the experimentaland the numerical case, the increase in calcium concentration started at 0.05 s.

38

4.2.2. Rate of force development

In this section, we will look at what happens to the permissivity when in steady statesuddenly all crossbridges are detached. In real muscles, this happens for example whenthe muscle undergoes a sudden length change.

(a) Numerical force transients after a sudden de-tachment of all crossbridges, by forcing thetransition from permissive to non-permissive(P → N ) for each myosin head. The differentcurves refer to different values of the calciumconcentration. Adapted from Regazzoni et al.(2018).

(b) Experimental tension transients after a suddendetachment of all crossbridges. The differentcurves refer to different values of the calciumconcentration, expressed in pCa (= − log(c), asin section 1.2). Tension (y-axis) is comparableto force. Adopted from Wolff et al. (1995).

Figure 4.5.: Numerical and experimental force transients after a sudden detachment of all crossbridges.In the bottom panels the force is normalised to the maximum amount of force/permissivityfor each calcium concentration separately. Tension (y-axis in (b)) is comparable to force.

We can simulate this by keeping the calcium concentration fixed until steady state isreached. Steady state permissivity is reached the earliest when the sarcomere length hasreached steady state. Since the SL is prescribed, it will not change anymore after it hasreached steady state. Therefore, we can just fix the sarcomere length at the steady statevalue of the SL transient. We, again, assume that steady state is always reached after

39

four seconds. After these four seconds, for all myosin heads, we force a transition fromthe permissive state P to the non-permissive state N . This imitates the detachment ofthe MHs. When we do this for several values of the fixed calcium concentration c, thisresults in the upper panel of Figure 4.5a. In the lower panel, we see the results afternormalizing the each curve to the maximum permissivity that is reached (the steadystate value). The rate of force redevelopment can best be investigated in the normalisedfigure. Here we see that for higher values of the calcium concentration, the permissivityis earlier back to its original level. This means that the rate of force development ishigher when the calcium concentration is higher.

Now we can compare our results with experimental results. Wolff et al. (1995) did suchan experiment, of which the outcomes can be seen in Figure 4.5b. In this experiment,the tension produced by the sarcomere was measured; this is comparable to the forcethat is generated by the sarcomere. Furthermore, the calcium concentration is expressedas pCa. As in Section, this is pCa = − log(c). In the lower panel of the figure, we againsee the normalised values of the tension. For each calcium concentration the tensionis normalised to the maximum value it reaches. Although indications of the calciumconcentration are missing in this graph, we can see (looking at the shape of the curves)that the lowest curve is for pCa = 5.7, the middle for pCa = 5.6 and the upper one forpCa = 4.5. This means that the rate of force development is highest for the smallestvalue of pCa. Hence, the rate of force development is highest for the highest value ofthe calcium concentration, which is also what our model predicts.

40

5. Discussion

In this thesis, a detailed description was given of the reduced ODE model that wasdeveloped by Regazzoni et al. (2018). First, we described the original model of Riceet al. (2003) with the modification of Washio et al. (2012). We saw that this modelhad 4N degrees of freedom, which is approximately 5 · 1021 for N = 36. Therefore, thismodel was computational too complex to be able to determine solutions. To reduce thenumber of degrees of freedom, we looked at events in terms of triplets of myosin headsinstead of at events in terms the full sarcomere. In the rewriting of the probabilities,we made use of a conditional independence hypothesis, stating that knowledge of thestate of an MH does not provide extra information about an MH three positions awaywhen the states of the two MHs in between are also known. This reduced the numberof degrees of freedom significantly to (N − 2) · 43, which is 2176 for N = 36.

Numerical tests showed that the force-length and force-calcium relationships were bothpredicted considerably well by the model. This supports the hypotheses on which themodel is based, such as the conditional independence hypothesis. However, the force-velocity relationship could not be derived from the model. The dynamical analyses, onthe other hand, showed good similarities with experimental results, verifying the use ofthe prescribed c and SL transients.

Compared to other reduction methods, the method of Regazzoni et al. (2018) has someadvantages. First, using the method of Regazzoni et al. (2018) allows us to also computesolutions that are not steady state solutions. As mentioned in section 1.3, reducing themodel to a so-called Ising problem as Rice et al. (2003) did, only steady state solutionscould be computed. Furthermore, different from the method of Washio et al. (2012),no time-consuming “off-line phase” (see section 1.3) is needed to compute coefficients.Moreover, the model of Regazzoni et al. (2018) is applicable to all conditions, insteadof only for the conditions the coefficients are fitted for. We do have to note, though,that the model of Washio et al. (2012) had only 144 degrees of freedom, which is lessthan in the method of Regazzoni et al. (2018). However, the off-line phase slows downthe method, making it slower than the method of Regazzoni et al. (2018). Furthermore,the approximation induces a larger error than the approximation of our reduced model(Regazzoni et al., 2018).

