Theoretical analysis of power‐law stress relaxation and calcium‐dependent passive mechanics in cardiac muscle

Abstract

Abstract

This study investigates the passive viscoelastic properties of cardiac muscle by introducing a theoretical model that explains the observed power‐law kinetics of murine cardiac trabeculae passive stress decay. The model accounts for two parallel processes contributing to passive mechanics: an elastic component and a viscoelastic component designed to simulate stress/strain‐mediated unfolding of serial domains in the titin molecule. Under stress, serial globular domains within the elastic region of the titin molecule reversibly unfold. This unfolding phenomenon contributes to both hysteresis (a lag in stress between loading and unloading) and preconditioning effects in simulated mechanics. Experimental evidence indicates that stress relaxation in cardiac muscle follows a power law and that the muscle's non‐linear stress–strain relationship and hysteresis behaviour are calcium‐dependent. To analyse these phenomena, we simulate the apparent viscous element as a mesoscopic‐scale ensemble of chains, each composed of serial globular domains that unfold in a stress‐dependent manner. The observed increase in passive tension with increased Ca²⁺ concentration is attributed to Ca²⁺‐mediated: (1) PEVK attachment to actin; (2) stiffening of the proximal element; and (3) stabilization of folded conformations of serial domains in the titin chain. Although the model was developed to represent the behaviour of titin, it equivalently represents any contributing process involving a linked series of domains that undergo stress‐mediated unfolding. By providing a unified basis for the observed viscoelastic and preconditioning effects, calcium dependency, and power‐law stress relaxation phenomena, this study offers a novel theoretical basis for understanding and simulating the role of titin in striated muscle mechanics.

Key points

• Passive stress relaxation of cardiac muscle follows a power‐law decay, a phenomenon that is explained using a theoretical model of dynamic unfolding of globular domains along polymer chain.

• The theoretical model simulates the behaviour of titin, a giant sarcomere protein linking myosin thick filaments to the Z disk and providing passive restoring force during muscle stretch.

• The theoretical model is able to account the observed effects of Ca²⁺ on the effective viscoelastic passive mechanics of cardiac muscle.

• This model provides a theoretical basis for understanding passive viscocelastic properties and titin's role in striated muscle mechanics.

Abstract figure legend Rapidly stretched cardiac trabeculae exhibit a viscoelastic tension response characterized by a power‐law decay. Increasing calcium concentration substantially increases the tension peak severalfold, even when active myosin contraction is eliminated. A mechanistic model attributes this calcium dependence to titin stiffening and the attachment of its PEVK segment to actin.

Introduction

Titin is a large filamentous protein that connects the Z‐disk to the myosin thick filament in sarcomeres and functions as a molecular spring, contributing to the passive mechanical properties of striated muscle (as illustrated in Fig. 1). Titin also plays a key role in organizing the sarcomeric structure, ensuring the alignment of contractile proteins, and aiding in the transmission of force during muscle contraction. Single‐molecule atomic force microscopy reveals that when the titin molecule is stretched, serial globular immunoglobulin‐like (Ig) and N2B domains in the titin chain unfold sequentially, resulting in a non‐monotonic force–extension relationship (Rief et al., 1997). The reversible domain folding behaviour is thought to contribute to governing the viscoelastic and hysteresis behaviours in passive stress–strain relationships in muscle (Nedrud et al., 2011).

Figure 1. Illustration of titin composition in sarcomere and schematic representation of the model.

A, the incorporation of titin into the sarcomere is illustrated, with the titin elastic region in the I‐band connecting the myosin thick filament to the Z‐disk (adapted from Giganti et al. (2018) under CC‐BY‐4 international license https://creativecommons.org/licenses/by/4.0/). B, diagram illustrating the model composition for the titin elastic region. Two components of the titin strand are considered: a component proximal to the PEVK segment and a component distal to the PEVK segment. This distal domain is modelled as a non‐linear elastance. The proximal domain is modelled as a series of domains (Ig and N2B domains) that can exist in folded or unfolded states. The lengths of the proximal and distal segments are denoted L _p and L _d. A non‐linear spring Θ_ss represents a parallel contribution to stress in the muscle. C, the relationship between stress in the proximal domain σ_p(n,ϵ) and strain of the proximal domain ϵ is plotted for different values of n, the number of domains in unfolded state in a chain. D, the relationship between unfolding rate U_{n → n +1} and strain of the proximal domain, ϵ, is plotted for different values of n. Both stress and unfolding rate increase with ϵ, and decrease with increasing n.

Because titin is the major sarcomeric structural element that provides a restoring force to oppose passive elongation, its mechanical properties are a fundamental determinant of myocardial diastolic function. At sarcomere lengths of roughly 2.1 µm and less, titin is estimated to contribute the majority of the total measured passive tension (Granzier and Irving, 1995; Helmes et al., 1999; Wu et al., 2000). Thus the passive pressure–volume relationship associated with diastolic filling of the chambers of the heart is governed, at the cellular and molecular levels, in part by titin mechanics (Chung et al., 2013; Granzier & Irving, 1995; Granzier & Labeit, 2002). Phosphorylations of titin residues via protein kinases A, D and G in the myocardium are associated with reductions in titin‐mediated passive stiffness, while phosphorylation of protein kinase C targets is associated with increased passive stiffness (Herwig et al., 2020; LeWinter & Granzier, 2010; Loescher et al., 2022). Thus, the passive mechanical behaviour of the myocardium is physiologically regulated via various signalling pathways that regulate the molecular mechanical properties of titin.

Calcium ion concentration is also known to play an important role in regulating the mechanical properties and function of titin. Labeit et al. (2003) found the stiffness of the PEVK segment of human titin is sensitive to Ca²⁺, which alters the apparent persistence length associated with the unstressed molecule. Increasing Ca²⁺ concentration lowers the persistence length or, equivalently, increases the passive elastic resistance to elongation. Kulke et al. (2001) showed that the PEVK domain binds to actin in a Ca²⁺‐dependent manner. Squarci et al. (2023) observed a muscle activation‐dependent mechanical ‘rectifier’ behaviour, interpreted as the PEVK segment reversibly attaching to the actin fibre in the intact muscle, as illustrated in Fig. 1.

