Original research

Slow force response and auto-regulation of contractility in heterogeneousmyocardium

V.S. Markhasin a,b, A. Balakin a, L.B. Katsnelson a,b, P. Konovalov a, O. Lookin a, Yu Protsenko a,O. Solovyova a,b,*

a Institute of Immunology and Physiology, Ural Branch of the Russian Academy of Sciences, Pervomayskaya Str., Bld 106, 620049, Yekaterinburg, RussiabUral Federal University, 19 Mira str, 620002, Yekaterinburg, Russia

a r t i c l e i n f o

Article history:Available online 21 August 2012

Keywords:Myocardial mechanicsCardiomyocyteStretchShorteningCooperativityExcitationecontraction couplingCardiac mechano-electric couplingElectromechanical mathematical model

a b s t r a c t

Classically, the slow force response (SFR) of myocardium refers to slowly developing changes in cardiacmuscle contractility induced by external mechanical stimuli, e.g. sustained stretch. We present evidencefor an intra-myocardial SFR (SFRIM), caused by the internal mechanical interactions of muscle segmentsin heterogeneous myocardium. Here we study isometric contractions of a pair of end-to-end connectedfunctionally heterogeneous cardiac muscles (an in-series muscle duplex). Duplex elements can be eitherbiological muscles (BM), virtual muscles (VM), or a hybrid combination of BM and VM. The VM imple-ments an EkaterinburgeOxford mathematical model accounting for the ionic and myofilament mecha-nisms of excitationecontraction coupling in cardiomyocytes. SFRIM is expressed in gradual changes in theoverall duplex force and in the individual contractility of each muscle, induced by cyclic auxotonicdeformations of coupled muscles. The muscle that undergoes predominant cyclic shortening shows forceenhancement upon return to its isometric state in isolation, whereas average cyclic lengthening maydecrease the individual muscle contractility. The mechanical responses are accompanied with slow andopposite changes in the shape and duration of both the action potential and Ca2þ transient in the car-diomyocytes of interacting muscles. Using the mathematical model we found that the contractilitychanges in interacting muscles follow the alterations in the sarcoplasmic reticulum loading in car-diomyocytes which result from the length-dependent Ca2þ activation of myofilaments and intracellularmechano-electrical feedback. The SFRIM phenomena unravel an important mechanism of cardiac func-tional auto-regulation applicable to the heart in norm and pathology, especially to hearts with severeelectrical and/or mechanical dyssynchrony.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Parmley and Chuck (1973) first showed that a 5e6% increase inmuscle length is not only followed by the immediate increase ofpeak force of contraction (the FrankeStarling low), but gives rise toa further gradual increase in isometric peak force, slowly devel-oping cycle-by-cycle for many beats, until a new enhanced steady-state is reached. This phenomenon is usually referred to as the slowforce response (SFR) in cardiac muscle. Mechanisms underlying thestretch-dependent SFR are still under consideration, usinga combination of experimental and computational modelling work(Calaghan and White, 2005; Cingolani et al., 2011; Niederer andSmith, 2007; von Lewinski et al., 2008; Ward et al., 2008).

Another manifestation of SFR to the change in the mechanicalload has been identified by Kaufmann et al. (1971). They observedslowly developing beat-to-beat increase in the peak shortening ofcardiac muscle in response to a switch from isometric to isotonicmode of contractions. After resuming the isometric conditions, themuscle developed an enhanced peak force (overshoot) over theisometric steady-state level during several beats, followed bya gradual force recovery. This experiment clearly showed thatcyclic shortening (not stretch) may also induce the SFR in cardiacmuscle. The afterload-dependent SFR has also been successfullyreproduced in computational representations (Solovyova et al.,2004).

Well-known expressions of SFR to a change in myocardialpacing rate are the Bowditch and Woodworth ‘staircase’phenomena (Bowditch, 1871; Woodworth, 1902). Several mathe-matical models address the pacing rate-dependent SFR andunderlying intracellular processes (Hinch et al., 2004; Luo andRudy, 1994; Markhasin et al., 1985; Rice et al., 2000).

* Corresponding author. Institute of Immunology and Physiology, UralBranch of the Russian Academy of Sciences, Pervomayskaya Str., Bld. 106, 620049Yekaterinburg, Russia.

E-mail address: O.Solovyova@iip.uran.ru (O. Solovyova).

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Progress in Biophysics and Molecular Biology 110 (2012) 305e318

All the above SFR types emerge as a consequence of sustainedchanges in external conditions (pacing rate, pre- or afterload), thatinduce the SR Ca2þ reloading in cardiomyocytes providing with thecontractility change (Kentish and Wrzosek, 1998; Noble and Seed,2011).

It has been shown that cardiomyocytes from different regions ofthe heart chambers (e.g. from the left ventricle) differ in theirelectrophysiological and mechanical properties, which ensuresfunctional heterogeneity of myocardium (see our recent review inMarkhasin et al. (2011)). To explore physiological consequences ofmyocardial heterogeneity, theoretical and experimental modelswere developed, the muscle duplexes, where two muscle samplesconnected in-series or in-parallel mechanically interact with oneanother (Markhasin et al., 2003; Tyberg et al., 1969). Recently, invirtual muscle duplexes based on our EkaterinburgeOxford inte-grative electromechanical model (EO model) of myocardium(Solovyova et al., 2003; Sulman et al., 2008), we observed a newexpression of SFR, caused by the mechanical interactions betweenduplex elements and manifesting in the slowly developing gradualchanges in the overall duplex force and in the individual contrac-tility of each coupled virtual muscle (Markhasin et al., 2004, 2011;Solovyova et al., 2006). In contrast to the above mentioned SFRinduced externally, this SFR resulted from internal dynamicmechanical interactions of contractile elements inside themyocardium, and which we call intra-myocardial SFR (SFRIM).

The main goal of this study was to identify SFRIM phenomena inthe muscle duplexes with different extent of the functionalheterogeneity. In earlier works we presented some theoretical andexperimental data on the slowly developing changes in themechanical activity of muscle duplexes and 1D in-series myocardialstrands (Markhasin et al., 2011; Solovyova et al., 2006). Here, alongwith partially self-reviewed results we present new experimentaldata on the intra-myocardial mechanical effects on action potential(AP) generation and Ca2þ transients (CaT) in coupled car-diomyocytes, and analyzed in detail the links between patterns ofdynamic deformations of coupled muscle segments during duplexcontractions and the functional changes in cardiomyocytesaccompanying the SFRIM phenomena.

2. Materials and methods

2.1. The muscle duplex

To address the effects of mechanical interactions betweenmyocardial tissue segments, here we use in-series duplexes wheremuscle segments are mechanically connected end-to-end. Differentsequences of muscle activation with varying time-lags (from 0 to100 ms) are used to simulate temporal aspects of propagatedexcitation throughout myocardial tissue. The physiological rele-vance of the duplex model stems from the fact that mechanicalsignal transduction occurs at the velocity of sound (in liquids about300 m s�1). This is about three orders of magnitude faster thanelectrical activation (travelling at a velocity of 0.3e0.5 m s�1

(Torrent-Guasp et al., 2005)), so mechanical effects (e.g. stretch)from early-activated myocardial segments are almost immediatelytransferred even to distant areas of later-activated tissue. This hasbeen confirmed in whole heart investigations, where earlier acti-vation of subendocardial muscular segments precondition sub-epicardial segments to the incoming electrical activation duringearly isovolumetric contraction (Ashikaga et al., 2009).

To mimic this setting, we evaluate the dynamic effects of end-to-end interactions between coupled muscles while keeping theouter ends of the duplex iso-positional. Imposing externallyisometric conditions on the whole duplex does not prevent internalelement length changes. Mechanical activity of duplex elements is

governed by coinciding yet opposite length changes, as they ‘pull’ ateach other until force in each element is balanced at any given time.Duplex element contractile activity is auxotonic, therefore, as is thecase in real tissue. The ensuing dynamic behaviour of coupledmuscle segments is conceptually similar to isovolumetric contrac-tion (or relaxation) of the ventricles, where internal length changesoccur at constant external dimensions (Ashikaga et al., 2009;Sengupta et al., 2006).

