Laser trap measurements of piconewton forces and nanometer steps of actomyosin interaction herald a new generation of powerful tools, both in their elegance and precision, with which the behavior of individual molecular events can be investigated. However, there have been some long-standing discrepancies about myosin step size, myosin force, accompanying velocity, and unforeseen problems associated with the experimental measurements, some of which are: (i) Inherent thermal fluctuations that add up to the measurements, (ii) compliance in the system, and (iii) limitations for studying coupled molecular motors. Because of these drawbacks, mathematical models will be of vital importance, providing potentially the only way to fully understand and interpret data from numerous experimental protocols. We use a Langevin-type stochastic model to examine some experimental results for nonprocessive myosin motor's interaction with actin in the laser trap. Our model predictions indicate that, in the detached state, the distribution of the ensuing displacements had approximately zero mean. Attachment events produced displacements with step sizes of approximately 8 nm, which is in agreement with some laser trap experimental results.
Increasing intracellular calcium concentration ([Ca]i) is the critical switch in activating cardiac contraction. Upon electrical activation sarcolemmal Ca current (ICa) is activated and Ca influx triggers release of additional Ca from the sarcoplasmic reticulum (SR). Ca can also enter via Na/Ca exchange (NCX), but this amount is small vs. that via ICa. Ca binds to the myofilaments to activate contraction. For relaxation to occur [Ca]i must decline, allowing Ca dissociation from the myofilaments. Four transporters are involved in this [Ca]i decline: 1) SR Ca-ATPase, 2) NCX, 3) sarcolemmal Ca-ATPase and mitochondrial Ca uniport. We have measured the contributions of these systems, and SR Ca-ATPase & NCX are by far dominant (although their relative contributions vary among species, cell types and in disease. For example, in heart failure (HF) the relative contribution of NCX increases and that of the SR Ca-ATPase decreases. This shift results in a reduction in the SR Ca load available for release, and this is a major factor in systolic dysfunction in HF. We have also used ionic currents as local biosensors to assess spatial gradients of [Ca]i and [Na]i that occur between the bulk cytosol and the junctional region (where the SR is close to the sarcolemma) and just beneath the rest of the sarcolemma. These gradients are produced by rapid transmembrane ion fluxes produced by the channels and transporters in the sarcolemma. The [Ca]i (& [Na]i) in these spaces rises to much higher levels and peaks much earlier than the globally measured Ca transient. The quantitative information gleaned from these types of experiments provides uniquely valuable data in constraining and improving mathematical models used to describe the electrophysiological, Ca handling and contractile properties of the cardiac myocyte and heart.
Many presynaptic terminals contain receptors for the transmitter molecules released from the terminal. Examples include metabotropic glutamate and GABA receptors. Binding to these receptors often causes depression of subsequent transmitter release, either through inhibitory actions on presynaptic calcium channels or activation of presynaptic potassium channels. Interestingly, the inhibitory affects on calcium channels can often be relieved by depolarization. We will discuss mathematical models of this process, coupled to models of transmitter release. Both minimal and more complex models will be described. The impact of autoinhibition on the synaptic filtering of information will be illustrated.
Calcium sparks (Cheng et al., Science 262:740 1993) are microscopic calcium release events inside muscle cells and probably reflect elementary stages in excitation-contraction coupling. While their detection provides information on the probability of calcium release from the sarcoplasmic reticulum (SR), more information could be obtained if we could also measure the SR release flux as this would give greater insight into properties of the junction in terms of the number of release channels open times their single channel current.
From simple estimates of buffering power and change in calcium concentration Cheng et al. (1993) suggested that the calcium spark arose from a SR calcium release flux of about 4pA but this estimate takes no account of the time course of SR release. The time course of Ca release might be measured by a chemical method where a high concentration of Ca buffer is used in combination with a low affinity Ca indicator to make the local Ca signal proportional to the release flux (Pape et al., J. gen. Physiol., 106:259 1995). However tests of this method using 3-dimensionally resolved flash photolysis in drops of test solution revealed that only moderate fidelity could be obtained.
A computational approach involving problem inversion is attractive and tests indicated that some useful information might be obtained despite the extremely low signal to noise ratios associated with the Ca spark. By evoking repeated Ca sparks from single identified within the cell signal averaging could be applied to improve noise statistics and calculations suggested that peak fluxes were somewhat larger than previously supposed. However resolution of release time course was still poor and noise dominated.