Another convenience of the method that Regazzoni et al. (2018) used to reduce thenumber of degrees of freedom, is that it is not only applicable for this specific model. Infact, it can be used for each model for a spatially explicit Markov chain with cooperativeinteractions. For a model with N units that are all described by a S-state CTMC, thisyields a system of ODEs with (N − 2) · S3 degrees of freedom instead of 4N .

41

5.1. Possible improvements of the model

Despite the good numerical results, there are some elements of the model that could beimproved. For example, the calcium-based regulation is in the model associated withthe MHs, while in fact, it is associated with the regulatory units (section 1.1). Regazzoniet al. (2018)1 motivated this choice by stating that the distance between two consecutiveMHs is comparable with the distance between two consecutive RUs (respectively about43 nm and 38 nm, e.g. Bers (2001); Keener and Sneyd (2009)). Furthermore, they statethat since the model contains the description of only one filament (instead of both), itdoes not matter which unit on the one filament faces which unit on the other filament.Finally, they argue that the length of the overlap region on the sarcomere is the samewhether you view it from the one filament or from the other. Therefore, this will notaffect the model.

Figure 5.1.: Steady state force-calcium (first column) and force-length (second column) relationships inthe physiological range of SL obtained using both modifications (each modification on adifferent row). Reprinted from Regazzoni et al. (2018).

Regazzoni et al. (2018) also described two ways of modifying the model, in whichthey incorporate M = 32 RUs in the actin filament instead of N = 36 MHs on the(half) myosin filament. Here again, the other half of the sarcomere can be obtained bysymmetry of the sarcomere. The steady state force-length (only for 1.6 µm ≤ SL ≤ 2.2µm) and force-calcium relationships obtained for both modifications can be found inFigure 5.1. For clarity, also the steady state force-length and force-calcium for the

1Actually, they stated that Washio et al. (2012) motivated this choice using these arguments, but Icould not find it in the article of Washio et al. (2012).

42

method used in the article, as obtained by Regazzoni et al. (2018) are presented here(Figure 5.2). Based on these figures, Regazzoni et al. (2018) conclude that the resultsare not significantly affected by the modifications. However, when looking at the figures,we would conclude that there is a difference between the results. For example, looking atthe force-calcium relationship, we see that for a sarcomere length of 1.5 µm the originalmodel predicts a permissivity of approximately 0.5 for c = 10. The models with themodifications, on the other hand, predict permissivities of approximately 0.2 and 0.3for the same calcium concentration, which is quite a big discrepancy. Yet, we do agreewith Regazzoni et al. (2018) that the qualitative properties of the curves are essentiallythe same for the models with and without modifications. Therefore, we think it wouldbe better to use a description of the RUs instead of the MHs. It will then have to beinvestigated which of the modifications produces the best results.

(a) Steady state force-calcium relationship as obtained by Regazzoni et al. (2018). Reprinted fromRegazzoni et al. (2018).

(b) Steady state force-length relationship as obtained by Regazzoni et al. (2018). Reprinted from Regaz-zoni et al. (2018).

Figure 5.2.: Steady state force-calcium and force-length relationships as obtained by Regazzoni et al.(2018). Presented here for easy comparison with the relationships derived with the modi-fications and for comparison with the results of the full model (dashed lines).

Moreover, because of the restriction of the model to the dynamics of the actin filament,the model does not incorporate velocity-dependent effects, which are linked to cross-bridge dynamics (Keener and Sneyd, 2009). Unfortunately, we saw that this even makes

43

it impossible to even derive the force-velocity relationship from the model. Therefore,Regazzoni et al. (2018) wrote that they plan on developing their model by implement-ing a more accurate description of the crossbridge dynamics. In fact, Dr. F. Regazzonialready produced the enriched version of the model and will soon write a paper about it(personal communication, June 14, 2019). Since the crossbridge dynamics are influencedby the energy-carrying molecules ATP, ADP and Pi, it would also be interesting to im-plement the effect of these molecules into the description of the crossbridge dynamics.Implementing these would for example allow to investigate the energy consumption ofthe contraction of the heart muscle.

Another point of consideration is that in the model of Regazzoni et al. (2018) boththe c and the SL transients are experimentally derived input functions of the model.This means that the functions are not based on a physical principle, and can thus beimproved. Furthermore, the sarcomere length being an input of the model is actuallya bit weird. Namely, when a muscle contracts, the sarcomere length changes. Thischange thus depends on the activation of the sarcomere, i.e. it depends on the calciumconcentration c. A possible improvement would therefore be to let the change in SLdepend on c instead of giving it as an input function. Of course, we did do somedynamical tests. However, in both tests we fixed the sarcomere length. Hence, we didnot do tests in order to verify the use of an SL transient as input of the model. On theother hand, simulating the twitch contraction let us to conclude that we could verify thechoice of the calcium transient as input of the model.