We explored three sets of duplex element combinations: (1)a biological duplex comprising of two isolated multicellularmyocardial preparation (biological muscles (BMs); i.e. thin papil-lary muscles or thin trabeculae); (2) a virtual duplex comprising oftwo computational models of the electromechanical activity ofcardiac muscle (virtual muscles (VMs), see below for details), VMs;or (3) a hybrid duplex comprising of one BM and one VM (Protsenkoet al., 2005). A schematic illustration of the three duplex settingscan be found in the on-line supplement (Fig. S1).

2.1.1. Biological duplexesFor these experiments, the two BMs were kept in separate

perfusion systems, and mechanical interaction was implementedusing a computer-controlled system to interrelate input/outputsignals of muscle force and length between both BMs, with aninternal time steps of 100 ms, while prescribing the duplexconstraints (opposite length changes, equal force). The principalfeatures of the experimental duplex method and a scheme of thecomputer-based control algorithm we developed to imitate real-time mechanical interactions between two muscles under theduplex constraints have been described in detail elsewhere(Markhasin et al., 2003; Protsenko et al., 2005). Additionally to themechanical activity of both muscles, we measured AP time-coursesin cells of either of the BM in biological duplexes, using floatingmicroelectrodes. Unfortunately, due to the dynamic nature of thepreparation, it was difficult to measure AP in both preparationssimultaneously or for prolonged periods of time, in particularduring switches from individually contracting BM elements toa coupled duplex, and back to the uncoupled state (complete APrecordings were obtained in 3 out of 40 experiments).

2.1.2. Hybrid duplexesA similar as above step-by-step control algorithm was used to

implement muscle interactions in hybrid duplexes, wherecomputer model calculations were synchronized with the signalexchange between BM and VM (the latter implemented asa computer model-driven mechanical actuator) in real time(Protsenko et al., 2005). One of the main benefits of the hybridduplex configuration is that it allows one to expose one and thesame BM to interaction with various VMs (e.g. by attaching a slowor a fast VM to the same BM, as described below). This can be usedto simulate a wider range of functional heterogeneities than isusually possible in the biological duplex setting. In addition tocalculating model parameters underlying the electromechanicalactivity in the VM (including: muscle force and sarcomere length;AP; concentration of intracellular Ca2þ ([Ca2þ]i)), and to recordingthe mechanical activity of the coupled BM, the hybrid duplex hasallowed us to simultaneously measure AP or CaT (monitored withfura-2) in the BM during its interactions with the VM.

2.1.3. Virtual duplexesIn the virtual duplex setting, interactions between coupled VMs

(i.e. between two individual models simulating one element each)are implemented using the same constraints (opposite lengthchanges and equal forces; see Solovyova et al., 2003 for details). Tosimulate the effects of VM interactions in heterogeneous ventric-ular tissue, where cardiomyocytes from distant regions differ in the

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electromechanical properties (e.g. transmural or baseeapexgradients; for review see Markhasin et al., 2011) we developedso-called fast and slow VM samples. These VMs differ in theirvelocity of contraction and relaxation, AP duration, and the rateconstants underlying CaT (for detail, see Solovyova et al., 2003).

2.1.4. Experimental protocolsTo explore slowly developing effects of mechanical interactions

between muscle elements in the duplex, we use the followingprotocol. First, we allow each of the two duplex elements tocontract in isolation in isometric mode at a length set to 0.9e0.95 ofLMAX (LMAX is the muscle length at which a maximal isometric peakforce is developed), until they reach a steady-sate. Parametersrecorded (muscle force and length, AP, CaT) in this setting serve asthe baseline to compare with that developed after duplex forma-tion. Then, the muscles are mechanically interconnected in-series,and effects of their interactions are monitored before approach-ing a new steady-state. Finally, duplex elements are disconnectedagain, and returned back to the initial isometric conditions, toassess changes in their contractile state.

Note that in this paper we address effects of myocardialmechanical interactions that result mainly from the temporalheterogeneity in the mechanical activity of coupled musclesegments within duplexes. This arises as a consequence of twofactors: (1) the time-lag in muscle electrical excitation (referred toas the “electrical asynchrony”), and (2) the original heterogeneity inthe timing of muscle force/shortening generation (referred to as the“mechanical asynchrony”, e.g. fast vs slow contracting musclesegment). So, three forms of the internal temporal heterogeneity ofmuscle duplexes were considered, accounting for either of theelectrical or mechanical asynchrony, or mixture of the two. If inbiological duplexes two BMs were similar in terms of thecontractionerelaxation rates, lengtheforce and forceevelocitydependences, temporal heterogeneity was introduced/enhancedvia a stimulation delay between BMs or/and by using differenttemperatures in the experimental bath chambers (warm BMdeveloped contraction faster, while generating lower peak force). Inhybrid duplexes, muscle heterogeneity could be modified viacombination of a BM with different VMs (e.g. fast or slow VM), andby imposing stimulation delays as well. In virtual duplexes, we usedfast and slow VMs and systematically explored different stimula-tion delays to enhance or decrease temporal heterogeneity. More-over, here we excluded additional heterogeneity factors such asdifferences in baseline peak force (weak vs strong) or initial lengths(short vs long) of duplex elements. In virtual or hybrid duplexes, wedefined a VM to have the same initial lengths and baseline peakforce as the other duplex element (whether VM or BM). In exper-iments with two BMs, this restriction was implemented usingrelative length and force amplifying coefficients in transferringmechanical outputs between the individual elements. This involvedapplying command input/output signals that simulated matchinglengths and peak forces (e.g. if one BM shortened by 1% of itsindividual LMAX, the second muscle was stretched by 1% of its ownLMAX). This allowed us to unify initial conditions for musclecoupling, and to compare results obtained across the differentduplex pairs.

2.2. Muscle isolation and measurements

The study conformed to the NRC Guide for the Care and Use ofLaboratory Animals and animal experiments were approved by theAnimal Welfare Committee of the Institute of Immunology andPhysiology. BM experiments were carried out using papillarymuscles harvested from right ventricles of rats (3e6 months),rabbits (5e6 months), and guinea pigs (1e4 months) of either sex.

After approved euthanasia of the animal, hearts were quicklyexcised and placed in modified Krebs solution containing (in mM):NaCl 118.5; KCl 4.2; MgSO41.2; CaCl2 2.5; glucose 11.1, and bufferingwith NaHCO3 and KH2PO4 and bubbling with 95%O2 þ 5%CO2 wereused to set the pH of target magnitude 7.35. During tissue prepa-ration, butanedione monoxime (30 mM) was used to reducemechanical and metabolic activity, and to prevent tissue damageduring muscle isolation; this was washed-out for subsequentexperimentation. Thin papillary muscles were dissected from theright ventricle and attached between a force transducer anda length servomotor in an experimental chamber superfused withmodified Krebs solution. In total, we studied SFRIM in 62 papillarymuscles used to form biological or hybrid duplexes, includingpreparations from rat (n ¼ 39, see Figs. 2 and 6e8), rabbit (n ¼ 11,see Fig. 1C and D) and guinea-pig (n ¼ 12, see an example of SFRIMin Fig. S2 in the on-line supplement). All measurements wereconducted at 25 �C, using a 3 s inter-stimulus interval.

At the beginning of each experiment, muscles were released toslack length (L0), then sequentially stretched with an increment ofw2e3% of L0, and allowed to approach steady-state contractions.The stretching ramps were stopped when no further increase inactive tension, or a significant increase in passive tension, wereobserved. This length was designated as LMAX. Working length (LW)was set to 95% of LMAX. Peak isometric force developed by a musclewas stable during the measurements (i.e. force was similar prior toduplex coupling and after recovery following duplex disconnec-tion; significant deviation served as an exclusion criterion for thesample concerned). BM force was measured using a precision forcetransducer (KG-4, Scientific Instruments GmbH, Heidelberg,Germany; range 0e50 mN, resolution 0.02 mN). Length changeswere computer-controlled and applied by fast servomotors witha total movement range of up to10 mm and <1 mm single step size(tailor-made equipment and MC1 servomotor, Scientific Instru-ments GmbH, Heidelberg, Germany). The servomotors allowed usto prescribe different mechanical environments to a BM prepara-tion, including isometric and isotonic modes of contraction, as wellas to apply dynamic length changes required to allow BMeBM orBMeVM interactions in biological and hybrid duplexes.