As an alternative, we applied a model fitting approach where a flexible basis function was used to describe SR Ca release time course as well as changes in its spatial dimensions. This function was incorporated into a model that described Ca diffusion and reaction with the major Ca binding sites in the cell. The model was then fit to averaged Ca sparks with the sole variables being those associated with the SR Ca release function. The fits were robust and also provide estimates of parameter sensitivity during the fitting process. From this analysis we suggest that the Ca spark arises from multiple SR ca release channels whose open probability declines with time. Given the rather low variability in Ca spark amplititude, this result suggests that the SR Ca release channels time course must be controlled by other factors than just intrinsic SR release channel gating.
Work done in collaboration with M.B. Cannell and C. Soeller.
The motor proteins myosin and kinesin move down their polymers in a cyclic interaction that involves alternate tight binding to the polymer and to nucleotides. Several lines of experimental evidence show that the energy released by the binding of the motor to the polymer is harnessed to produce mechanical work. In a recent study we varied the strength of the actomyosin bond and found that the free energy available to produce force was proportional to the free energy released in the formation of the actomyosin bond. Kinesin is a two headed motor protein that walks down a microtubule. Each head is connected to the coiled-coil stalk by a 15 amino acid region known as the neck linker. Our previous work has shown that the neck linker has 2 conformations, one in which it is docked to the head, and one in which it is undocked. Alternate docking and undocking of the neck linkers of the two heads bias the binding of the unbound kinesin head to the next site towards the plus end of the microtubule. The mechanical energy required to reach the next site is actually produced by thermal fluctuations, which are captured by the tight binding of the kinesin head to the next site. Thus for both motor proteins we conclude that the free energy driving the production of mechanical work is directly coupled to the formation of the motor-polymer bond. Because these are entropically driven reactions, work is performed by a "thermal ratchet" in which a thermal fluctuation is captured by bond formation. In many models of motor mechanisms, such as proposed for myosin by A. F. Huxley in 1957, the motors function via thermal ratchet mechanisms. These mechanisms place restraints on the energetics of the force producing steps. These mechanisms are now sufficiently well defined to allow calculation of the energetics of many of the steps. These estimates can lead to more quantitative models for the motility of both motors.
Fluctuations due to the stochastic nature of biochemical reactions are an inherent property of all biochemical networks. This talk will provide an introduction to the mathematical and computational methods used to understand stochastic effects in signaling pathways. Different mechanisms for converting a graded response to a binary ('all or none') response will be discussed, and the mating pheromone signal in yeast will be used to illustrate these ideas.
Regulation of contraction is generally considered a permissive mechanism in which the regulatory proteins, troponin (Tn) and tropomyosin (Tm), move over the surface of the thin filament following calcium binding to troponin and expose actin sites to which cross bridge attach. It is known, too, that the maximum ATPase rate of unregulated acto-S-1 is the same as that of regulated acto-S-1. However in both motility assays and single fibers whose thin filaments have been reconsitituted, it has been found that Tm and Tn binding to actin in HMM-containing in vitro systems increases both the unloaded thin filament sliding speed (Smax) and the isometric force exerted by HMM on thin filaments attached to microneedles (Po) by about 50%. These effects could be produced by the regulatory proteins increasing either the unloaded cross bridge step size (Smax) and/or unitary isometric force output (Fu) or by increasing the rate of negatively strained cross bridge detachment (Smax) or the number of cross bridges attached during isometric force generation (Po). To examine the effects regulatory proteins on Smax, the variation of thin filament sliding speed in in vitro motility assays was measured at various [MgATP] and [MgADP] concentrations. Plots of the speed versus substrate and/or product concentration revealed that the addition of Tm and Tn to thin filaments markedly increased Smax and significantly increased the Kd for MgADP at physiological concentrations. These results imply that the regulatory proteins speed the rate of release of MgADP from the acto-S-1-ADP complex, thereby increasing Smax. To examine the effects of regulatory proteins on the cross bridge unitary force production, a three-bead optical trap assay system was used. This novel system introduced a bead-position feedback mechanism which greatly reduced the effect of the bead-actin compliance and maintained the position of the actin filament with respect to the myosin cross bridge subsequent to cross bridge attachment. This feedback technique permits measurement of cross bridge force events approaching isometric conditions at a range of ATP and Pi concentrations. At pCa 9, thin filaments containing Tm and Tn did not interact with HMM on the pedestals. However when the calcium contraction was raised to levels sufficient to saturate the Tn calcium-binding sites (pCa=5), thin filaments containing Tm and Tn sustained significant numbers of force producing events. In these latter measurements, we found that histograms of the number of events versus either isometric force exerted during the events or the duration of the events themselves in thin filaments containing only actin did not differ from regulated actin filaments at pCa 5. These latter results suggest that the unitary cross bridge force production by the AM.ADP.Pi complex is not altered by the presence of Tm/Tn. Thus the effects of Tm and Tn on force seen at physiological MgATP concentrations must arise from either an increase in the force exerted by AM.ADP complexes or an increase in the number of cross bridges attached to the thin filaments during isometric contractions at physiological [MgATP]. (Supported by AR30988 [EH], AR45990 [HS and YT], and DH38834 and HL67734 [LST]).