Titin's mechanical behaviour arises from unique structural features and functional mechanisms. Its structure is characterized by repeating Ig and fibronectin (Fn) domains, and an N2B domain, akin to molecular beads on a string. This modular arrangement allows for extensibility, as certain domains can independently unfold and refold, acting as molecular springs (Kellermayer et al., 1997; Helmes et al., 1999; Granzier and Labeit, 2002). The extensibility of titin is particularly prominent in the I‐band region, which is rich in these domains. Given the fundamental role of titin in determining the passive mechanical properties of muscle, and given the fact that titin mechanical function is dynamically regulated by post‐translational modifications and intracellular Ca²⁺, it is not surprising that titin mutations are linked to multiple myopathies (Emig et al., 2021), including restrictive cardiomyopathy (Peled et al., 2014), dilated cardiomyopathy (Herman et al., 2012) and muscular dystrophies (Hettige et al., 2022). Understanding the molecular–mechanical function of titin is crucial to understanding the mechanisms underlying these mutations.

Baker et al. (2025) assessed the Ca²⁺‐dependence of the stress response to stretch of permeabilized murine right ventricular trabeculae, showing that addition of Ca²⁺ increases the apparent viscous component of the stress response. To characterize the passive myocardial properties, permeabilized mouse trabecular fibres were stretched over half‐sarcomere lengths from 0.95 to 1.175 µm at ramp speeds ranging from 0.00225 to 2.25 µm s⁻¹, and then allowed to relax for several seconds to minutes. While the rate of tension increase during the stretch phase of the experiment was, unsurprisingly, determined by the speed of the stretch, the rate of tension decay following stretch was, remarkably, observed to following a time scale‐independent power‐law time course. In the presence of Ca²⁺ the peak tension increased with Ca²⁺ concentration, even with cross‐bridge contraction fully inhibited by para‐nitroblebbistatin and mavacamten (Mava).

Here, data from Baker et al. (2025) are analysed to develop and identify a theoretical model of the viscoelastic passive mechanical response to stretch in these fibres. The model makes the simplifying assumption of representing passive axial force in the muscle as the sum of two components: a non‐linear elastic component and a viscoelastic component modelled based on the biophysical properties of titin (Fig. 1). The model integrates unfolding of a series of globular elements and a calcium‐dependent PEVK–actin binding mechanism. The model is able to effectively capture the calcium‐ and stretch‐dependent peak tension as well as the tension decay kinetics observed at different Ca²⁺ levels. The power‐law nature of tension decay is predicted to emerge as a property of the sequence of foldable elements (e.g. Ig and N2B domains), each capable of unfolding and introducing a finite amount of slack into the chain. Equivalently, each unfolding event increases the persistence length of the chain. With each unfolding event, the tension sensed by folded elements in the chain decreases, reducing the effective rate of unfolding. As a result, the time scale of decay increases with decreasing tension during the decay process.

The model developed to simulate these processes is a novel tool for investigating passive myocardial mechanics and representing titin mechanics at a mesoscopic scale, bridging molecular and tissue scales. It also serves as a means to test and refine hypotheses regarding the function and regulation of passive myocardial properties, and acts as a foundation for integrated modelling of myocardial cell and tissue behaviour.

Methods

Ethical approval

No animal experiments were conducted for the theoretical analyses and computational experiments reported here.

Data analysed in this study were obtained from the study of Baker et al. (2025). Experiments conducted by Baker et al. (2025) were approved by the Animal Care and Use Subcommittee of the San Francisco Veterans Affairs Medical Center and conformed to the Guide for the Care and Use of Laboratory Animals published by the National Institutes of Health (Revised 2011). That institution is accredited by the American Association for the Accreditation of Laboratory Animal Care (Institutional PHS Assurance Number is A3476‐01).

Experimental data on passive muscle mechanics

In brief, right ventricular demembranated trabeculae from 12‐week‐old male (n = 3) and female (n = 3) mice were stretched from 95% to 117.5% of reference muscle length. The reference length (L ₀) was set by adjusting the muscle until the sarcomere length reached 2.0 µm. Muscle dimensions, and active and passive forces were measured at L ₀ and at 21°C. The ramp length extension times range from t _r  = 0.1 s to 100 s, corresponding to velocites of 2.25 to 0.00225 L ₀/s. Following extension, the muscle was held fixed at muscle length of 1.175L ₀ for at least 60 s. Muscle stress response to this strain protocol was observed in Ca²⁺‐free relaxing solution (denoted pCa = 11), at saturating Ca²⁺ conditions (pCa = 4.51) to measure maximal stress, and in solution with 50 µM para‐nitro‐blebbistatin and 50 µM mavacamten (PNB + Mava), fully deactivating the cross‐bridge force generation. The strain protocol was repeated with PNB + Mava present at five different Ca²⁺ concentrations: pCa = 4.51, 5.5, 5.75, 6 and 6.25. For details, refer to Baker et al. (2025). Experiments comparing responses under relaxing conditions without PNB + Mava to response under relaxing + PNB + Mava conditions showed no substantial differences in peak stresses, suggesting that PNB and Mava do not affect passive mechanical properties under Ca²⁺‐free conditions.

An example time course of stress measured of a muscle in relaxing solution (pCa = 11) during this protocol is illustrated in Fig. 2, showing that the peak stress is highest for the most rapid ramp speed. Peak stress is approximately 14 kPa for the fastest ramp, and approximately 7 kPa for the slowest ramp. Regardless of peak stress, the stress relaxes to approximately 6 kPa at 60 s following the ramp end.

Figure 2. Representative set of measured muscle stress during ramp‐up and decay (dataset 1, male) from Baker et al. (2025) .

A, during the ramp increases in length, muscle length was extended from 0.95L ₀ to 1.175L ₀ for ramp times of 0.1, 1, 10 and 100 s (velocity 2.25 × 10⁻³–2.25L ₀/s). B, a detail of muscle length and stress Θ as functions of time for the fastest ramp (0.1 s). Stress increases in a concave‐up manner during the ramp increase in length, achieves a peak value at the end of the ramp increase, and then decays.