APs in cardiomyocytes of BMs were recorded using the floatingmicroelectrode technique and an intracellular electrometer (IE-210,Warner Instruments Corp., Hamden, CT, USA). Time-dependentchanges in [Ca2þ]i, were monitored using the fluorescent dyeFura-2(AM) (5 mM in 5% w/v Pluronic F-127). The dye was excitedusing a halogen light source at 340 nm or 380 nm, and emittedfluorescence was collected through an FLUAR 10X/0.50 objectiveand a wide-band filter set #02 (excitation peak at 365 nm, beamsplitting at 395 nm, and 420 nm long-pass emission filter, Carl ZeissJena, Germany) on a photomultiplier unit (PHOM-1, ScientificInstruments GmbH, Heidelberg, Germany) for ratiometric [Ca2þ]i,signal analysis. This was mounted to an inverted microscope(Axiovert-200, Carl Zeiss Jena, Germany). Details of the system andthe dye loading procedure have been described in detail elsewhere(Lookin and Protsenko, 2011).

Muscle force/length were measured simultaneously with eitherAP or [Ca2þ]i. Mechanical and AP data were sampled at 10 kHz, andfluorescence at 400 Hz, using an analog-to-digital PCI-1716S card(Advantech Co., Ltd., Taiwan), using tailor-made software run underHyperKernel (ArcSystems Ltd., Japan), a real-time integrated envi-ronment. The interaction protocols and length interventions wereapplied via the same fast PCI-1716S interface, with time-intervals of100 ms. Data were processed off-line, using in-house software tocalculate steady-state forceelength and forceevelocity relationsand to get the parameters of force, AP, and ratiometric [Ca2þ]i,signals measured in the steady-state conditions and during slowresponses.

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Fig. 2. Experimental recordings of SFRIM in a biological duplex, comprising of two papillary muscles from right ventricle of rat. Both BM produce near identical isometriccontractions in isolation (panel B, left). The duplex is made heterogeneous by activating muscle 2 with a stimulation delay of 80 ms at pacing interval of 3 s, once mechanicallyconnected in-series. The panel layout in A and B is the same as in Fig. 1CeD.

Fig. 1. SFRIM registered in virtual (AeB) and biological (CeD) myocardial duplexes. A: Example SFRIM in a heterogeneous virtual duplex, comprising of a slow (muscle 1) and a fast(muscle 2) virtual muscle (VM) with a stimulation delay of the fast VM of 30 ms at a pacing interval of 3 s. Phases (1)e(5) show transitions of the peak force developed by the VM inisolation, once coupled in-series, and after duplex disconnection. Force is normalized to the baseline peak force (which is indicated by a dotted line). Each bar shows the average ofthe peak force during two consequent cycles. B: Left traces (labelled “uncoupled”) show the time-course of steady-state isometric forces during a single contraction cycle, developedby the VMs before coupling (normalized to the peak force) and superimposed taking into account the stimulation delay to be imposed after muscle coupling. Middle traces(“coupled”) show the time-course of steady-state force development by the two VM once connected in-series (at phase (3) in panel A). Right traces (“coupled”) show oppositelydirected cyclic length changes in the two VM (expressed as fractions of the initial muscle length) during every steady-state beat of the duplex. CeD: Experimental recordings ofSFRIM in a biological duplex, comprising of two papillary muscles from rabbit right ventricle at 25 �C (muscle 1) and 30 �C (muscle 2) with matching individual peak forcesdeveloped in isolation. The muscle 2 is activated with a stimulation delay of 40 ms at a pacing interval of 3 s, once muscles are mechanically connected in-series. The panel layout inCeD is the same as in AeB, with the exception that absolute magnitudes of force and deformations are shown. Note similarity of experimentally-observed (CeD) andcomputationally-predicted (AeB) behaviour.

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All chemicals were purchased from SigmaeAldrich (USA),except fura-2/AM (Fluka Biochemika, Switzerland).

2.3. Computational cardiac muscle model

We used the EO model, combining the Noble’98 ventricularcardiomyocyte electrophysiology model (Noble et al., 1998) withthe description of Ca2þ handling and mechanical activity inventricular myocardium developed by Solovyova et al. (2003). The

EO model comprises w30 ordinary differential equations and wasused to simulate the electrical and mechanical activity of VM in thedifferent experimental conditions. The CellML model representa-tion can be found at CellML model repository at http://models.cellml.org/e/b9/ and run with using Cellular Open Resource athttp://cor.physiol.ox.ac.uk/ (Garny et al., 2009). A key feature of themodel is inclusion of the cooperative dependence of thin filamentCa2þ activation, particularly the dependence on the concentrationof attached cross-bridges. We have previously shown with the

Fig. 3. Dependence on the relative peak force over/undershoot produced by BMs in biological and hybrid duplexes upon duplex uncoupling (SFRIM excess, normalized to thebaseline peak force developed in each element before duplex formation) on (A) the phase-dependent deformation integral (see text) and (B) the amplitude of the respectivedeformation phase. Deformations are expressed in % of the initial working muscle length.

Fig. 4. Effects of command cyclic deformations on SFR in rat isolated papillary muscles. Panel A: Command deformations with different amplitudes of the first phase of shorteningare shown. Panel B: Recordings of the force developed by a papillary muscle under the cyclic deformations from the left panel A are shown. Panel C: Slow peak force transitionsdeveloped by the papillary muscle after exposure to the cyclic deformations and after their cancellation. Panels DeE: SFR excess developed upon resuming baseline isometricconditions after a series of deformations is shown against the deformation amplitude (panel D) and against the number of cycles with imposed deformations (panel E). Dashed linesshow linear regression, with coefficients correlation of �0.61 (panel D) and þ0.73 (panel E) respectively, p < 0.01. Dotted lines show the boundary between force over- orundershoot. Different symbols are indicative of different individual experiments at pacing interval of 3 s and mechanical interventions started shortly after electrical stimulationwith a delay of 10e30 ms.

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model that the cross-bridge-induced increase in the affinity oftroponin C (TnC) to Ca2þ is a key mechanism and vital contributorto the effects of cardiac excitationecontraction coupling andmechano-electric feedback (Solovyova et al., 2003, 2004; Sulmanet al., 2008). These computational models describe fast and slowVM and were used to govern VM activity, both in hybrid and virtualduplex studies.

3. Results

3.1. Intra-myocardial SFR (SFRIM) in heterogeneous muscle duplexes

3.1.1. Model predictions on SFRIM in virtual duplex (‘dry’)experiments

First we studied effects of mechanical element interactions inheterogeneous virtual duplexes (Fig. 1A and B). As described above

in Section 2, we explored effects of the temporal heterogeneity onthe activity of coupled VMs, and used a fast and slow VM thatproduced equal peak isometric forces when in isolation, but dis-played a prominent mechanical asynchrony (see Fig. 1B, left). Theyalso showed respective differences in their forceelength andforceevelocity relationships, as explored before (Solovyova et al.,2003). Moreover, the faster VM displayed shorter AP and CaT,along with higher SR Ca2þ loading, higher CaT amplitude and end-systolic [Ca2þ]i levels, compared to the slower VM. Here (in Fig. 1),we introduced an electrical asynchrony with 30ms delay of the fastmuscle stimulation to mimic normal activation sequence in theventricular wall where subendocardial cardiomyocytes (moreslowly contracting in isolation) are activated earlier than sub-epicardial ones (faster contracting in isolation).

Each VM approached steady-state isometric contractions at thesame initial muscle length before duplex formation (see phase (1)

Fig. 5. Effects of the mechanical interactions on the electrical activity and Ca2þ handling in a virtual heterogeneous duplex. The traces demonstrate gradual cycle-by-cycle changesin APD90 (panel A), CaT amplitude ([Ca2þ]i, max; panel B), CaD90 (panel C), and diastolic SR Ca2þ loading ([Ca2þ]SR; panel D) in cardiomyocytes of the shortening VM1 and stretchingVM2 shown in Fig. 1A, as they develop before, during and after mechanical muscle duplex interactions. Arrows indicate the onset and termination of muscle coupling, and values arenormalized to their respective baseline magnitudes in cardiomyocytes of the VMs during isometric contractions before coupling.