We should also remark that multiple parameter values are chosen such that there isagreement with experimental data. For example, we saw in section 2.2.1 that the value ofaR in the geometrical function χRA(SL, i) was chosen such that it agrees with the force-length relationship. However, the model loses its predictive value this way. Intuitively, itfeels obvious that the model can predict for example the force-length relationship whenaR is chosen this way. Therefore, it would be better to base the values of parameters onphysical principles, for instance, in stead of basing them on fitting.

Finally, the choice of triplets in the reduction is not scientifically supported. Therefore,it would be nice to find scientific evidence that justifies this choice. Furthermore, itwould be interesting to look at, for example, at quintuplets of MHs (groups of fiveconsecutive MHs). This would increase the number of degrees of freedom: for N = 36from (N − 2) · 43 = 2176 to (N − 4) · 45 = 32.768 (N − 4 since there are no quintupletscentered in the first two and last two MHs, 45 since the quintuplet consists of 5 MHs).However, it might improve the quality of the results of the model.

5.2. What else Regazzoni et al. (2018) have done

A very important aspect of the reduced model that we did not describe in this thesis, iswhether or not the reduced model indeed reduced computation time. Furthermore, wedid not compare the outcomes of the original model and the reduced model. However,Regazzoni et al. (2018) did investigate this. They found that using the same computerplatform the simulation of one second of physical time took about 15.9 s for the reduced

44

model, while it took more than 72 hours for the original model using the so-called MonteCarlo method (Regazzoni et al., 2018). This is a huge reduction in computational time.

Furthermore, they compared the results of both models by looking at the predictions ofboth models for, for example, the force-length and force-calcium relationships. In figure5.2 results for both the reduced and the full model are plotted. In both relationships wesee that at some points there is a small discrepancy between the results, but in generalthe graphs look almost the same. Therefore, we conclude that the method of reducingthe number of degrees of freedom is quite good.

45

Acknowledgements

First of all, I would like to thank Dr. R. Planque for supervising me during my thesis.Thank you for helping me finding a suitable article that contained enough mathematics,for having discussions about for example the number of degrees of freedom in the model,and for providing me with feedback. I would also like to thank Prof. Dr. R. W. J.Meester for being my second examiner. Furthermore, a big thank you is needed for Dr.F. Regazzoni. You helped me a lot by giving detailed answers to my questions. Finally,I want to thank my family and friends for supporting me, advising me and providing mewith feedback.

46

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Bhagavan, N. (2002). Medical Biochemistry (Fourth Edition). Elsevier Inc., Philadelphia,Pennsylvania.

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Crewther, B. T., Cronin, J., and Keogh, J. W. L. (2005). Possible Stimuli for Strengthand Power Adaptation. Sports Medicine, 35(11):967–989.

Dobesh, D. P., Konhilas, J. P., and de Tombe, P. P. (2002). Cooperative activation incardiac muscle: impact of sarcomere length. American Journal of Physiology-Heartand Circulatory Physiology, 282(3):H1055–H1062.

Gordon, A., Huxley, A., and Julian, F. (1966). The variation in isometric tension withsarcomere length in vertebrate muscle fibres. The Journal of Physiology, 184(1):170–192.

Gordon, A. and Pollack, G. (1980). Effects of calcium on the sarcomere length-tensionrelation in rat cardiac muscle. Implications for the Frank-Starling mechanism. Circu-lation Research, 47(4):610–619.

Hessel, A. L., Lindstedt, S. L., and Nishikawa, K. C. (2017). Physiological Mechanismsof Eccentric Contraction and Its Applications: A Role for the Giant Titin Protein.Frontiers in Physiology, 8(70).

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Huxley, H. E. and Hanson, J. (1954). Changes in the cross-striations of mus-cle during contraction and stretch and their structural interpretation. Nature,173(4412):973–976.

Janssen, P. M. L. and Hunter, W. C. (1995). Force, not sarcomere length, correlateswith prolongation of isosarcometric contraction. American Journal of Physiology,269:H676–H685.

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Keener, J. and Sneyd, J. (2009). Mathematical Physiology, I: Cellular Physiology.Springer-Verlag New York Inc., New York, NY, United States.

Kentish, J. C., ter Keurs, H. E., Ricciardi, L., Bucx, J. J. J., and Noble, M. I. M. (1986).Comparison between the sarcomere length-force relations of intact and skinned tra-beculae from rat right ventricle. Influence of calcium concentrations on these relations.Circulation Research, 58(6):755–768.