For fitting the computational model, measured data (relaxed and PNB + Mava with various Ca²⁺ concentrations) were corrected for drift of the force transducer signal by subtracting a spline interpolation of zero‐force level, which was assessed by short slack after each ramp. Subsequently, to mimic a normal single cell response, all forces were normalized to maximal force measured in activating solution at pCa = 4.51 prior to PNB treatment, averaged across multiple experiments and finally rescaled to mean force. This way we avoid error in estimating cross‐sectional area and variable fraction of non‐contracting actors, as we focus only on passive properties of contractile elements. For model identification, each ramp was subsampled to 100 time points with logarithmic distribution to ensure the transient peak is captured.

Computational model

The mathematical model of I‐band mechanics takes the form of an ensemble of titin chains, with a single member of the ensemble illustrated schematically in Fig. 1. The structure of the model is similar to the 0D simplification of Heidlauf et al. (2016, 2017). The single titin chain comprises a proximal unfolding domain (e.g. Fn and Ig‐like domains), distal domain, and a proline‐, glutamic acid‐, valine‐ and lysine‐enriched (PEVK) segment, connected in series. This distal domain is modelled as a non‐linear elastance. The proximal domain is modelled as a series of springs (Ig and N2B domains) that can exist in folded or unfolded states. The PEVK domain of the titin strand is assumed to reversibly attach to the actin filament in a Ca²⁺‐sensitive manner, while its compliance is distributed into proximal or distal domains. The actin chain is assumed to be effectively rigid. Any potential stiffness of the actin chain can apply only in the case of PEVK attachment and is embodied in the distal domain stiffness. The total length of the protein is denoted L = L _p + c, with L _p and L _d denoting length in the proximal and distal segments. Strain in the proximal domain, denoted as ϵ, is defined relative to the reference muscle length L ₀:

$$s=L_{\mathrm{p}}-0.95L_0$$

where L ₀ = 1 µm is a reference half‐sarcomere length, and 0.95L ₀ is the initial half‐sarcomere length in the ramp extension experiments used to identify the model. The model explicitly tracks the ensemble distribution of strain ϵ and number of unfolded domains n in the proximal chain. As a computational simplification, unfolding events are not considered in the distal chain, as it is in series with a significantly more compliant proximal chain, minimizing the impact of distal unfolding. An alternative model, where the distal chain unfolds while the proximal unfolding is not considered in the proximal chain, was tested but failed to fit the data. The titin strand is assumed to attach to and detach from a neighbouring actin filament at the PEVK site, located between the proximal and distal chain domains. The actin‐bound configuration is referred to as the attached state. The unbound configuration is referred to as the unattached state.

The variables in the model are listed and summarized in Table 1 and parameters of the model are listed in Table 2. The total stress of the muscle arises from contributions from unattached states, attached states, and contributions from parallel non‐titin components. The function p _u(n,ϵ) is the ensemble probability density of n, number of domains in unfolded state per chain, and ϵ, stretch, in unattached chains, such that

$$\sum _{n=0}^{N_{\mathrm{g}}}\underset{0}{\overset{\infty }{\int }}{p}_u\left(n,\epsilon \right)\mathrm{d}s$$

represents the fraction of titin chains that are in the unattached state. N _g is the total number of domains in the chain that can exist in either a folded or unfolded state. N _g does not represent a physical quantity, but rather a numerical discretization. Similarly, the function p _a(n,ϵ) is the ensemble probability density of n, number of domains in unfolded state per chain, and ϵ, stretch, in attached chains, such that

$$\sum _{n=0}^{N_{\mathrm{g}}}\underset{0}{\overset{\infty }{\int }}{p}_{\mathrm{a}}\left(n,\epsilon \right)\mathrm{d}s$$

represents the fraction of chains in the attached state. These fractions obey

$$1=\sum _{n=0}^{N_{\mathrm{g}}}\underset{0}{\overset{\infty }{\int }}{p}_u\left(n,\epsilon \right)\mathrm{d}s+\sum _{n=0}^{N_{\mathrm{g}}}\underset{0}{\overset{\infty }{\int }}{p}_{\mathrm{a}}\left(n,\epsilon \right)\mathrm{d}s$$ (1)

Table 1.

Variables in model

Variable	Description	Units
Independent variables
t	Time	s
ϵ	Strain in proximal chain	µm
n	Number of domains in unfolded state in proximal chain	Discrete, unitless
Time dependent variables
p u(n,ϵ)	Probability density for (n,ϵ) in unattached chains	µm−1
p a(n,ϵ)	Probability density for (n,ϵ) in attached chains	µm−1
σp(n,ϵ)	Stress in proximal titin chain	kPa
σd(n,ϵ)	Stress in distal titin chain	kPa
Θp(n,ϵ)	Average stress in ensemble of proximal titin chains	kPa
Θd(n,ϵ)	Average stress in ensemble of distal titin chains	kPa
Θ	Total stress in muscle fibre	kPa
L	Length of 1/2 sarcomere	µm

Table 2.

Model parameters

Parameter	Description	Value	Units
Adjustable parameters (Relaxed, Ca2+‐free, pCa ≈ 11)
k p	Proportionality constant for proximal chain length–stress relation	512.3	kPa
k d	Proportionality constant for distal chain length–stress relation	40 × 103	kPa
αU	Proportionality constant for unfolding rate	2.668 × 107	s−1
n p	Exponent for proximal chain length–stress relation	2.37	Unitless
n d	Exponent for distal chain length–stress relation	2.74	Unitless
n U	Exponent for unfolding rate	9.035	Unitless
∆U	Slack length associated with unfolding one unit	0.165	µm
k ss	Proportionality constant of the parallel spring	1.0168 × 109	kPa
n ss	Exponent for the parallel spring	12.8	Unitless
μ	Viscosity of dashpot element	0.678	kPa s µm−1
Fixed parameters
N g	Number of discretized unfolding domains in a chain	10	Unitless
L 0	Reference half‐sarcomere length	1.0	µm
Ca2+‐sensitive parameters (activated, pCa = 4.51)
αU	Proportionality constant for unfolding rate	2.668 × 107	s−1
n U	Exponent for unfolding rate	9.035	Unitless
k A	Rate of attachment of PEVK element to actin	5.381 × 10−3	s−1
k D	Rate of detachment of PEVK element from actin	0.383	s−1

The stress in an individual proximal titin chain domain is assumed proportional to stretch ϵ

$$\begin{array}{ccc}\hfill {\sigma }_{\mathrm{p}}\left(n,\epsilon \right)& =& k_{\mathrm{p}}·\max {\left(0,\left(\epsilon -n{\mathrm{\Delta }}_{\mathrm{U}}\right)/L_0\right)}^{n_{\mathrm{p}}},\hfill \\ \hfill n& =& 0,1,\text{…}{N}_{\mathrm{g}}\hfill \end{array}$$ (2)

where ∆_U is the slack length associated with the unfolding of one proximal domain unfolding element, L ₀ is a reference half‐sarcomere length taken to be 1.0 µm and n _p is a parameter. The relationship between σ_p(n,ϵ) and ϵ is illustrated in Fig. 1C .