Work done in collaboration with Yasuharu Takagi, Larry S. Tobacman, Henry Shuman, and Earl Homsher.
Calcium sparks are thought to be the most elementary events in coupling the electrical excitation of the heart to contraction. They are caused by stochastic opening of calcium channels. A computational model has been developed to explore the basic mechanisms behind calcium sparks in the heart. The model is a set of stochastic differential equation solved numerically through Monte Carlo simulation. The model has been expanded into a spatial model describing the sarcomere, the basic structural unit of contraction in the heart. The model integrates both elements of the the sarcomere morphology and biophysics and suggests mechanims governing spark dynamics.
Synaptic facilitation is a ubiquitous form of short-term synaptic plasticity, elicited with just a few action potentials, and decaying on time scales of 10s to 100s of ms. Although facilitation is known to depend on the presynaptic accumulation of residual Ca2+, its precise mechanisms are still largely unknown. Here we explore the hypothesis that facilitation may result from the progressive saturation of endogenous Ca2+ buffers. According to this mechanism, gradual reduction of endogenous free buffer concentration during stimulation causes the AP-evoked Ca2+ transients to grow, even if the Ca2+ influx remains constant from pulse to pulse, and in the absence of significant accumulation of residual Ca2+ in free form. Proposed on purely theoretical grounds by Klingauf and Neher (1997), such mechanism has been recently shown to play a role at calbindin-containing neocortical and hippocampal synapses (Blatow et al., 2003; Jackson and Redman, 2003), and Purkinje dendrites (Maeda et al, 1999). Using computational modeling, we systematically explore the conditions on endogenous buffering properties necessary to produce significant facilitation of Ca2+ transients (FCT). In particular, we will show that the buffer mobility is the crucial parameter for synaptic facilitation: interestingly, achieving significant FCT requires endogenous buffers to be either very mobile, or completely immobilized. In the former case FCT results from the global saturation of the buffer in the entire presynaptic terminal, while in the fixed buffer case FCT is caused by the local buffer saturation (Ca2+ influx trapping) in the vicinity of the active zone. Further, we show that the FCT magnitude depends non-monotonically on the total buffer concentration, in agreement with the "pseudofacilitation" effect observed by Rozov et al. (2001). Finally, we will compare our modeling results with the properties of SF recorded at the crayfish neuromuscular junction (NMJ).
Modeling in cardiology gives one of the most exciting and important examples of application of methods of applied and computational mathematics to medicine and biology. Such modeling started more than 40 years ago, form the famous Hodgkin-Huxley model for propagation of excitation waves in nerve cells. Later, the modeling was extended to cardiac tissue (Noble 1962), where in our days we have the most important medical applications. It turned out, that abnormalities in wave propagation in the heart underlay the most dangerous cardiac arrhythmias and sudden cardiac death, accounting for about 1 death in 10 in industrialized countries. I will discuss how the models used for large scale computational projects in electrophysiology have changed over the years and will present our recent work on developing models for human cardiac cells for anatomically based model of human heart.
