Effects of Extracellular Matrix Deformation on Autocrine Signaling

A Model for Mechanotransduction in Cardiac Muscle: Effects of Extracellular Matrix Deformation on Autocrine Signaling

, IVA N V. M ALY,1 RICHARD T. LEE,1 2 and DOUGLAS A. LAUFFENBURGER1 1Biological Engineering Division, Massachusetts Institute of Technology, Cambridge, MA and 2Cardiovascular Division, Brigham and Women’s Hospital, Boston, MA

(Received 26 May 2004; accepted 27 June 2004)

Abstract—We present a computational model and analysis of the growth factor receptor (EGFR). G protein-coupled recep- dynamic behavior of epidermal growth factor receptor (EGFR) tors can mediate activation of shedding of heparin-binding signaling in cardiac muscle tissue, with the aim of exploring epidermal growth factor (HB-EGF) from the surface of transduction of mechanical loading into cellular signaling that 3 could lead to cardiac hypertrophy. For this purpose, we integrated cardiomyocytes, as they do in other cell types, in response recently introduced models for ligand dynamics within compliant to the pressure overload. The released autocrine HB-EGF intercellular spaces and for the spatial dynamics of intracellular binds to the EGFR and, through the downstream effects of signaling with a positive feedback autocrine circuit. These kinetic receptor activation, triggers the hypertrophic response.1,17 models are here considered in the setting of a tissue consisting In the present study, we performed a computational anal- of cardiomyocytes and blood capillaries as a structural model for the myocardium. We show that autocrine EGFR signaling ysis of the behavior of the EGFR signaling network in can be induced directly by mechanical deformation of the tissue cardiomyocytes, with the aim of exploring transduction of and demonstrate the possibility of self-organization of signaling mechanical loading into cellular signaling that could lead that is anisotropic on the tissue level and can reﬂect anisotropy to cardiac hypertrophy. of the mechanical deformation. These predictions point to the Recently we have found autocrine signaling through the potential capabilities of the EGFR autocrine signaling circuit in mechanotransduction and suggest a new perspective on the cardiac EGFR involved also in mechanosignaling in lung epithe- hypertrophic response. lium, which is possibly implicated in the transformation of this tissue in response to asthmatic bronchoconstriction.32 The central event of the translation of the mechanical stress Keywords—Spatially distributed models, EGFR, Signaling into the cellular signaling in this case was compression of networks. the intercellular space into which the autocrine ligands of EGFR were shed. A several-fold reduction of the intercellu- INTRODUCTION lar volume was registered by two-photon microscopy, and Heart muscle adapts to increases in mechanical load a mathematical model of the shedding and diffusion in the such as high blood pressure in part by undergoing growth, compliant intercellular space predicted that the compres- or hypertrophy, which may serve to normalize stress in sion would lead to an equivalent several-fold increase of the tissue. Although this can be an adaptive physiologi- the concentration of the ligands at steady state. The elevated cal process, for example in response to exercise, chronic concentration of ligands to which the receptors on the cell surface are exposed then sets off the intracellular signaling overload-induced hypertrophy such as that induced by hy- 32 pertension can progress to heart failure.16 At the molecular cascade. level, the hypertrophic response often occurs with reex- In our new work here, we proceed further to consider the pression of embryonic genes and increased synthesis of potential involvement of a similar mechanism of mechan- contractile proteins,11 but the signaling events that lead to otransduction in the heart muscle by means of modeling the these downstream changes are relatively poorly understood. distribution of autocrine EGFR ligands in the heart muscle In particular, it remains largely unclear what mechanism, tissue subjected to mechanical stress. Our recent model for spatial self-organization of autocrine signaling in a single or rather multiple mechanisms, translate the mechanical 18 load into cell signaling. One of these mechanosignaling isolated cell incorporated the effects of spatial distribution pathways involves autocrine signaling through epidermal of all the components of the EGFR signaling network, as well as its nonlinear kinetic properties. Figure 1 shows the elements and processes involved in the EGFR autocrine sig- Address correspondence to I. V. Maly, Biological Engineering Divi- sion, NE47-320, Massachusetts Institute of Technology, Cambridge, MA naling network according to our model. They include shed- 02139. Electronic mail: [email protected] ding of EGF-like ligand from the cell surface, its diffusion 1319

0090-6964/04/1000-1319/1 C 2004 Biomedical Engineering Society 1320 MALY et al.