The average stress associated with the ensemble of proximal domains is computed

$${\mathrm{\Theta }}_{\mathrm{d}}=\sum _{n=0}^{N_{\mathrm{g}}}\underset{0}{\overset{\infty }{\int }}{\sigma }_{\mathrm{p}}\left(n,\epsilon \right)\left(p_{\mathrm{a}}\left(n,\epsilon \right)+p_{\mathrm{u}}\left(n,\epsilon \right)\right)\mathrm{d}s.$$ (3)

Here titin molecules in both the unattached and attached states contribute to the total force.

The distal titin domain is assumed to behave like a non‐linear spring. The stress in this element is assumed proportional to stretch of the element L − ϵ − 0.95L ₀:

$${\sigma }_{\mathrm{d}}\left(n,\epsilon \right)=k_{\mathrm{d}}·\max {\left(0,\left(L-\epsilon -0.95L_0\right)/L_0\right)}^{n_{\mathrm{d}}}$$ (4)

Proximal and distal forces in an individual chain are assumed to quickly reach balance, that is σ_d(n,ϵ) = σ_p(n,ϵ). This force balance is achieved in the model by deforming the proximal chain with velocity.

$$v_{\mathrm{p}}\left(n,\epsilon \right)=\left({\sigma }_{\mathrm{d}}\left(n,\epsilon \right)-{\sigma }_{\mathrm{p}}\left(n,\epsilon \right)\right)/\mu ,$$ (5)

where μ is a viscosity factor, which also helps to achieve an effective force balance throughout the simulations.

Unfolding of a single domain, associated with a chain transition from state n to n + 1, occurs at a rate that is proportional to the strain on the globular chain:

$$U_{n\to n+1}\left(n,\epsilon \right)={\alpha }_{\mathrm{U}}{\left(\frac{\epsilon -n{\mathrm{\Delta }}_{\mathrm{U}}}{L_0}\right)}^{n_{\mathrm{U}}}$$ (6)

where ∆_U is the slack associated with one globule unfolding. The unfolding rate U_{n → n +1}(n,ϵ) is plotted as a function of ϵ in Fig. 1D . Refolding events – transitions from state n to n−1 – are not considered in the current model.

With the rates of stretching and unfolding defined by Eqns (5) and (6), the governing equation for p _u(n,ϵ) – the probability density of n and ϵ for unattached chain states – is

$$\begin{array}{ccc}\hfill \frac{\partial p_{\mathrm{u}}\left(0,\epsilon \right)}{\partial t}& =& -v_{\mathrm{p}}\left(0,\epsilon \right)\frac{\partial p_{\mathrm{u}}\left(0,\epsilon \right)}{\partial \epsilon }\hfill \\ & & -U_{0\to 1}\left(0,\epsilon \right)N_{\mathrm{g}}{p}_{\mathrm{u}}\left(0,\epsilon \right)-k_{\mathrm{A}}{p}_{\mathrm{u}}\left(0,\epsilon \right)\hfill \\ & & +k_{\mathrm{D}}{p}_{\mathrm{a}}\left(0,\epsilon \right)-v_{\mathrm{p}}\left(n,\epsilon \right)\frac{\partial p_{\mathrm{u}}\left(n,\epsilon \right)}{\partial \epsilon }\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{\partial p_{\mathrm{u}}\left(n,\epsilon \right)}{\partial t}=-U_{n\to n+1}\left(n,\epsilon \right)\left(N_{\mathrm{g}}-n\right)p_{\mathrm{u}}\left(n,\epsilon \right)\hfill \\ & & +U_{n-1\to n}\left(n-1,\epsilon \right)\left(N_{\mathrm{g}}-n+1\right)p_{\mathrm{u}}\left(n-1,\epsilon \right)\hfill \\ & & -k_{\mathrm{A}}{p}_{\mathrm{u}}\left(n,\epsilon \right)+k_{\mathrm{D}}{p}_{\mathrm{a}}\left(n,\epsilon \right),n=1\text{…}{N}_{\mathrm{g}}\hfill \end{array}$$ (7)

Similarly, the governing equation for p _a(n,ϵ) – the probability density of n and ϵ for attached chain states – is

$$\begin{array}{ccc}\hfill \frac{\partial p_{\mathrm{a}}\left(0,\epsilon \right)}{\partial t}& =& -U_{0\to 1}\left(0,\epsilon \right)p_{\mathrm{a}}\left(0,\epsilon \right)+k_{\mathrm{A}}{p}_{\mathrm{u}}\left(0,\epsilon \right)\hfill \\ & & -k_{\mathrm{D}}{N}_{\mathrm{g}}{p}_{\mathrm{a}}\left(0,\epsilon \right)\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{\partial p_{\mathrm{a}}\left(n,\epsilon \right)}{\partial t}=-U_{n\to n+1}\left(n,\epsilon \right)\left(N_{\mathrm{g}}-n\right)p_{\mathrm{a}}\left(n,\epsilon \right)\hfill \\ & & -U_{n\to n+1}\left(n,\epsilon \right)\left(N_{\mathrm{g}}-n\right)p_{\mathrm{a}}\left(n,\epsilon \right)\hfill \\ & & +U_{n-1\to n}\left(n-1,\epsilon \right)\left(N_{\mathrm{g}}-n+1\right)p_{\mathrm{a}}\left(n-1,\epsilon \right)\hfill \\ & & +k_{\mathrm{A}}{p}_{\mathrm{u}}\left(n,\epsilon \right)-k_{\mathrm{D}}{p}_{\mathrm{a}}\left(n,\epsilon \right),n=1\text{…}{N}_{\mathrm{g}}.\hfill \end{array}$$ (8)

Equations (7) and (8) assume that attachment and detachment of the PEVK domain of the titin chain to actin occur at rates k _A and k _D. For Ca²⁺‐free conditions (pCa = 11) k _A  = 0 and no attachment occurs.