Muscle Physiology Laboratory (Last updated 22 December 2008). Fundamental func-tional properties of skeletal muscle. Retrieved from http://muscle.ucsd.edu/

musintro/props.shtml.

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Regazzoni, F., Dede, L., and Quarteroni, A. (2018). Active contraction of cardiac cells:a reduced model for sarcomere dynamics with cooperative interactions. Biomechanicsand Modeling in Mechanobiology, 17(6):1663–1686.

Rice, J. J., Stolovitzky, G., Yuhai, T., and de Tombe, P. P. (2003). Ising Model ofCardiac Thin Filament Activation with Nearest-Neighbor Cooperative Interactions.Biophysical Journal, 84(2):897–909.

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Washio, T., Okada, J.-I., Sugiura, S., and Hisada, T. (2012). Approximation for Coop-erative Interactions of a Spatially-Detailed Cardiac Sarcomere Model. Cellular andMolecular Bioengineering, 5(1):133–126.

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48

A. Insight in the matrix AIn this appendix, we will show step by step how the matrix A is created for N = 1 andN = 2. This way we will get more insight in what the matrix looks like exactly. Wewill first look at the case where N = 1. This case is not so interesting, since there areno cooperative interactions with only one MH, but it is small enough to work out thedetails. We start with working out the probability that the MH is in state α at timet+ ∆t, starting from the last expression in equation (2.5). The other possible states forthe MH are β, γ, and δ. For readability we will just write Aαβ for the transition ratefrom β to α. This gives the following expression:

P((α)t+∆t

)= P

((α)t

)−∆t · P

((α)t

)·∑

ξ∈S\{α}

Aξα + ∆t ·∑

ξ∈S\{α}

[P((ξ)t)· Aαξ

]+O(∆t2)

= P((α)t

)−∆t · P

((α)t

)· [Aβα + Aγα + Aδα]

+ ∆t ·[P((β)t

)· Aαβ + P

((γ)t

)· Aαγ + P

((δ)t)· Aαδ

]+O(∆t2)

= P((α)t

)+ ∆t ·

[(−Aβα − Aγα − Aδα)P

((α)t

)+ AαβP

((β)t

)+ AαγP

((γ)t

)+AαδP

((δ)t)]

+O(∆t2).

We can do the same for the probabilities that the MH is in state β, γ, and δ at timet+ ∆t, this gives

P((β)t+∆t

)= P

((β)t

)+ ∆t ·

[(−Aαβ − Aγβ − Aδβ)P

((β)t

)+ AβαP

((α)t

)+ AβγP

((γ)t

)+AβδP

((δ)t)]

+O(∆t2),

P((γ)t+∆t

)= P

((γ)t

)+ ∆t ·

[(−Aαγ − Aβγ − Aδγ)P

((γ)t

)+ AγαP

((α)t

)+ AγβP

((β)t

)+AγδP

((δ)t)]

+O(∆t2),

P((δ)t+∆t

)= P

((δ)t)

+ ∆t ·[(−Aαδ − Aβδ − Aγδ)P

((δ)t)

+ AδαP((α)t

)+ AδβP

((β)t

)+AδγP

((γ)t

)]+O(∆t2).

Now we let

p(t) =

P ((α)t)P ((β)t)P ((γ)t)P ((δ)t)

.

49

Then we see that we can indeed write

p(t+ ∆t) = p(t) + ∆tAp(t) +O(∆t2),

where

A =−Aβα − Aγα − Aδα Aαβ Aαγ Aαδ

Aβα −Aαβ − Aγβ − Aδβ Aβγ AβδAγα Aγβ −Aαγ − Aβγ − Aδγ AγδAδα Aδβ Aδγ −Aαδ − Aβδ − Aγδ

.

To get more insight in the matrix A, we also want to look at this matrix for biggerN . However, since the matrix has size 4N × 4N , the matrix becomes very big very fast.Therefore, we will just have a look at the case where N = 2. First, we work out theprobability that the two MHs are in state α1 and α2 at time t+ ∆t. The other possiblestates for the MHs are β1, β2, γ1, γ2, δ1, and δ2. Again we use the shorter notation forthe transition rate for readability. Starting from the last expression in equation (2.5),this gives

P((α1, α2)t+∆t

)= P

((α1, α2)t

)−∆t · P

((α1, α2)t

)· [Aβ1α1 + Aγ1α1 + Aδ1α1 + Aβ2α2 + Aγ2α2 + Aδ2α2 ]

+ ∆t ·[P((β1, α2)t

)· Aα1β1 + P

((γ1, α2)t

)· Aα1γ1 + P

((δ1, α2)t

)· Aα1δ

+P((α1, β2)t

)· Aα2β2 + P

((α1, γ2)t

)· Aα2γ2 + P

((α1, δ2)t

)· Aα2δ2

]+O(∆t2)