Fig. 6. Effects of mechanical interactions on the electrical activity of cardiomyocytes in a BM of biological duplex. A biological duplex was formed of two rat BMs, activated witha 100 ms stimulation delay at pacing interval of 2 s (see panel B, left). Mechanical activity in each muscle was measured simultaneously with APs in a cardiomyocyte of one of theinteracting BM. Panel A: SFRIM is developed in the BM during mechanical interactions within the duplex (see force transition during phases 2e3, where this muscle experiencessequential shorteningelengthening on each beat shown in panel B, right), and after resuming individual isometric contractions after duplex uncoupling (see phases 4e5). Note theforce overshoot developed by the muscle upon disconnection, this muscle was presumably shortened during the interaction in duplex (black lines on panel B). Layout of panel B isthe same as in Figs. 1 and 2. Panel C: changes in APD80, measured in one and the same cardiomyocyte of this muscle during the force transitions shown in panel A. Arrows indicatethe onset of muscle coupling and uncoupling, respectively. Panel D: Superposition of AP traces measured in the cardiomyocyte at different phases (1)e(5) of the SFR, identified in A.The thin dotted line shows the resting potential, which did not change during the experiment.

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Fig. 7. Effects of mechanical interactions in a hybrid muscle duplex on the shape and duration of CaT in a rat papillary BM, interacting with a VM, activated with 100 ms delay atpacing interval of 3 s. The mechanical activity in the BM was measured simultaneously with fura-2/AM fluorescence signal. Panel A: SFRIM is shown in the BM during mechanicalinteractions within the duplex (phases 2e3), and after resuming isometric contractions following duplex uncoupling (phases 4e5). Note force overshoot developed by the BM uponuncoupling (compare 4 to 3). Panel B: Time-courses of the force (top, normalized to the baseline peak force), deformation (middle, normalized to the initial muscle length), and fura-2 fluorescence intensity (bottom, 340/380 nm excitation wavelength ratio) are shown at phases (1)e(5) of the SFRIM (black lines), compared to baseline (grey lines) registeredduring steady-state isometric contractions of BM preceding coupling to the VM. Numerical labels on the traces indicate percentage changes in characteristic values: peak force, peakshorteningelengthening, duration of CaT at 80% of decay, compared to baseline.

Fig. 8. Changes in CaT amplitude in a BM of hybrid duplex. A hybrid duplex was formed from a rat BM and a VM activated with a 50 ms stimulation delay at pacing interval of 3 s(panel B, left). The mechanical activity in the BM (panels AeB) was measured simultaneously with fura-2/AM fluorescence (panel C, normalized to the baseline peak CaT). The SFRIM

(panel A) is shown along with gradual changes in CaT amplitude (square marks, panel C) in the BM. Dotted lines show baseline levels of peak force and peak fura-2 fluorescence.

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in Fig. 1A). VM length was set to 0.9 LMAX, corresponding toa sarcomere length of 2 mm.Model parameters were fitted tomatchsteady-state isometric peak forces at this initial length (Fig. 1B, left).Then we imposed kinematic restrictions imitating end-to-endcoupling between the two VM with stimulation delay of the fastVM. The temporal heterogeneity of individual contractile activitiesalong with the activation timing allowed VM to contract aux-otonically, with opposite length changes (Fig. 1B, right) in thepresence of equal total forces (passive þ active) developed by eachof the coupled VM (Fig. 1B, middle).

The electrical andmechanical asynchrony (Fig.1B, left) gives riseto their dynamic deformations with two phases of shortening/lengthening in eachmuscle during every contraction of the coupledduplex (Fig. 1B, right). The first-activated slow VM starts shorteningand pulls the later-activated (almost passive at that point) fast VMduring the early phase of duplex contraction for about 50 ms. Herethe peak of the initial shortening of the earlier-activated slow VM(and the matching pre-stretch of the later-activated fast VM) isabout 2%. Then, as the fast VM becomes active; it shortens for100 ms until individual activity in both muscles overlap. For thenext 250ms, the force-generating potential in the slow VM exceedsthat of the fast VM, which relaxes faster (see time-course of indi-vidual force traces before coupling, Fig. 1B, left). This causessecondary shortening of the slow VM for most of its active cycle,with a peak shortening of about 6%, equivalent to a minimalsarcomere length of 1.94 mm. The late phase of fast VM shorteningoccurs after its own activity cycle is completed, thus reflecting itspassive relaxation from the stretched length to the initial one. Notethat during the contraction cycle, the length of the slow VMremained below the initial length, i.e. this muscle produced short-ening, whereas the length of the fast VM stayed above the initiallength, i.e. this VM underwent dynamic stretch during the cycle.

Mechanical interactions of fast and slow VMs cause an imme-diate decrease in peak force after duplex formation, here by�16% ofbaseline peak force in each VM prior to coupling (see Fig. 1A,transitions 1e2). After the initial fall, a gradual increase in forcedevelopment can be observed by about 2% of baseline (see transi-tions 2e3 labelled ‘coupled’ in Fig.1A), reaching a new steady-state.This is accompanied by a small (about 1%) increase in the amplitudeof individual VM deformations.

To reveal whether contractility of each muscle changed duringtheir mechanical interaction, we explored VM responses todisconnection (see phases 4e5 labelled ‘uncoupled’ in Fig.1A). Bothmuscles showed pronounced transitions of their peak force upondisconnection, indicative of a changed contractile state that theyreached while contracting in the heterogeneous myocardialsystem. Note that the patterns of force transition are qualitativelydifferent in the two VMs after uncoupling. The slow VM, thatpredominantly shortened during the active cycle, developeda pronounced overshoot by about þ10% of peak force beyond thebaseline levels before duplex formation. This indicates ‘contractilitygain’ in the slow VM due to the mechanical interaction with its in-series duplex partner. In contrast, the fast VM, that was stretchedduring the contractile cycle of the duplex, showed an undershootby about �4% in peak force production upon disconnection, indi-cating a ‘contractility loss’. Note that the force overshoot developedin the slow VM was more pronounced than the respective under-shoot in the fast VM.

Heterogeneous virtual duplexes predict that quantitative char-acteristics of the SFRIM, i.e. the magnitude of slow changes in thepeak force developed by coupled VMs, and the magnitude of forceincrement (overshoot/undershoot) produced by either of the VMsafter uncoupling, may vary significantly. For example, variations ofthe stimulation delay between the above fast and slow VMsaffected SFRIM. If these VMs were stimulated simultaneously (which

increased the time-lag between the peaks of individual muscleactivity) SFRIM was remarkably enhanced (Markhasin et al., 2004).Under these conditions, the slow force increase developed bycoupled muscles reached up to þ12%, while the overshootproduced by the predominantly shortening slow VM increased upto þ56%, while the force undershoot in the cyclically stretched fastVM was about �7%.

3.1.2. Physiological (‘wet’) experimentsTo test the model’s predictions, we performed experimental

studies using biological and hybrid muscle duplexes. In biologicalduplexes, two BMs isolated from one and the same right ventriclewere used. Temporal and functional heterogeneities betweenmuscle elements were introduced as described in Section 2.

The experimental data shown in Fig. 1 were obtained froma “slow” (25 �C) and a “fast” (30 �C) papillary muscle from rightventricle of rabbit, activated with a stimulation delay of 40 ms forthe fast BM. After an immediate force decrease by about�10% uponmuscle coupling (against the baseline force prior to duplexformation), there is a gradual increase up to 5% in duplex peak forcegeneration (Fig. 1C, phases 2e3). Upon disconnection, the earlier-activated slow BM that shortens by �9.5% during the contractilecycle within the duplex (Fig. 1D, right), shows a prominent peakforce overshoot by þ12% (Fig. 1C, phase 4, similar to the respectiveslow VM in Fig. 1A). In contrast, the second, later-activated fastercontracting BM that is stretched during the active cycle developsa force undershoot by about�8% upon disconnection (Fig.1C, phase4, similar to the later-activated fast VM in Fig. 1A).