Model parameter definitions and estimated values are listed in Table 2. The force proportionality constants k _p, the PEVK attachment rate k _A and unfolding rate are assumed to be Ca²⁺‐dependent. The value of N _g is set to 10, a value that represents the numerical discretization of the chain and is not meant to represent the true number of unfolding domains in the proximal chain.

The total passive muscle stress is computed from a sum of contributions from titin and from a parallel non‐linear spring that represents the steady state stress Θ_ss at a given length:

$$\mathrm{\Theta }\left(t\right)={\mathrm{\Theta }}_{\mathrm{d}}\left(t\right)+k_{\mathrm{ss}}{\left(\frac{L-0.95L_0}{L_0}\right)}^{n_{\mathrm{ss}}}$$ (9)

For the ramp extension experiments simulated here the initial state of the model is p _u(n,ϵ) = δ(ϵ)δ_{0, n} and p _a(n,ϵ) = 0, assuming no attachment to actin with zero strain and all globular elements in the folded state. Prior to initiating the ramp‐up, the model is run to reach an initial steady state. Computer codes for simulating the model are available at https://github.com/beards‐lab/TitinViscoelasticity.

Parameter identification

A global optimization was performed to find the best match to the measured tension responses to ramps across all the different Ca²⁺ levels. The objective of this optimization process was to simultaneously minimize the least squares differences between model outputs and data and the number of parameters that are assumed to vary with Ca²⁺ concentration. In practice we first identified the model for the high‐Ca²⁺ case and then a depth‐first recursive search was performed to explore all possible parameter subsets that could be adjusted to match the low‐Ca²⁺ data. Out of all possible subsets, we found that at least four parameters are needed for a satisfactory fit. Although there were five plausible combinations of four parameters, each subset contained proximal segment stiffness k _p, one of PEVK attachment or detachment rates (k _A or k _D) and either a term affecting unfolding rate (either α_U or n _U) or proximal chain length–stress exponent n _p. The lowest‐cost combination of four parameters was k _A, k _p, α_U and n _U, which were varied as functions of Ca²⁺ concentration. To fit the intermediate Ca²⁺ levels, we adjusted these four parameters only, while enforcing monotonicity with Ca²⁺ concentration.

Results

Properties of the stress relaxation

Average stress time course data from experiments of Baker et al. (2025) at pCa = 11 are plotted in Fig. 3 on linear, semilog‐y, and log–log scales. To plot the data on semilog‐y and log–log scales a common long‐time steady‐state stress level Θ_∞ was estimated and the data were shifted in time so that the decaying tails of the stress data are optimally overlain by minimizing ${\sum }_i{[({\mathrm{\Theta }}_i-{\mathrm{\Theta }}_{\infty })-A{(t-t_{r_i}-{\tau }_i)}^{-\beta }]}^2$ for all four ramp durations t _{r i} ∈ 0.1, 1, 10, 100 s, where Θ _i are the measured stresses, τ _i individual ramp time offsets, A is the common power law amplitude and β is the common power law exponent. For these data the estimated value of Θ_∞ is 4.84 kPa, and the total stress minus the steady‐state stress, Θ − Θ_∞, is plotted on the y‐axes in the semilog and log–log panels in Fig. 3. The time shifts τ _i used to overlay the decay tails are indicated in the figure.

Figure 3. Analysis of measured stress decay in relaxed muscle fibres.

A, stress versus time for the four ramp speeds on a linear scale. B and C, stress versus time for the four ramp speeds on semilog‐y and log–log scales. The semilog (B) and log–log (C) plots show a power‐law function fitting of the decay curves for all ramps. The ramp times t _r  = 0.1 s to 100 s and time shifts τ _i for the semilog and log–log plots are indicated in the figure inset.

Figure 3 illustrates that the stress decay process does not follow exponential kinetics (does not follow a straight line on the semilog‐y plot). Rather, the decay follows a power law with Θ ∼ t ^{−0 . 20}. This power‐law decay is independent of the ramp speed or peak height. Moreover, the observed power‐law decay, with an exponent of −0.20, suggests that stress decay may extend over extraordinarily long time scales. This result also compares remarkably well with behaviour observed for human myocardium, which shows a power‐law decay, with an exponent of −0.184 (Nordsletten et al., 2021). As illustrated in Fig. 3, the extrapolation of the power‐law decay predicts apparent viscous contribution of 1 kPa to the passive stress 1000 s after the peak stress, while a decrease to 0.1 kPa at this rate would theoretically take over 800 years. Thus, the estimated Θ_∞ of 4.84 kPa is lower than the apparent steady‐state stress observed 60 s after the peak stress reported by Baker et al. (2025). In contrast, Ca²⁺‐treated muscle shows a more complex response that cannot be fit using power‐law behaviour alone (see below).

Model analysis of passive muscle dynamics under Ca ²⁺ ‐free conditions

Figure 4 shows optimal model fits to stress data collected under Ca²⁺‐free (pCa = 11) conditions. The ramp times t _r indicated in the figure denote the length of time each ramp takes. The greatest peak stress is associated with the fastest ramp, t _r  = 0.1 s. The computational model is able to correctly match not only the peak heights for different ramp durations, but also the stress increase during ramp up and the decay time course. Figure 4 B shows that the model is also able to capture the observed power‐law decay in stress, with stress decay in the model following Θ ∼ t ^{−0 . 24} over three decades in time.

Figure 4. Model fit to stress data following ramp increases in muscle length at Ca²⁺‐free (pCa = 11) solution.