Defining the conformational changes that occur at the nucleotide site of motor proteins, and their relationship to the generation of force and motion is a fundamental goal of the motility field. One surprise that came from the x-ray crystallographic structures of myosin (an actin-based motor), and kinesin (a microtubule-based motor) was that the significant structural homology between the two classes of proteins, suggesting a common evolutionary ancestor protein. One structural difference between the proteins is in the triphosphate-binding domain at the nucleotide site. In the original structures of myosin, the triphosphates are tightly encased in a structure termed the "phosphate tube". It has three structural elements, the P-loop, switch 1, and switch 2. The switch notation derives from homologous elements in the G-proteins. In kinesin-family motor structures, switch 1 is displaced from the nucleotide, opening the phosphate tube. Given that structural similarity can imply functional similarity, we asked whether myosin structures could help to define previously un-characterized structures in related proteins. Using myosin as a template, the switch 1 domain of the kinesin-family motor, ncd, was deformed to close the phosphate tube. Molecular dynamics simulations then implied that the modified structure was thermodynamically stable. Furthermore, the simulations suggested that this previously uncharacterized state may be crucial for motor function. We conclude that analyses of one class of motor (myosin or kinesin), can provide insights into the function of the other.
We discuss a unifying stochastic model for single motor protein movements and cytoskeletal filament polymerizations. We show that in stationary states, these models can be classified into either equilibrium or nonequilibrium steady-state (NESS) in which free energy transduction occurs. We study the laws of thermodynamics and efficiencies of the energy transductions.
Cardiac cell models have been developed for over 30 years; however, constructing accurate representations of the myofilaments (MF) has lagged far behind. The insufficient representations both hamper the use of model for predicting force development and the effect of force on Ca buffering. We postulate that the difficulty in accurately modeling MF result from the intrinsic cooperative mechanisms that rely on spatial interaction and cannot be directly calculated with ordinary differential equations (ODEs) as with most components of myocyte models. Here we propose an ODE-based model of the MF system that approximates the behavior of spatially explicit models. Our model differs from others by avoiding a mean-field approximation that produces unphysiological Force-Ca (F-Ca) relations. The mean-field approximation tries to represent the entire ensemble of regulatory proteins by a single number (i.e. the mean activation) that ignores any spatial variability. Briefly, our approach assumes thin filament activation is a steeply nonlinear function of [Ca] to represent phenomenologically the effects of nearest-neighbor interactions along the thin filament. Explicit Monte Carlo modeling of this process supports this assumption. Moreover a novel, feature of the model is that Ca binding to troponin is decomposed into "regulatory Ca binding" that activates the thin filament and "apparent Ca binding" that is sensed by the cell. In the real system these binding quantities are equivalent, but here they are separated to avoid the deleterious effects of the mean-field approximation. Our MF model can recapitulate experimentally measured F-Ca relations and twitches, including the length dependent effects that are thought to be the cellular basis of the Frank-Starling Law. Additionally the model is coupled to the Bers-Puglisi rabbit cardiomyocyte model to illustrate the compatibility with existing ODE-based cardiac models and the effects of mechanical effects on Ca buffering.
Work done in collaboration with Fei Wang, Donald M. Bers, and Pieter de Tombe.
The application of mathematical modelling to elucidate the mechanisms of muscle contraction has spanned a wide spectrum of techniques and spatial scales. Models range from the phenomenological to biophysically based, molecule to whole organ. Using the heart as an example the challenge of linking these different frameworks will be explored such that an effect of altering the kinetics at one spatial scale can be included in the behaviour at another spatial scale. For example, the blocking of coronary vessels in the heart disrupts cellular metabolism which inhibits the molecular acto-myosin ATPase and in turn reduces the pumping capacity of the heart.
At the molecular level a stochastic model of actin and myosin protein binding, which incorporates protein filament compliance, will be introduced and shown in the limit of rigid filaments to reduce to the modelling framework based on partial differential equations first proposed by Huxley.
The next step is to use this framework to obtain a computationally efficient model of cellular tension generation which is suitable for embedding in tissue models to predict cardiac mechanical behaviour. Using the distributed Moment Approximation, an inverse approach is developed where a root-finding technique is employed to determine the values of the Gaussian variables such that stiffness, tension and energy dynamics in the cross-bridge model are identical to those in phenomenological models which are used to characterise experimental data.
This calculated active tension is combined with passive constitutive laws and the equations of finite deformation to predict whole heart mechanics. Finally, the computational and mathematical modelling implications of predicting organ level behaviour from disturbed protein interactions for muscle mechanics in heart disease are reviewed.