= P((α1, α2)t

)+ ∆t ·

[(−Aβ1α1 − Aγ1α1 − Aδ1α1 − Aβ2α2 − Aγ2α2 − Aδ2α2)P

((α1, α2)t

)+Aα1β1P

((β1, α2)t

)+ Aα1γ1P

((γ1, α2)t

)+ Aα1δ1P

((δ1, α2)t

)+Aα2β2P

((α1, β2)t

)+ Aα2γ2P

((α1, γ2)t

)+ Aα2δ2P

((α1, δ2)t

)]+O(∆t2)

(A.1)

Since our A matrix will be 4N × 4N = 42 × 42 = 16× 16, we won’t look at the completematrix, but only at the first row. Looking at equation (A.1), we see that this first rowwill be:

50

−Aβ1α1 − Aγ1α1 − Aδ1α1 − Aβ2α2 − Aγ2α2 − Aδ2α2

Aα2β2

Aα2γ2

Aα2δ2

Aα1β1

000

Aα1γ1

000

Aα1δ1

000

T

Here we already see the zeros appearing for transitions such as γ1 → β1, since the firstMH is not in state γ1 at time t.Furthermore, there will be even more zeros, since the transitions 0N ↔ 1P and 1N ↔0P are not possible. So when for example α1 = 0N and δ1 = 1P , then Aα1δ1 = 0.

51

B. Matlab code

In this appendix the code I used for obtaining my numerical results is presented. Thiscode is an adopted version of the code developed by Regazzoni et al. (2018). Theoriginal code can be found as supplementary material in the online version of theirarticle (https://doi.org/10.1007/s10237-018-1049-0).

In the first section, the code that was used for determining the force-length and force-calcium relationships, is given. In the subsequent sections, the changes that are madeto the code for each of the other simulations, will be presented.

B.1. Force-length and force-calcium relationship

The code that was used for deriving the force-length and force-calcium relationships ispresented here.

f unc t i on [ SL plot , t p l o t , PfracODE plot , Ca plot , tt , SLtt , Catt ] =S imu la t i e s1 ( SLconst , Caconst )

TimeInit = t i c ( ) ;

%% Options o f execut ion

nchains = 1000 ; % Number o f Monte Carlo samples ( a l s o used tocompute the i n i t i a l c ond i t i on f o r the reduced ODE)

i n i t S t a t e s = [ 1 ] ; % Vector conta in ing the s t a t e s used to i n i t i a l i z ethe h a l f sarcomere (1=0N 2=1N 3=1P 4=0P)

dt = 2 .5 e−5; % Reference time step [ s ]T = 4 ; % Time hor i zon [ s ]

ODEPeriod = 10 ; % The time step employed in the reduced ODE modeli s g iven by ODEPeriod∗dt

ProbUpdatePeriod = 10 ; % Trans i t i on r a t e s are computed with a per iod o fProbUpdatePeriod∗dt

PlotPer iod = 1000 ; % Output p l o t s are updated with a per iod o fPlotPer iod ∗dt

%% Macroscopic parameters

l 0 = 2 . 2 ; % [ micro m]

%% Ca evo lu t i on

52

c0 = . 1 ; % [ micro M]cmax = 1 . 1 ; % [ micro M]tau1 = . 0 2 ; % [ s ]tau2 = . 1 1 ; % [ s ]t0 = 0 . 1 ; % [ s ]

beta = ( tau1 / tau2 ) ˆ(−1/( tau1 / tau2 − 1) ) − ( tau1 / tau2 ) ˆ(−1/(1 − tau2 / tau1 ) );

i f Caconst >= 0Ca = @( t ) 0∗ t + Caconst ; % constant c value

e l s eCa = @( t ) c0 + ( t>=t0 ) .∗ ( ( cmax − c0 ) / beta ∗ ( exp(−( t−t0 ) / tau1 ) −

exp(−( t−t0 ) / tau2 ) ) ) ; % c value changing accord ing to calc iumt r a n s i e n t

end

%% SL evo lu t i on ( v a l i d only without fo r ce−l enght coup l ing )