For all time courses the muscle is lengthened from 0.95L ₀ to 1.175L ₀ in a linear ramp and then held at the final length to observe the stress decay. A, stress data and optimal model fit on a semilog‐x plot. B, model‐predicted (simulated) stress decay on a log‐log plot illustrating the power‐law behaviour, Θ ∼ t ^{−0 . 24}. C–F, the peak and initial stress decay on linear scale plots for each ramp time, from t _r  = 0.1 s to t _r  = 100 s. The data are presented as an average of six individual responses, with error bars indicating the standard deviation.

The probability state space of the simulated ensemble of elastic chains p _u(n,ϵ) during the 1‐s ramp stretch and decay is illustrated in Fig. 5, which shows the state probability as heat maps in the (n,ϵ) space at different time points in the experiment. In the initial state all domains are in a folded state (n = 0), there is no strain in the system, and thus p _u(n,ϵ) = δ(ϵ)δ_{0, n} . When the chain is rapidly stretched by 0.225 µm, at time t = 0.1 s, p _u(n,ϵ) has a peak at approximately ϵ = 0.18 µm and n = 2, meaning that the proximal part of the chain is stretched by approximately 0.19 µm on average and the distal part by 0.03 µm. At this time point there is non‐zero probability for states n = 1…4, meaning that chains in the simulated ensemble have begun to unfold. During the relaxation phase of the experiment, the peak in the probability density shifts to higher values of n, meaning additional domains unfold. At the final time point in the simulation (t = 100 s), the peak of p _u(n,ϵ) is at n = 9 and ϵ ≈ 0.22 µm, corresponding to a strain in the proximal chain of 0.22 µm with nine domains in the chain unfolded. Since each unfolding introduces ∆_U slack in the chain, the final state is associated with roughly 9∆_U  = 0.15 µm of slack in the chain.

Figure 5. Probability space of the simulated chain under Ca²⁺‐free conditions (pCa = 11) demonstrates how the model represents an increase of unfolded domains on applied stress.

A, the probability density p _u(n,ϵ) as a heat map (as a percentage) at four different times in the simulation (t = 10⁻³, 1, 2 and 100 s) of the 1‐s ramp extension experiment, with contours drawn at 1%. B, the simulated stress time course and the four time points (t = 10⁻³, 1, 2 and 100 s) for which the probability density p _u(n,ϵ) is illustrated in A. There is no attachment considered at pCa = 11, and thus p _a(n,ϵ) is always zero and is not shown.

Model analysis of passive muscle dynamics under high‐Ca ²⁺ conditions

Figure 6 shows optimal model fits to stress data collected under high‐Ca²⁺ (pCa = 4.51) conditions. Similar to the stress response at low Ca²⁺, the greatest peak stress is associated with the fastest ramp, t _r  = 0.1 s. The peak stresses at each extension speed are all markedly higher at pCa = 4.51 compared to peak stresses at pCa = 11. The model captures this Ca²⁺‐induced increase in peak force through the Ca²⁺‐dependence of chain stiffness, unfolding rate and PEVK domain attachment to actin. The simulated stress decay time courses on a log–log scale in Fig. 6 B show that predicted long‐time decay behaviour at high Ca²⁺ follows the same power‐law behaviour at low Ca²⁺.

Figure 6. Model fit to stress data following ramp increases in muscle length at high Ca²⁺ (pCa = 4.51).

For all time courses the muscle is lengthened from 0.95L ₀ to 1.175L ₀ in linear ramp and then held at the final length to observe the stress decay. A, stress data and optimal model fit on a semilog‐x plot. B, model‐predicted stress decay on a log–log plot, with second‐half of the decay (30–60 s) fitted with a power law. C–F, the peak and initial stress decay on linear scale plots for each ramp time, from t _r  = 0.1 s to t _r  = 100 s. The data are presented as an average of six individuals, with error bars indicating the standard deviation.

At high Ca²⁺ the simulated titin elastic chain adopts conformation in both the p _u(n,ϵ) and p _a(n,ϵ) spaces, as illustrated in Fig. 7, which shows the state probability as heat maps in the (n,ϵ) space at different time points in the experiment. At time t = 10⁻³ s, the binding and unbinding to actin are in an equilibrium balance, with approximately 1.4% of conformations in the attached state. In the initial state all chains are folded with no strain on the titin chain, and thus p _u(n,ϵ) + p _a(n,ϵ) = δ(ϵ)δ_{0, n} . When the chain is rapidly stretched by 0.225 µm, at time t = 1 s, the peak in p _u(n,ϵ) moves to ϵ ≈ 0.16 µm, predicting that the proximal titin chain is relatively less stretched at this time point in the simulation compared to behaviour at low Ca²⁺ (Fig. 5). At this time point (t = 1 s), the majority of unattached chains have already unfolded while attached chains remain mostly in the folded conformation at zero strain. The attachment of the titin chain to the actin filament reduces strain in the proximal domain in response to the global strain and thus slows the unfolding. Later in the decay process both attached and detached states exist primarily in highly unfolded conformations, reducing the PEVK attachment contribution.

Figure 7. Probability space of simulated chain at high Ca²⁺ (pCa = 4.51) demonstrates how the model represents an increase of unattached and actin‐attached unfolded domains on applied stress.

A, the probability density of the unattached states p _u(n,ϵ) as a heat map at four different times in the simulation (t = 10⁻³, 1, 2, and 100 s) of the 1 s ramp extension experiment, with contours drawn at 1%. B, the probability density of the attached states p _a(n,ϵ). C, the simulated stress time course and the four time points (t = 10⁻³, 1, 2 and 100 s) for which the probability densities p _u(n,ϵ) and p _a(n,ϵ) are illustrated in A and B.

Figure 8 shows stress decay after a ramp‐up, as measured by Baker et al. (2025), and simulated using the proposed model over a range of Ca²⁺ concentrations, demonstrating that the model effectively captures the Ca²⁺‐sensitive mechanisms. The peak stress gradually increases from about 25 kPa at low Ca²⁺ (pCa = 6) to around 45 kPa at pCa = 5.5, indicating high sensitivity to Ca²⁺ in the micromolar range. With decreasing Ca²⁺ concentration the stress decay tends to be more similar to a power law, as discussed above. Model parameters across a range of Ca²⁺ concentrations are well fit using a Hill curve (Fig. 9). The estimated Hill coefficients for the Ca²⁺ dependence of these four parameters are remarkably similar: K _A  = 5.87 ± 0.09.