Cardiac excitation-contraction (E-C) coupling centers on the process that links sarcolemmal Ca2+ influx via L-type Ca2+ channels (DHPRs) to Ca2+ release from the sarcoplasmic reticulum (SR) via ryanodine receptors (RyRs). Since the mid 1980s Ca2+ induced Ca2+ release has been the cornerstone mechanism to mediate the E-C coupling signal transduction process. However, this process has proven experimentally difficult to dissect so that a complete description of how the cell achieves both stable and sensitive intracellular calcium release is still lacking. Mathematical modeling can provide a framework to test hypotheses for how E-C coupling can be both highly responsive and yet deterministic while the triggering signal (Ca2+ influx) is the same as the output signal (SR Ca2+ release). The close apposition of surface and SR membranes in the region where E-C coupling takes place profoundly alters free Ca2+ levels so that 'local control' of E-C coupling is now a central theme in modeling efforts. Nevertheless, most models do not explicity consider local Ca2+ gradients or their temporal evolution. To address this problem we carried out detailed calculations of Ca2+ changes in the diad and several non-linear and unique properties were found. In particular, the large local surface area to volume ratio combined with electrostatic and phospholipid Ca2+ binding lead to an apparent volume-expansion of the cleft space. Simulations of RyR gating showed that RyR activation should be possible on the physiological time scale for realistic values of DHPR Ca2+ influx. In addition, we suggest that there is an optimal DHPR gating time that maximizes responsitivity of the RyRs while minimizing Ca2+ entry. To analyze possible interactions between RyRs we have used Monte Carlo approaches for various geometries. While termination of CICR is problematic for simple models, introduction of allosteric interactions between RyRs increases both responsivity as well as reliability of release termination. It is also notable that a sub-microscopic wave can propagate across arrays of RyRs which will alter the time course of SR release. Current theories lean heavily on the idea that either RyRs exhibit time- and Ca2+ dependent inactivation or that the SR depletion during release plays a role. Although there is experimental evidence for both of these processes mathematical analysis shows that SR release termination can be achieved without either. This result suggests that more caution needs to be applied to the interpretation of experimental results which purport to show either support or refute particular mechanisms.
Work done in collaboration with Mark B. Cannell.
In mammalian cardiomyocytes during excitation-contraction coupling, calcium influx through L-type calcium channels (DHPRs) activates calcium release from juxtaposed ryanodine receptor (RyR) calcium release channels of the sarcoplasmic reticulum. Indirect evidence suggests that individual calcium release events are triggered by single DHPR openings. To clarify the coupling fidelity between DHPRs and RyRs, we inspected the potency of calcium influx through DHPRs to activate calcium release. The responses of calcium currents and local calcium transients to short voltage prepulses were measured in rat ventricular myocytes using whole-cell patch clamp and confocal microscopy and analyzed by equations derived from the law of mass action.
The extent of test pulse calcium current inactivated by the prepulse was directly proportional to the fraction of dyads that exhibited calcium release induced by the prepulse. Therefore, the extent of ICa inactivation by prepulse-induced calcium release was used as the measure of calcium release activation. To describe the relationship between calcium influx and the resultant calcium release, a new variable, relative potency of calcium influx, representing the fraction of calcium influx sensed by the RyRs, had to be introduced. The relative potency of DHPR openings was larger for tail calcium currents following the prepulse than for calcium currents during the prepulse. Moreover, it was larger for the DHPR reopenings than for the first openings during tail current.
Altogether, our analysis suggests that solitary DHPR openings have surprisingly low potency to activate RyRs and trigger calcium release. The potency is dramatically increased if DHPR openings are clustered due to the potentiating effect of the preceding openings on the subsequent openings, which may occur by increasing the basal calcium level and/or prolonging the duration of Ca2+ signals at the RyR sensing sites. In short, the fidelity of DHPR-RyR coupling depends on the recent history of calcium influx.
Support: HHMI HHMI 55000343, VEGA 2/1082/21 (to A. Zahradnikova), VEGA 2/4153/04 (to I. Zahradnik), NIH FIRCA 1 R03 TW05543-01 (to S. Gyorke)
Keywords: calcium signaling, cardiac myocytes, excitation-contraction coupling, calcium current, mathematical modeling
Work done in collaboration with Z. Kubalová, A. Zahradníková, jr., E. Poláková, M. Dura, J. Pavelková, I. Zahradník and S. Györke.