SL0 = l 0 ;SL1 = l 0 ∗ . 9 3 ;SLt0 = . 1 5 ;SLt1 = . 5 5 ;SLtau0 = . 0 5 ;SLtau1 = . 0 2 ;

i f SLconst >= 0SL = @( t ) 0∗ t + SLconst ;

e l s eSL = @( t ) SL0+ (SL1−SL0) ∗ (max(0 ,1− exp ( ( SLt0−t ) /SLtau0 ) ) − max(0 ,1−

exp ( ( SLt1−t ) /SLtau1 ) ) ) ;end

%% Sarcomere parametersLA = 1 . 2 ; % [ micro m]LM = 1 . 6 5 ; % [ micro m]LB = . 1 ; % [ micro m]nu = 36 ; % [− ]Kon = 80 ; % [ micro M ˆ −1 ∗ s ˆ −1]Koff = 80 ; % [ s ˆ −1]Q0 = 3 ; % [− ]SLQ = 2 . 2 ; % [ micro m]alphaQ = 1 . 4 ; % [ micro m ˆ −1]Kbasic = 10 ; % [ s ˆ −1]mu = 10 ; % [− ]gamma = 40 ; % [− ]aR = . 1 ; % [ micro m]aL = . 1 ; % [ micro m]

x i = @( i ) (LM−LB) ∗ . 5 ∗ i /nu ;

53

xAZ = @(SL) (SL−LB) /2 ;xLA = @(SL) LA − xAZ(SL) − LB;xRA = @(SL) xAZ(SL) − LA;

ChiRA=@(SL , i ) ( x i ( i ) <= xRA(SL) ) .∗ exp(−(xRA(SL)−x i ( i ) ) . ˆ2 /aRˆ2) + . . .( x i ( i ) > xRA(SL) ) .∗ ( x i ( i ) < xAZ(SL) ) + . . .( x i ( i ) >= xAZ(SL) ) .∗ exp(−( x i ( i )−xAZ(SL) ) . ˆ2 /aRˆ2) ;

ChiLA=@(SL , i ) ( x i ( i ) <= xLA(SL) ) .∗ exp(−(xLA(SL)−x i ( i ) ) . ˆ2 /aLˆ2) + . . .( x i ( i ) > xLA(SL) ) ;

Q = @(SL) Q0 − alphaQ ∗(SLQ−SL) ∗(SL<SLQ) ;

Kpn0=Kbasic∗gamma∗gamma;Kpn1=Kbasic∗gamma∗gamma;Knp0 = @(SL) Q(SL) ∗Kbasic /mu;Knp1 = @(SL) Q(SL) ∗Kbasic ;K1on=Kon ;K1off=Koff /mu;

%% I n i t i a l i z a t i o n

xMC = randi ( i n i t S t a t e s , nu , nchains , ’ int8 ’ ) ;xODE = ze ro s (nu−2 ,4 ,4 ,4) ; %xODE( i , a , b , c ) = P( ( X { i −1} , X i , X { i +1}) = ( a , b ,

c ) )

f o r i =1: nchainsf o r j =2:nu−1

xODE( j −1,xMC( j −1, i ) ,xMC( j , i ) ,xMC( j +1, i ) )=xODE( j −1,xMC( j −1, i ) ,xMC( j, i ) ,xMC( j +1, i ) )+1/nchains ;

endend

% checkf o r j =1:nu−2

i f ( abs (sum(sum(sum(xODE( j , : , : , : ) ) ) ) − 1) > 1e−10)e r r o r ( ’ Error ! ’ ) ;

endend

PC = ze ro s (nu , 4 , 4 , 4 , 4 ) ; %PC( i , a , b , c ,D) = P( ( a ,D, c ) i ˆ t+dt | ( a , b , c ) i ˆ t )% PL = ze ro s (nu , 4 , 4 , 4 , 4 ) ; %PL( i , a , b , c ,D) = P( ( a , b , c ) i ˆ t+dt | (D, b , c ) i ˆ t

)% PR = ze ro s (nu , 4 , 4 , 4 , 4 ) ; %PR( i , a , b , c ,D) = P( ( a , b , c ) i ˆ t+dt | ( a , b ,D) i ˆ t

)PhiC = ze ro s (nu−2 ,4 ,4 ,4 ,4) ; % ( i , a , b , c ,D)PhiL = ze ro s (nu−2 ,4 ,4 ,4 ,4) ; % ( i , a , b , c ,D)PhiR = ze ro s (nu−2 ,4 ,4 ,4 ,4) ; % ( i , a , b , c ,D)

jMat = repmat ( ( 1 : nu ) ’ , 1 , 4 , 1 , 4 ) ;expMat=repmat ( permute ( [ 0 0 1 1 ; 0 0 1 1 ; 1 1 2 2 ; 1 1 2 2 ] , [ 3 , 1 , 4 , 2 ] ) ,nu

, 1 , 1 , 1 ) ;

54

maxiterMicro = c e i l (T/dt ) ;maxiterPlot = c e i l ( maxiterMicro / PlotPer iod ) ;

t t=l i n s p a c e (0 ,T,1000 ) ;

t p l o t = ze ro s (1 , maxiterPlot ) ;t p l o t (1 ) = 0 ;PfracODE plot = ze ro s (1 , maxiterPlot ) ;PfracODE plot (1 ) = 0 ;PfracVarianceODE plot = ze ro s (1 , maxiterPlot ) ;PfracVarianceODE plot (1 ) = 0 ;

i t e r P l o t = 1 ;