Figure 8. Comparison of measured and simulated stretch response across a range of Ca²⁺ concentration from highest Ca²⁺ with pCa = 4.51 (A) to Ca²⁺‐free (F) (pCa = 11).

For intermediate Ca²⁺ concentrations only the fastest ramp data were acquired.

Figure 9. Parameters varied with Ca²⁺ concentration required to fit ramp responses over a range of Ca²⁺ concentrations shown in Fig. 8 .

Parameters were identified allowing only monotonous transition and were subsequently fit using Hill curve.

Simulation of cyclic loading

To explore preconditioning behaviour we simulated stress response to cyclical sinusoidal loading, with the initial condition of the model in the fully folded state: p _u(n,ϵ) = δ(ϵ)δ_{0, n} . Cyclical loading was simulated by imposing a half sarcomere length of L(t) = L ₀ + (A/2)(cos(ωt − π) + 1), with amplitude A = 0.225 µm and frequency ω/(2π) = 1 s⁻¹. Predicted stress response to this length time course is illustrated in Fig. 10 for the Ca²⁺‐free condition. Simulations show that during the initial extension, stress increases to a peak of approximately 12 kPa. The peak stress decays with each subsequent cycle of stretch. The right panel of the figure illustrates this hysteresis behaviour in a plot of stress versus stretch.

Figure 10. Simulation of Ca²⁺‐free cyclical loading.

A, imposed half‐sarcomere length L(t) and resulting model‐predicted stress Θ(t) (B) for loading frequency of 1 Hz. C, the predicted length–stress relationship demonstrating loading–unloading difference and preconditioning of the relaxed muscle. The non‐smooth behaviour seen in the stress–strain plot is a numerical artifact of the state space discretization.

The predicted stress–strain behaviour, showing a drop in peak force from one cycle to the next, is similar to experimental data, for example, from biaxial extension of myocardium (Nedrud et al., 2011; Sommer et al., 2015). The ability to capture this behaviour suggests that the model provides a mechanistic basis for simulating preconditioning phenomena in striated muscle tissue. Unlike experimental results on intact tissue, the simulated hysteresis loop collapses after several loading cycles. This is because the current model does not account for refolding of unfolded domains in the chain. Future investigations into titin‐mediated preconditioning and hysteresis behaviour will require the current model to be extended to account for refolding events.

Discussion

We developed and analysed a mathematical model of the passive mechanics of myocardium represented with an elastic component and a component representing titin, in which titin chains are represented as an ensemble of chains with a probabilistic distribution of strain and degree of unfolding of serial elements in the chains. The model captures a broad range of important phenomena over multiple Ca²⁺ concentrations that contribute to the non‐linear passive mechanics of myocardial tissue.

In fitting model output to data from Baker et al. (2025), the model accurately reflects the observed stress response to a ramp increase in strain, with a concave‐up increase in stress during the extension, followed by a decay. As observed by Baker et al. (2025), the peak stress achieved in the ramp extension experiment increases with increasing extension speed, reflecting an apparent viscous component of the stretch response. This phenomenon is predicted to be associated with the dependency of rates of serial titin domain unfolding events on strain.

At relatively slower extension rates, unfolding domains have more time to unfold during extension, increasing the persistence length of the chain, and resulting in less stress compared to stress at higher extension rates. Regardless of extension speed, when the muscle is held at a fixed length following a ramp extension, the stress decays with a time course that follows t ^−β, where β is estimated to be 0.2 from the raw experimental data and 0.24 from model simulations. This power‐law decay emerges as a property of the sequence of globular elements in the titin chain. Each element is capable of unfolding and introducing a finite amount of slack into the chain. Equivalently, each unfolding event increases the persistence length of the chain. As each unfolding event introduces slack into the chain, the stress sensed by unfolded elements in the sequence decreases, reducing the effective rate of unfolding. Thus, the effective rate of decay decreases as the decay proceeds, resulting in the slow power‐law decay.

This unfolding of serial foldable elements in the titin chain is also predicted to contribute to the hysteresis and preconditioning in the muscle stress–strain relationship. The mechanism underlying hysteresis and preconditioning phenomena in the model follows from the fact that the effective stiffness in the elastic I band region is greatest in the fully folded state. Muscle stretching leads to unfolding, which reduces the effective stiffness of the muscle.

In sum, the unfolding beads‐on‐a‐chain titin model captures the following features of passive myocardial mechanics:

1. The dependency of maximum stress on the rate of muscle extension – the apparent viscous component of the stress response – in a muscle‐stretch experiment, as illustrated in Figs 4, 6 and 8.

2. The power‐law kinetics of stress relaxation, as illustrated in Figs 3 and 4.

3. Preconditioning in the passive viscoelastic muscle stress–strain behaviour, as illustrated in Fig. 10.

Furthermore, by including the Ca²⁺‐dependent PEVK domain attachment to actin and Ca²⁺‐dependent chain stiffness, the model captures the observed dependence of peak stress on Ca²⁺ concentration, as illustrated in Fig. 6. This Ca²⁺ dependence may be important in vivo as Ca²⁺ stimulates cross‐bridge activation and force generation. The model is able to match the high‐Ca²⁺‐treated muscle response by stiffening the proximal domain, stabilization of folded conformations of serial domains in the titin chain, and attachment of the PEVK domain of titin to the actin filament.

Figure 11 shows simulations of the model under high‐Ca²⁺ conditions (pCa = 4.51), but (1) with disabled PEVK attachment and (2) with disabled PEVK attachment while the parameters were reoptimized for optimal fit. These results demonstrate that although only a small fraction of PEVK domains are predicted to be attached under high‐Ca²⁺ conditions, the PEVK attachment is associated with a substantial contribution to the stress achieved at the fastest ramp speed at high Ca²⁺ (compare to Fig. 6). This interpretation is in line with observations of Squarci et al. (2023). In addition, while retuning parameters reduces the overall error, the viscoelastic chain model is unable to capture the observed dynamics with the PEVK mechanism excluded.