Note that in experimental biological and hybrid duplexes,similar to in-silicomodel predictions, characteristics of SFRIM highlyvaried as well. An experimental recording, demonstrating anexample of quantitatively prominent SFRIM in a rat biologicalduplex is shown in Fig. 2 (see also Fig. S2 in the on-line supplementfor SFRIM recordings in guinea-pig). Both BMs showed similar time-courses of the force development during isometric contractions inisolation (Fig. 2B, left). Once mechanically connected in-series, theBMs were stimulated with a delay of 80 ms (an example duplexwith electrical asynchrony only). Comparing to the baseline forcebefore connection, an immediate �53% decrease in peak forcegeneration is followed by a gradual peak force increase up to 20% ofthe first duplex contraction (Fig. 2A, phases 2e3). Upon discon-nection, the earlier-activated BM, which predominantly shortenedduring the early contraction within the duplex (Fig. 2B, right)showed a prominent peak force overshoot by up to 40% (Fig. 2A,phase 4). In contrast, the later-activated BM, which mainly expe-rienced stretch in early systole (Fig. 2B, right) developed a forceundershoot by about �12% (Fig. 2A, phase 4) upon disconnection.

This kind of SFRIM scenario was frequently observed in thebiological and hybrid duplexes with apparent intrinsic mechanicalsynchrony (in terms of similarity in individual isometric contrac-tions of muscles), where electric asynchrony was introduced bystimulation delays (see Figs. 7 and 8, and Markhasin et al., 2011).Other examples of SFRIM in an “apparently-homogeneous” hybridduplex with imposed electrical asynchrony are presented inSolovyova et al. (2006), and a consistent example of SFRIM ina virtual duplex composed of identical VMs can be found in the on-line supplement (see Figs. S3 and S4).

3.1.3. Relations between the deformation patterns and SFRIM inbiological and hybrid duplexes

The dynamic deformations registered in each interacting BMsshowed different deformation patterns with one (Fig. 1C and D),two (Figs. 2, 7 and 8), or multiple (Fig. 6) phases of shortening/lengthening of different magnitudes during every contractile cycle.To quantify the role of either dominating shortening or lengthening

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of individual duplex elements in either overshoot or undershoot inforce production upon duplex disconnection, we integrated defor-mation signals over the entire contractile cycle for each duplexelement (see Fig. S1 in the on-line supplement for detail). Theresulting cyclic deformation integral (CDI) signifies an averagedeformation value, and its polarity shows which part of deforma-tion (shortening or stretch) dominates. When CDI is negative ina coupled muscle, this means that on average this muscle con-tracted at shorter lengths during systole (compared to diastole). Asa rule, a negative CDI coincided with a higher amplitude of short-ening. In-series muscle elements have CDIs of opposite signs. Asummary of observations on negative or positive CDI among ratBMs (total number n ¼ 39) used in biological and hybrid duplexes,and their respective contractility changes, is presented in Table 1.

Out of all BMs with a negative CDI (n ¼ 31), 77% developeda positive overshoot in peak force production upon duplexdisconnection (see Table 1). Thus, dominating shortening duringthe contractile cycle (negative CDI) of a muscle in the duplex ispredictive of a ‘contractility gain’ (peak force overshoot) uponduplex disconnection, with a sensitivity of 89% (the frequency ofnegative CDI among all BMs that develop a force overshoot, n ¼ 27)and a specificity of about 42% (the frequency of lack of negative CDIamong BMs that develop a force undershoot, n ¼ 12). On the otherhand, 63% out of all BMs with a positive CDI (n ¼ 8) developeda force undershoot upon disconnection (see Table 1), suggesting thedominating stretch of interacting muscle as indicative ofa ‘contractility loss’, with a rather low sensitivity of 42%, but a highspecificity of 89% (see Table 1).

Thus, the CDI correlations suggest that muscle shortening moreprobably predetermines the contractility gain, whereas stretchingmay lead to the contractility loss. In all duplex experiments illus-trated in this paper force overshoot/undershoot upon duplexdisconnection followed the above tendency (see CDI polarity in thelegends of Figs. 1, 2 and 6e8). In consistency with this hypothesis,we found a significant correlation between the magnitude of theforce excess (overshoot/undershoot) after uncoupling and therespective value of the phase-dependent deformation integral (anaverage value) calculated either over the shortening phase forovershooting BMs or over the stretching phase for undershootingBMs (correlation coefficient of �0.85, p < 0.01, Fig. 3A). A similarlygood correlation (correlation coefficient of �0.81, p < 0.01) wasfound between the magnitude of the force overshoot/undershootand the amplitude of the respective shortening/stretching phaseduring the cycle (Fig. 3B).

3.1.4. Role of parameters of cyclic mechanical deformations in SFRof cardiac muscle

To verify more directly whether characteristics of the twophases of contraction (shortening and stretch) distinctly influencethe contractile state of cardiomyocytes, we performed experimentsin which a single papillary muscle was exposed to cycling lengthchanges that mimic patterns observed in a duplex (Fig. 4, compareto Figs. 2 and 7).

For simplicity reasons we imposed a combination of two sinus-like half-phases of deformation for every contraction cycle, whereinitial muscle shortening of varying amplitude was followed bymuscle lengthening (or vice versa). Like in a number of therespective experimental duplex recordings (e.g. those presented inFigs. 2 and 7), the duration of the first phase was slightly shorterthan that of the second (150 ms vs 250 ms, Fig. 4A). The amplitudeof the first phase of imposed deformations varied from �15%to þ10% (negative values mean shortening, positive values meandistension) of theworking length (LW). The amplitude of the secondphase was set to �6e9% of LW, and was opposite in sign to the firstphase (Fig. 4A).

Before the interventions, muscles approached steady-stateisometric contractions. Exposure to the various prescribed defor-mation altered muscle force development. Peak force under inter-ventions was inversely related to the extent of shorteningpermitted, i.e. showed inactivation by shortening (Fig. 4B). SFRmanifested both in a small gradual increase in peak force produc-tion during the imposed length change protocols, and in a morepronounced overshoot (or undershoot, not shown) after theircancellation (Fig. 4C). To evaluate quantitatively the effect ofdeformation amplitude on muscle contractility, we calculated therelative excess (overshoot or undershoot) of the peak force uponcancellation of the intervention. As expected from the data previ-ously obtained in muscle duplexes, increased muscle shorteningduring the first phase of contraction resulted in a more pronouncedforce overshoot upon cancellation of forced deformations (Fig. 4D,negative deformation range). Muscle stretch during the first phaseof imposed contractions either had no effect, or gave rise to anundershoot in peak force production after cancellation of inter-ventions. This decrease was the more pronounced, the higher thestretch amplitude applied (Fig. 4D, positive deformation range).The dependence of the relative force response on the amplitude ofthe deformations was nearly linear for every muscle tested (n ¼ 6,not shown), and linear for the data pooled from all experiments(Fig. 4D, dashed line).

In two further experiments we applied cyclic deformations (firstshortening to �15% of LW, then lengthening up to 5% of LW) overa variable number of contraction cycles (1, 5, 10, 20, 40 and 100) totest whether there is a cumulative effect on contractility. The forceovershoot observed with the given protocol generally developedwithin the first 20e40 beats (Fig. 4E).

3.2. Slowly developing responses of the electrical activity and Ca2þ

handling to the mechanical interactions in heterogeneous duplexes

3.2.1. Model predictions (in-silico ‘dry’ experiments)Both the model predictions and the experimental data above

suggest that the ‘gain’ or ‘loss’ in the contractile state of interactingmusclesmay reflect the respective slowchanges in the Ca2þ loadingof cardiomyocytes. In the virtual duplex shown in Fig. 5 (whosemechanical activity is presented in Fig. 1A and B), the gain in thecontractility of the VM that shortened during most of thecontraction cycle (see Fig. 1B) was accompanied with dynamicincrease in the CaT amplitude and duration at 90% of decay (CaD90),AP duration at 90% of repolarisation (APD90), and the diastolic SRCa2þ loading above baseline levels observed before muscle inter-action. Changes of opposite polarity were seen in the VM that isstretched during most of the contraction cycle (see Figs. 5 and 1B).