TimeTotProbUpdate=0;TimeTotAdvanceODE=0;

% PC i n i z i a l i z a t i o nones nu414 = ones (nu , 4 , 1 , 4 ) ;gammaPlusExp = gamma. ˆ expMat ;gammaMinusExp = gamma.ˆ−expMat ;PC( : , : , 2 , : , 1 ) = Koff ∗ ones nu414 ;PC( : , : , 4 , : , 1 ) = Kpn0∗gammaMinusExp ;PC( : , : , 3 , : , 2 ) = Kpn1∗gammaMinusExp ;PC( : , : , 3 , : , 4 ) = K1off ∗ ones nu414 ;

%% Execution

f o r i t e rMi c ro =1: maxiterMicro%% I t e r a t i o n i n i t i a l i z a t i o nt = i t e rMic ro ∗dt ;

i t e r p l o t = mod( i te rMicro , PlotPer iod ) == 0 ;iter ODE = mod( i te rMicro , ODEPeriod ) == 0 ;

Ca t = Ca( t ) ;SL t = SL( t ) ;

%% P r o b a b i l i t i e s updatei f (mod( i t e rMicro , ProbUpdatePeriod ) == 0 | | i t e rMi c ro == 1)

timerProbUpdate = t i c ( ) ;

ChiRAmat = ChiRA( SL t , jMat ) ;ChiLAChiRAgammaPlus = ChiRAmat .∗ChiLA( SL t , jMat ) .∗ gammaPlusExp ;

PC( : , : , 1 , : , 2 ) = Kon .∗ChiRAmat∗Ca t ;PC( : , : , 2 , : , 3 ) = Knp1( SL t ) .∗ChiLAChiRAgammaPlus ;PC( : , : , 4 , : , 3 ) = K1on .∗ChiRAmat∗Ca t ;PC( : , : , 1 , : , 4 ) = Knp0( SL t ) .∗ChiLAChiRAgammaPlus ;

TimeTotProbUpdate = TimeTotProbUpdate + toc ( timerProbUpdate ) ;

55

end

%% ODE advancei f ( iter ODE )

timerAdvanceODE = t i c ( ) ;

dtAdvanced = 0 ;s m a l l e s t d t = dt ;whi l e ( dtAdvanced<dt )

xODE2 = sum(xODE, 4 ) ;xODErep = repmat (xODE, [ 1 1 1 1 4 ] ) ;

PhiC = PC( 2 : ( nu−1) , : , : , : , : ) .∗ xODErep ;

PhiL ( 2 : ( nu−2) , : , : , : , : ) = repmat ( . . .permute (sum( PhiC ( 1 : ( nu−3)

, : , : , : , : ) , 2 ) , [ 1 , 3 , 4 , 6 , 5 , 2 ] ). . .

. / repmat (xODE2( 2 : ( nu−2) , : , : ), 1 , 1 , 1 , 1 , 4 ) . . .

, 1 , 1 , 1 , 4 ) ;

PhiL ( i snan ( PhiL ) ) =0;PhiL ( i s i n f ( PhiL ) ) =0;PhiL ( 1 , : , : , : , : ) = repmat ( permute (PC( 1 , 1 , : , : , : ) , [ 1 , 3 , 4 , 6 , 5 , 2 ] )

, 1 , 1 , 1 , 4 ) ;

PhiL = PhiL .∗ xODErep ;

PhiR ( 1 : ( nu−3) , : , : , : , : ) = repmat ( . . .permute (sum( PhiC ( 2 : ( nu−2)

, : , : , : , : ) , 4 ) , [ 1 , 6 , 2 , 3 , 5 , 4 ] ). . .

. / repmat ( permute (xODE2( 2 : ( nu−2) , : , : ) , [ 1 , 4 , 2 , 3 ] ), 1 , 1 , 1 , 1 , 4 ) . . .

, 1 , 4 ) ;

PhiR( i snan (PhiR) ) =0;PhiR( i s i n f (PhiR) ) =0;PhiR(nu− 2 , : , : , : , : ) = repmat ( permute (PC(nu , : , : , 1 , : )

, [ 1 , 6 , 2 , 3 , 5 , 4 ] ) , 1 , 4 ) ;PhiR = PhiR .∗ xODErep ;

NotAdvanced = 1 ;dtCurr = dt−dtAdvanced ;whi l e ( NotAdvanced )

xODEnew = xODE + dtCurr ∗ ODEPeriod ∗ sum( permute (PhiC, [ 1 , 2 , 5 , 4 , 3 ] ) − PhiC . . .