Figure 11. Model simulations of stress response following ramp increases in muscle length at high Ca²⁺ (pCa = 4.51) with PEVK attachment disabled (PEVK knockout) while other parameters were kept at the same values as used in Fig. 6 and with PEVK attachment disabled and reoptimized for best fit (PEVK knockout reoptimized).

Results illustrate the contribution mediated by the PEVK attachment and that the PEVK attachment mechanism is important to describe the observed response to the fastest ramp‐up stretches.

Model limitations

The model formulation invokes a number of simplifying assumptions, including the simplification that domains may unfold only in the titin segment proximal to the PEVK domain. Moreover, the model does not distinguish specific unfolding domains, such as Ig, Fn and N2B elements, which may be dominant at lower stresses (Sun et al., 2024). The validity of this simplification is apparent in the ability of the model to match the observed stress relaxation data. However, to represent different behaviour‐associated site‐specific mutations, domain deletions or post‐translational modification, a more detailed model formulation may be required. Since the attachment of the PEVK domain to actin does not come into play in low‐Ca²⁺ (pCa = 11) conditions, the assumption that unfolding events occur in only one segment of the chain does not influence the behaviour of the model at zero or very low Ca²⁺, and incorporating distal domain unfolding would make the model more complex. Chung et al. (2011) analysed the effect of PEVK domain deletion in relaxed and Ca²⁺‐activated mouse papillary muscle and found significantly lower viscous stress in the PEVK deletion, with no significant differences in mechanical response over the range of pCa from 9 to 6.8. The lack of a substantial effect of Ca²⁺ over this range of pCa is consistent with our findings (Fig. 9). However, our model matches the relaxed muscle data (pCa = 9) without any PEVK–actin interaction, seemingly inconsistent with the observations of Chung et al. (2011). Thus, our assumption that PEVK‐mediated effects contribute only at high Ca²⁺ might be a simplification that is not valid under all conditions and for all applications.

For applications demonstrated here, the model does not account for refolding of unfolding domains in the titin chain and thus the model is unable to capture persistent hysteresis during cyclical loading. To simulate this phenomenon, as well as to simulate the contribution of titin mechanics to passive myocardial mechanics in vivo, it would be necessary to incorporate reversible refolding into the model. Preliminary tests suggest that adding a constant refolding rate results in hysteresis similar to those observed experimentally. Simulations (not shown) predict if refolding occurs only at very low forces or while slackened, refolding would not affect the experimental observations used here for model identification. Thus, robust incorporation of refolding kinetics would require additional experiments, which will be a focus of follow‐up experiments.

Although model parameter identification is based on a set of relatively rich and informative data sets, the estimated parameter values likely do not represent a unique global solution. This lack of confidence in the global uniqueness of the parameter estimates does not mean that the predictions of the model are flawed, but rather that the parameters are not necessarily well identifiable and their values should be considered valid only in mutual combination and within the context of this model.

The experiments used to identify the model were conducted at 21°C. In future it would be valuable to attain relevant data at 37°C to investigate how the mechanical phenomena explored here depend on temperature. To be able to maintain intracellular Ca²⁺ concentration, all experiments were performed with permeabilized membranes. While this process reduces passive stiffness by about a third, it does not affect the processes studied (Chung & Granzier, 2011).

In the proposed model, the increased tension with increased Ca²⁺ concentration is attributed to (1) PEVK attachment to actin, (2) stiffening of proximal element, and (3) stabilization of domain unfolding rates (by decreasing rate and rate exponent). For other possible candidates (with slightly worse fit, data not shown) it was sufficient to vary (1) and (2) and proximal and distal chain exponents (n _p and n _d), but keeping the unfolding rates constant. With the given data we cannot effectively rule these possibilities out, so we simply used the lowest cost combination of Ca²⁺ dependencies.

Attribution of apparent viscous component to titin

In this work we developed a model of viscoelastic passive muscle mechanics based on model of titin as a series of elements that unfold in a stress/strain‐dependent manner in parallel with a non‐linear elastic element. Yet analysis of mechanics data obtained under myofilament‐extracted conditions from Baker et al. (2025) suggests that a component of the apparent viscoelastic response is due to non‐myofilament structures. Our analysis of the data from Baker et al. predicts that the parallel elastic component contributes roughly 5 kPa to the peak stress of approximately 15 kPa observed at the fastest ramp speed observed under zero‐Ca²⁺ conditions. Of the remaining ∼10 kPa viscous component, as much as 4 kPa may be attributed to non‐myofibrillar structures. Thus, the apparent viscous component of the stretch response at low Ca²⁺ is not expected to be entirely due to titin. Indeed, there exist additional passive mechanical mechanisms that are not included in this model, including an effective viscoelasticity of microtubules (Caporizzo et al., 2018). Nevertheless, the general form of our theoretical model – of a linked series of domains that undergo stress‐mediated unfolding – may equivalently represent contributions from non‐titin‐mediated processes to the power‐law stress relaxation of the myocardium. Moreover, as demonstrated in Baker et al., the Ca²⁺‐dependent component of the viscous response is entirely attributable to myofibrillar structures.

Competing interests

The authors declare no competing interests.

Author contributions

F.J., D.A.B, D.A.N. and A.J.B. conceptualized the study, F.J. and D.A.B. designed and implemented the model and drafted the manuscript. All authors contributed to interpreting the theoretical model and editing and revising the manuscript. All authors have read and approved the final version of this manuscript and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

Funding

This work was supported by Department of Veterans Affairs Merit Review Award I01BX000740 (AJB) and National Heart, Lung and Blood Institute Grants R01 HL154624 (A.J.B., D.A.B.) and R01 HL173346 (D.A.B.).

Supporting information

Peer Review History

Biography

Filip Ježek earned his PhD in Cybernetics and Artificial Intelligence from Czech Technical University in 2019. Currently a postdoctoral researcher at the University of Michigan's, he specializes in computational modelling of cardiovascular system, cardiac muscle mechanics and patient‐specific simulations.

Data availability statement

All data and codes to reproduce figures in this paper are available at https://github.com/beards‐lab/Titin Viscoelasticity.