The patterns of gradual change in the above parameters of Ca2þ

handling and electrical activity differ from one another, and fromthe pattern of peak force transitions (Figs. 5 and 1A). For instance,the SR Ca2þ content changes gradually from baseline to a newsteady-state, showing either gradual accumulation in the muscledeveloping force overshoot after disconnection or loss in the

Table 1Frequencies of negative and positive CDI for rat BMs (total number n ¼ 39) frombiological and hybrid duplexes that developed either overshoot or undershoot uponduplex disconnection.

Overshoot Undershoot Total Overshootfraction (%)

Undershootfraction (%)

Negative CDI in BM [n] 24 7 31 77 23Positive CDI in BM [n] 3 5 8 38 63Total 27 12 39Negative CDI [%] 89 58Positive CDI [%] 11 42

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muscle that shows force undershoot upon disconnection (Fig. 5D).At the same time, CaD90 and APD90 undergo more complex, non-monotonous transitions during the period of muscle interactions(Fig. 5A and C).

3.2.2. Physiological (‘wet’) experimentsTo assess model predictions on possible effects of the mechan-

ical activity on both Ca2þ handling and the electrical activity incardiomyocytes of coupled muscles, we performed experimentalstudies on biological and hybrid duplexes.

The experimental recordings shown in Fig. 6 were obtained ina biological duplex, formed of two rat BMs. An excitation delay of100 ms between the coupled muscles caused multiphase defor-mations during the contractile cycle (Fig. 6B, right), and resulted inSFRIM. This SFRIM was accompanied by a pronounced increase inAPD80 by up to 30% in cardiomyocytes of the muscle that demon-strated predominant shortening (a negative CDI) during duplexcontractions followed by a peak force overshoot upon disconnec-tion (Fig. 6AeC). Afterwards, AP shape and APD80 in this musclegradually returned to baseline level seen before duplex formation(Fig. 6C and D). These changes in AP shape and duration (Fig. 6C andD) we registered in one and the same cell for the entire period ofexperiment. The data clearly show that mechanical deformation ofthe muscle during contraction while connected in the duplexaffects the electrical activity of cardiomyocytes.

In experiments performed on hybrid duplexes, we registeredSFRIM simultaneously with the changes in the shape and durationof CaT in a BM interacting with a VM. An example of such experi-mental recording is shown in Fig. 7. The BM, which developeda peak force overshoot upon duplex disconnection, showeda significant change in both shape and duration of CaT just uponduplex formation, and a non-monotonous recovery to baselinefollowing disconnection.

The amplitude of Ca2þ transients also changed in the BM duringmechanical duplex interaction. For example, Fig. 8 demonstratesa transition in CaT amplitude that correlated with SFRIM in a ratpapillary muscle that interacted with a VM whose stimulation wasdelayed for50ms.TheBMpredominately shortenedduringeachbeat(having a negative CDI), and showed a peak force overshoot upondisconnection. Similar one to another patterns of correspondingchanges in SFRIM and CaT amplitude were observed in this BM.

4. Discussion

4.1. SFRIM and the role of polarity of cyclic deformations in coupledmuscles

The present study demonstrates that mechanical interactionsbetween two end-to-end connected myocardial segments formingmuscle duplexes cause slow beat-to-beat changes in individualmuscle contractility, along with slow changes in the total duplexforce development. We termed this SFR as the intra-myocardial SFR(SFRIM), to stress that it was induced by internal behaviour of duplexelements, governed by electrical asynchrony in their activation andintrinsic mechanical asynchrony in their individual activity. It isnoteworthy that the SFRIM were first found in virtual duplexes, andthen the model predictions were confirmed in the respectivephysiological experiments in biological and hybrid duplexes (seee.g. Fig. 1). Qualitatively similar manifestations of SFRIM registeredin BMs from different animal species (rat, rabbit, guinea-pig)suggest that the effects are not species-specific but representa general feature of any heterogeneous myocardial system.

In most of the duplex experiments, one muscle developed forceovershoot, whereas the other showed an undershoot upon duplexdisconnection, reflecting the respective “gain” or “loss” in their

individual contractility (Figs. 1 and 2). These observations recon-firm our previous reports on biological, hybrid and virtual duplexes(Markhasin et al., 2004, 2011; Solovyova et al., 2006). Lessfrequently, we observed also synergistic overshoots (or under-shoots) in both muscles after termination of duplex interactions(Solovyova et al., 2006).

Along with the mechanical manifestations of SFRIM, in-silicoexperiments revealed slowly developing beat-to-beat changes inboth the AP and CaT shape and duration in cardiomyocytes ofinteracting muscles (Fig. 5, see also our previous reports(Markhasin et al., 2004; Solovyova et al., 2006)). In virtual duplexes,individual SFRIM in interacting VMs followed the respective gradualgain (or loss) in the SR Ca2þ loading (Fig. 5D). In agreement with in-silico predictions, AP duration increased in BMs that produceda force overshoot in biological duplexes (compare Figs. 6C and 5A).Pronounced changes in CaT amplitude, shape and duration werealso seen in BMs interacting with VMs in hybrid duplexes (Figs. 7and 8). The changes in AP and CaT parameters recorded in BMs(Figs. 6e8) also evidenced the mechanical effects on the cellularelectrical activity and Ca2þ handling, suggesting the respectivechanges in the SR Ca2þ loading in interacting muscles.

The SFRIM in coupled muscles were accompanied with theiropposite auxotonic deformationsunder isometric constrain imposedon thewholeduplex. The patterns of individualmuscle deformationsevolved from the duplex asynchrony varied considerably betweenexperimental preparations. Unipolar shortening or lengthening ofcoupledmuscles (Fig.1), biphasic deformationpatterns (Figs. 2, 7 and8) and more complex length oscillations (Fig. 6) during everycontractile cycle were observed. The latter complex deformationsmay explain mechanical waves observed in the canine hearts afteracute coronary ligation (Weyman et al., 1984).

In spite of the diversity in individual SFRIM dynamics, we foundthe following general tendency apparent in experimental data andpredicted from simulation outputs. The coupled BMs from biolog-ical and hybrid duplexes that undergo predominant cyclic short-ening, quantitatively estimated as negative CDI, more frequently (in77% of the cases) showed force enhancement upon return to itsisometric state in isolation, whereas average cyclic stretching(positive CDI) was often (in 67% of the cases) accompanied witha decrease in the individual muscle contractility (see Table 1). Inagreement with this tendency, we found a rather high correlationbetween the extent of the force overshoot/undershoot and either ofthe amplitude or the average value (assessed as the phase-dependent deformation integral) of shortening/stretching (Fig. 3).

The role of the dominating dynamical shortening or lengtheningin the changes of muscle contractility was further confirmed ina separate series of experiments, where papillary muscles wereexposed to command deformations (Fig. 4) mimicking thoseregistered in biological duplexes. Here, the magnitude of peak forceovershoot/undershoot increased near-linearly with the amplitudeof shortening/lengthening, and depended on the number of inter-vention cycles (Fig. 4D and E). These results are in keeping withearlier reports on a contribution of dynamic shortening toincreased contractile activity of papillary muscles upon switchingfrom isometric to isotonic modes of contraction (Kaufmann et al.,1971), which has been reproduced by us in computational model-ling (Solovyova et al., 2004).

4.2. Intracellular mechanisms behind the SFRIM: mechanisticinsights from modelling

4.2.1. Mechanisms of the SR loading in the dynamically shorteningmuscle in virtual duplexes

Only computational model provided a means to unravel intra-cellular links between the mechanical activity and the SR Ca2þ

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reloading in cardiomyocytes of interacting muscles consistent withtheir contractility gain or loss.

The SFRIM example in virtual duplex shown in Fig. 1 is mostconvenient for such analysis because it follows unipolar deforma-tions in interacting muscles, with only shortening of one VM andonly stretch of the other VM.