56

+ permute ( PhiL, [ 1 , 5 , 3 , 4 , 2 ] ) −PhiL . . .

+ permute (PhiR, [ 1 , 2 , 3 , 5 , 4 ] ) −PhiR , 5 ) ;

i f (min (xODEnew ( : ) )<0 | | max(xODEnew ( : ) )>1)dtCurr = dtCurr / 2 ;

e l s eNotAdvanced = 0 ;dtAdvanced = dtAdvanced+ dtCurr ;xODE = xODEnew;i f ( dtCurr < s m a l l e s t d t )

s m a l l e s t d t = dtCurr ;end

endend

end

TimeTotAdvanceODE = TimeTotAdvanceODE + toc ( timerAdvanceODE ) ;end

%% Ploti f ( i t e r p l o t )

%% margina ls computationmarginalsODE = [ sum(sum(xODE( 1 , : , : , : ) , 3 ) , 4 ) ; . . . %

f i r s t MHpermute (sum(sum(xODE, 2 ) ,4 ) , [ 1 3 2 ] ) ; . . . %

c e n t r a l MHspermute (sum(sum(xODE( end , : , : , : ) , 2 ) , 3 ) , [ 1 4 2 3 ] ) ] ; %

l a s t MH

PermissiveODE = marginalsODE ( : , 3 ) + marginalsODE ( : , 4 ) ;PfracODE = mean( PermissiveODE ) ;

PfracVarianceODE = sum( PermissiveODE .∗(1−PermissiveODE ) ) / nu ˆ2 ;

% checkf o r j =1:nu−2

i f ( abs (sum(sum(sum(xODE( j , : , : , : ) ) ) ) − 1) > 1e−8)d i sp ( s t r c a t ( ’ I t e r : ’ , num2str ( i t e rMic ro ) , ’ , The s t a t e xODE

v i o l a t e s the conse rva t i on o f p r o b a b i l i t y b y ’ , num2str (abs (sum(sum(sum(xODE( j , : , : , : ) ) ) ) − 1) ) ) ) ;

endend

i t e r P l o t = i t e r P l o t + 1 ;t p l o t ( i t e r P l o t ) = t ;

PfracODE plot ( i t e r P l o t ) = PfracODE ;PfracVarianceODE plot ( i t e r P l o t ) = PfracVarianceODE ;

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endend

toc ( TimeInit )

SL plot = SL( t p l o t ) ;Ca plot = Ca( t p l o t ) ;SLtt = SL( t t ) ;Catt = Ca( t t ) ;

B.2. Force-velocity relationship

For the modelling of the force-velocity relationship, we made three changes to the code:

• Two extra input parameters, Vshort and t ss, were added:

[ SL plot , t p l o t , PfracODE plot , Ca plot , tt , SLtt , Catt ] =Simulat ions2 ( SLconst , Caconst , Vshort , t s s )

• The value of T was changed to 6:

T = 6 ;

• The sarcomere length was changed such that it was constant until t ss and changedaccording to Vshort after that:

SL = @( t ) (0∗ t + SLconst ) . ∗ ( t < t s s ) + ( SLconst − Vshort ∗( t−t s s) ) .∗ ( t >= t s s ) ;

B.3. Twitch contraction

For the modelling of the twitch contraction, only two small changes where made to thecode:

• The value of T was changed to 1:

T = 1 ;

• The value of cmax was changed to 2.0:

cmax = 2 . 0 ;

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B.4. Rate of force redevelopment

For the modelling of the force redevelopment, two changes were made to the code:

• The value of T was changed to 6:

T = 6 ;

• The following code was added (right after the ‘while(NotAdvanced)’ loop, line 235)for the purpose of forcing a transition from P to N for every MH after four seconds:

i f t == 4xODE( : , : , 1 , : ) = xODE( : , : , 1 , : ) + xODE( : , : , 4 , : ) ;xODE( : , : , 4 , : ) = 0 ;xODE( : , : , 2 , : ) = xODE( : , : , 2 , : ) + xODE( : , : , 3 , : ) ;xODE( : , : , 3 , : ) = 0 ;

xODE( : , : , : , 1 ) = xODE( : , : , : , 1 ) + xODE( : , : , : , 4 ) ;xODE( : , : , : , 4 ) = 0 ;xODE( : , : , : , 2 ) = xODE( : , : , : , 2 ) + xODE( : , : , : , 3 ) ;xODE( : , : , : , 3 ) = 0 ;

xODE( : , 1 , : , : ) = xODE( : , 1 , : , : ) + xODE( : , 4 , : , : ) ;xODE( : , 4 , : , : ) = 0 ;xODE( : , 2 , : , : ) = xODE( : , 2 , : , : ) + xODE( : , 3 , : , : ) ;xODE( : , 3 , : , : ) = 0 ;

end

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