Let us consider briefly the shortening VM having a negative CDI(Fig.1B), which demonstrates cycle-by-cycle increase in the SR Ca2þ

loading (Fig. 5D) and the respective increase in its individualcontractility (Fig. 1A). Sarcomeres in the VM dynamically shortenbelow the control values registered during the isometric contrac-tion in isolation (see Fig. S5 in the on-line supplement). This evolvesthe sarcomere length- and velocity-dependent inactivation ofcontractile proteins, revealing in a decrease in the fraction of force-generating cross-bridges (see Fig. S5 and Izakov et al., 1991). Thisvia cooperative feedback effects of cross-bridges on the affinity ofTnC for Ca2þ induces faster Ca2þ release from TnC (see Fig. S5).Concurrent decrease in the amount of cross-bridges and Ca2þeTnCcomplexes nonlinearly affects their kinetics, enhancing furtherboth myofilament inactivation by shortening and Ca2þ release fromTnC (see Fig. S5). The latter is revealed in cycle-by-cycle increase inthe amplitude and duration of CaT in the dynamically shorteningVM (Fig. 5B and C, and Fig. S5 in the on-line supplement). Particularchanges in the peak CaT of the shortening VMwere similar to thoseregistered in our experiments (Fig. 8) and consistent with earlierexperimental findings (Lab et al., 1984).

Mechanically-induced surplus in [Ca2þ]i in turn affects Ca2þ

fluxes in and out of the cytosol. The SR Ca2þeATPase (SERCA) andNaþeCa2þ exchanger (NCX) are major contributors to Ca2þ removalfrom cytosol (Bers, 2001). The SR Ca2þ uptake flux (Jup) is activatedmuch earlier than the forward mode NCX current (iNCX) during thecycle. In the model, the Jup peak coincides with the [Ca2þ]i peak,while the peak of the forward mode iNCX is reached later during thelate phase of AP repolarisation. Thus, SERCA removes Ca2þ from thecytosol most effectively at rather high levels of CaT, while theforward mode iNCX contributes to Ca2þ uptake during the late partof CaT decline where [Ca2þ]i is rather small. In the model, theamplitude of Jup is more than 2 times higher compared to theamplitude of Ca2þ flux via NCX, and SERCA takes up about 70%while the NCX takes 30% of the total amount of Ca2þ removed fromthe cytosol, close to experimental estimates (Bers, 2001).

In the shortening VM, Jup nonlinearly increases straightwaywiththe respective increase in [Ca2þ]i (see Fig. S6 in the on-linesupplement), while iNCX depends not only on [Ca2þ]i but also onAP, which duration also increases in the shortening muscles asregistered both in models and experiments (Figs. 5A and 6). Thus inour model, the increase in both [Ca2þ]i and APD leads to a timedelay in activation of the forward mode iNCX, an increase in itsamplitude, andmore fast inactivation of the current as compared tothe isometric cycle (see Fig. S6 in the on-line supplement). Thus inthe shortening VM, all the above factors allow the SERCA to capturea major fraction of Ca2þ surplus against the forward mode of iNCX(see Figs. S6 and S7 in the on-line supplement) providing with anincrease in the SR Ca2þ loading as compared to the controlisometric conditions (Fig. 5D). At the same time we founda decrease in the amount of Ca2þ extruded via NCX during theentire cycle in this VM (see Figs. S6 and S7).

We also calculated changes in the amount of Ca2þ entering thecell via different transmembrane currents and found a decrease inthe integrated reverse mode of iNCX, a small increase in the inte-grated flux via L-type Ca2þ channels (see iCaL, Fig. S7), anda decrease in the integrated potential-dependent backgroundcurrent. The total sum of the entering fluxes showed a smalldecrease in the overall Ca2þ income to the cytosol during entirecycle in the shortening VM. The changes in the potential-

dependent currents were consistent with the changes in APshape and duration observed in the VM (Fig. 5D).

This analysis shows that at the particular deformation valuesobserved in the shortening VM during auxotonic contractions invirtual duplex, the increase in the SR Ca2þ content occurredexceptionally due to the increased SERCA uptake. As we showedpreviously, an increase in the deformation amplitude, e.g. due toswitch from the isometric to low-loaded isotonic mode ofcontractions, may lead to a significant increase in APD and conse-quent increase in Ca2þ income via iCaL, that additionally contributesto the SR Ca2þ loading in the shortening VM (Solovyova et al.,2004).

The positive feedback from muscle shortening on the SR Ca2þ

loading that we derived from the model analysis has obviousmechanical limitations, which do not allow the interacting VM toshorten over a certain limit within the muscle duplex. On the onehand, a cycle-by-cycle increase in the SR Ca2þ content and conse-quent SR Ca2þ release compensates the higher inactivation byshortening in this VM. On the other hand, the more shortening VMstretches the second in-series connected muscle, which impedesmore strongly via an increase in its passive elasticity and in itscontractile potential due to sarcomere distensions. That is whya dynamical equilibrium is approached slowly cycle-by-cycle untilthe mechanical activities of interacting muscles adjust to therespective higher level of steady-state SR Ca2þ content in theshortening VM as compared to the control isometric conditions.

4.2.2. Mechanisms of the SR unloading in the transiently stretchedmuscle in virtual duplexes

Intracellular events occurring in the other VM, which undergoesprevailing distensions (shows positive CDI) throughout thecontractile cycle of the virtual duplex (Fig. 1), are “mirror reflec-tions” to that found in the shortening VM. Comparing behaviour ofthe dynamically stretching VM with that observed during thecontrol isometric cycle, an increase in the sarcomere length causesan increase in the affinity of TnC for Ca2þ and slows down Ca2þ

release from TnC (see Fig. S5 in the supplement). This causesa decrease in [Ca2þ]i, manifesting in a decrease in the both ampli-tude and duration of CaT in the stretching VM (Fig. 5B and C). Thedecrease in [Ca2þ]i consequently decreases the overall SERCAuptake (see Figs. S6 and S7) ensuring the SR Ca2þ unloading. Theforward mode iNaCa in the stretched VM is activated earlier andinactivated more slowly due to a decrease in [Ca2þ]i and a decreasein APD (Fig. 5A and B). This allows NCX to compete more effectivelywith SERCA for the Ca2þ uptake and to extrude a larger amount ofCa2þ from the cell (see Figs. S6 and S7). The mechanically-inducedmodulations in Ca2þ fluxes result in a gradual beat-by-beatreduction in SR Ca2þ content (Fig. 5D), and a consequent forceundershoot in the stretching VM upon duplex disconnection.

Thus, we showed that cooperative length-dependent mecha-nisms of Ca2þ activation of myofilaments play a key role in theSFRIM, providing with the links between mechanical deformationsand the SR Ca2þ loading in cardiomyocytes of interacting myocar-dial segments.

4.2.3. Predictive ability of the EO modelBesides the case of unipolar deformations (Fig. 1), we observed

multiphasic deformations of interacting muscles in heterogeneousduplexes, with altering phases of shortening and lengthening(Figs. 2 and 6e8). In the framework of virtual duplexes, individualSFRIM in the VMs are entirely defined by the cooperative length-dependent mechanisms recruited at each particular phase ofsarcomere shortening or lengthening during muscle deformations.

From the total number of the BMs in biological and hybridduplexes taken for the statistical analysis (n ¼ 39, see Table 1), 24

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out of 39 (61%) BMs showed positive SFRIM along with negative CDI,and 5 out 39 (13%) BMs showed negative SFRIM along with positiveCDI. So, these 74% of experimental BMs showed SFRIM whichcorroborates the abovemodel predictions on the length-dependentintracellular mechanisms underlying such SFRIM manifestations.

Other 3 out of 39 (8%) BMs showed the force overshoots withpositive CDI, and last 7 out of 39 (18%) of BMs showed the forceundershoot with negative CDI. Our model was able to reproducethe peak force overshoot along with prevailing muscle distensions(for an example see Fig. 2, panel (a) in Solovyova et al. (2006)). Inthis case, the velocity of the dynamic sarcomere stretch may alsocontribute to additional Ca2þ gain in the SR, as rather fast sarco-mere lengthening may also facilitate cross-bridges detachmentwith the following feedback effects on Ca2þ handling (Izakov et al.,1991).

In all the cases where CDI and the contractility change have thesame polarity (i.e. do not correspond to each other as expected), theparticular timing of shortening or lengthening during thecontractile cycle may play the most essential role than the averagedeformation. It is known that the effects of mechanical stimuli onthe electrical and mechanical activity of myocardium cruciallydepend on the intervention timing and on the phase of the cyclewhere it is imposed (Izakov et al., 1991; Kohl et al., 1998; Lab, 1982;Solovyova et al., 2004; Zabel et al., 1996). Certainly, the CDI isa rather rough index of deformation, which has to be improvedwith introducing some “weights” for different phases ofshorteningelengthening during the entire cycle. Moreover, the EOmodel version used in present study did not account for othermechanisms of cardiac mechano-electric coupling (MEC), as acti-vation of mechano-sensitive ionic channels, stretch-mediatedactivation of the NaþeHþ exchanger, or stretch-induced effects onthe SR Ca2þ uptake and release (Calaghan and White, 2004; Iribeand Kohl, 2008; Iribe et al., 2009; Kohl et al., 2006; Perez et al.,2011), that could also contribute to the SR Ca2þ loading balanceand play a role in the SFRIM. Particularly, all successful simulationsof the SFR to the sustained stretch bymeans of mathematical modelshave been based on accounting for stretch-activated ion channels(Bluhm et al., 1998; Niederer and Smith, 2007; Rice et al., 1998; Taviet al., 1998). Therefore, further study has to be done to unravel therole of these MEC mechanisms in SFRIM.

4.2.4. SFRIM and intra-myocardial auto-regulation inheterogeneous myocardium

The electromechanical manifestations of slow changes in thefunctional state of interacting muscle elements that we observed inheterogeneous duplexes allow us to hypothesize specific mecha-nisms of intra-myocardial auto-regulation of heterogeneousmyocardium contractility. We found the following positive andnegative feedback regulation of the SR Ca2þ loading in car-diomyocytes by the cyclic dynamical deformations of interactingmuscles elements of heterogeneous duplexes. Duplex elementwhich length mainly decreases compared to the initial lengthundergoes a transient myofilament inactivation by shorteningduring every contraction cycle versus baseline isometric conditions.This cyclic inactivation is gradually compensated via increased beat-by-beat Ca2þ loading of the SR, providing the means of mainte-nance of the muscle contractility at the mechanical conditions. Atthe same time, the in-series coupled duplex element is on averagestretched during contraction and thus activated via the stretch-dependent effects on myofilaments. In this case, a beat-by-beatCa2þ unloading of the SR suggests the means for confining intra-myocardial heterogeneity to keep efficiency of the whole system.Both adaptation mechanisms act jointly to adjust local contractilefunctions to the mechanical environment in the heterogeneousmyocardium and to support an optimization of its global function.

These auto-regulation mechanisms revealed itself in the muscleduplex models of heterogeneous myocardial system as oppositechanges in individual contractility of interacting muscles providingwith the positive global SFRIM of the whole duplex which showeda slow gradual beat-to-beat increase in force production.

The mechanisms of the intra-myocardial auto-regulation of localcontractility in interacting muscle segments appears to be relevantto the intact ventricle, which is a highly heterogeneous myocardialsystem (discussed in more detail in Markhasin et al. (2011)).Regional differences in electromechanical properties of car-diomyocytes were observed both in the transmural and the base-apex directions. These coexist with transmural and base-apexgradients in activation timing, which result in heterogeneouslocal strain dynamics in the ventricular wall (Ashikaga et al., 2009;Sengupta et al., 2006). In particular, there is significant dispersion ofregional myocyte shortening between endo- and epicardial layers,as early shortening of the earliest activated tissue regions pre-stretches later-activated ones (in particular during the during iso-volumic phase of contraction), while dispersion of myofiberrelaxation occurs in the opposite direction (Ashikaga et al., 2007).

Our muscle duplex model, in spite (or perhaps because) of itssimplicity addresses key features of heterogeneous myocardium inthe beating heart. Duplex examples shown in Fig. 1 accounting forthe mixture of electrical and mechanical asynchrony in myocar-dium can mimic the transmural heterogeneity patterns in theventricular wall. Because cardiac fibres spread continuouslythroughout the depth of ventricular wall from subendo to subepisurfaces (Streeter, 1979), the coupled subendo- and subepicardialsegments of a fibre can be considered as in-series connected fora rough approximation. In another group of in-series duplexes (seeFigs. 2 and 6e8), only electrical asynchrony is introduced forapparently homogeneous muscle elements (in terms of similarityin their individual contractions before coupling). Such duplexsettings are thought to reflect the mechanical interactions betweenapical and basal myocardial segments, where differences betweenthe electromechanical properties of isolated cardiomyocytesappear to be small in comparison to transmural differences (Nget al., 1998; Wan et al., 2003). Mechanical interactions betweenthese regions activated with a certain time-lag are essential for thelongitudinal peristaltic base-to-apex movements of the ventricle(Ballester-Rodes et al., 2006).

Thus duplex models can help to unravel the physiological rele-vance of myocardial heterogeneity, and allow one to exploremechanisms of local auto-regulation in the simplest heterogeneoussystems. This auto-regulation may contribute to the stability andoptimization of the inotropic state of myocytes in different parts ofthe intact heart. In the case of pathological heterogeneities (such asthe mechanical interaction of normal and ischemic segments of theventricular wall) this local auto-regulation may allow the ventriclesto adapt to the pathological conditions and to maintain contractileperformance.

In the heart, interacting myocardial segments are exposed topermanent mechanical interactions which may trigger epi-genetic(long-term) effects, caused by the mechanical stimuli and thechanges in intracellular Ca2þ kinetics (Berridge, 2003; Ross andBorg, 2001). Recently, essential contribution of MEC in the elec-trical remodeling in myocardium was demonstrated in responsethe altered activation sequence of the LV (Jeyaraj et al., 2007) andsimulated with a mathematical model (Kuijpers et al., 2008), sug-gesting the role of mechanical asynchrony in the well-knownphenomenon of cardiac memory (Rosenbaum et al., 1982). Weassume that the SFRIM arising from the electrical and mechanicalasynchrony in heterogeneous myocardium may provide witha substrate for furthermyocardial remodeling. Our findings can alsohelp to shed a light on the mechanisms evolved from the cardiac

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resynchronization therapy (CRT). Frequently, patients receivingCRT do not respond clinically to this therapy and do not showreverse LV remodeling (Yu et al., 2005). This may be may related todisturbances in the physiological sequence of myocardium activa-tion and consequentmechanical dyscoordination (Kirn et al., 2008),in which SFRIM and auto-regulation mechanisms may play a role.That is why adequate myocardial re-coordination rather thanresynchronization should be the target for successive therapy (Kirnet al., 2008).

5. Conclusions

We described a new type of SFR, termed as the intra-myocardialSFRIM, which arises from the heterogeneity of myocardium andreflects mechanisms of auto-regulation of its contractility. TheSFRIM phenomena were initially found theoretically by means ofmathematical modelling, and subsequently confirmed in physio-logical experiments. This study demonstrates that mathematicalmodelling is not only a valuable tool allowing one for mechanisticinsights into physiological phenomena, but it provides a means ofdiscovering new phenomena, and thus helps to acquire knowledge.

Editors’ note

Please see also related communications in this issue by Dabiriet al. (2012) and Hales et al. (2012).

Acknowledgements

We are indebted to Professor Peter Kohl for the helpful discus-sions throughout this project, and to Dr. Nico Kuijpers for invalu-able comments on the manuscript. This work is supported by theUral Branch of Russian Academy of Sciences (12-N-14-2009, 12-U-4-1067) and Russian Basic Research Foundation (10-04-00601-a,11-04-00785-a). Pilot studies further benefited from a WellcomeTrust UK-Russia collaboration grant.

Appendix A. Supplementary material

Supplementary material related to this article can be found athttp://dx.doi.org/10.1016/j.pbiomolbio.2012.08.011.

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