FIGURE 1. Signaling interactions in the EGFR autocrine network. The shaded area is the cytoplasm; the white area is extracellular space.

in the extracellular space, and binding to a transmembrane pate in the signaling complex formation.15 For the pur- receptor. A ligand-bound receptor recruits the cytoplasmic poses of studying mechanosignaling on the tissue level, this adaptor protein Grb2, which in turn recruits Sos, the actual model complements the model for deformation-dependent activator of signaling downstream from the receptor.25 The accumulation of autocrine ligands in the intercellular fully assembled signaling complex of EGF-family ligand, spaces.32 EGF receptor, Grb2, and Sos causes phosphorylation of Raf Our present model addressing myocardial mechan- on the cytoplasmic side of the membrane. This transforma- otransduction examines potential effects of combining the tion of Raf is reversed in the cytoplasm. The phosphory- previous model for EGFR ligand dynamics in the compliant lated Raf is in its active form, and itself phosphorylates intercellular spaces32 with the spatial kinetic model of intra- cytoplasmic MEK sequentially at two sites. Each phos- cellular EGFR signaling with a positive feedback circuit,18 phorylation step is reversible. The double-phosphorylated now in the setting of a tissue consisting of cardiomyocytes MEK similarly activates ERK, and the positive feedback and blood capillaries. We show that the signaling can be loop is completed by activation of the ligand shedding by triggered directly by mechanical deformation of the tis- , , ERK.7 4 19 There is also a negative feedback loop in this sue, and demonstrate the possibility of self-organization of model, which operates through phosphorylation of Sos that autocrine signaling that is anisotropic on the tissue level is promoted by ERK and renders Sos unable to partici- and reﬂects the anisotropy of the mechanical deformation. Mechanotransduction in Cardiac Muscle 1321

These results of a computational analysis of spatial organization of the EGFR signaling on the tissue level point to the potential capabilities of the EGFR signaling network and suggest a novel perspective for studying not only the cardiac hypertrophic response but more generally other tissue mechanotransduction applications.

MODEL

The nonspatial aspects of the model for the EGFR autocrine circuit that we analyze here are the same as in our previously described model for self-polarization of a single autocrine cell.18 This nonspatial kinetic description is built, as from modules, from models that were developed by Kholodenko and colleagues for the assembly of the receptor signaling complex14 and the phosphorylation cascade.12 The model for the complex assembly was op- timized against experimental measurements. For the pur- poses of spatial modeling,18 the essential feature displayed by the model for the phosphorylation cascade is hyper- sensitivity of the output to the input signal.9 This type of signal transduction through the MAP kinase cascade, of which the Raf-MEK-ERK cascade concerned here is representative, has been observed in experiments and modeled similarly.10 These modules are embedded, providing more mechanistic detail necessary for spatially distributed modeling, in the overall kinetic framework for autocrine EGFR signaling with the positive and negative feedback that was developed by Shvartsman and colleagues.28 The latter model incorporated the measured kinetics of the receptor–ligand interactions and its nonlinear behavior was validated against experiments on tissue culture cells. No parameter values have been changed here compared to our previous model for an individual cell. However, it should be noted that as the size of cardiomyocytes is quite different from the size of the cells at which our previous modeling was aimed, the total concentrations of proteins in the cell are assumed to be conserved between the two models rather than the total numbers of proteins per cell. The total concentration in the cell is defined as the number of protein molecules of a given species in the cell di- vided by the volume of the cytoplasm. While this quantity can be expressed in nM, for example, it does not imply that the spatial distribution of this protein in the cell is uniform. A unique aspect of the model presented here is that the FIGURE 2. Structure of the myocardium. (A) Fluorescence kinetics of EGFR autocrine signaling network is consid- microphotograph of normal mouse heart muscle. Blue, ered as distributed in space under the constraints of the cell nuclei stained with DAPI; red, intercellular spaces geometry of a heterogeneous cellular tissue, the cardiac stained with maleimide-Texas red; green, blood capillaries stained with fluorescein4-lectin. (B) An idealization of the muscle. A micrograph of a section through cardiac muscle three-dimensional structure of the myocardium. Grey paral- in Fig. 2(A) reveals the structure of this tissue as consist- lelepipeds are cardiomyocytes; red cylinders are blood cap- ing of three basic elements: cardiomyocytes, intercellular illaries. Intercellular gap width is exaggerated for illustration only. (C) A section through the idealized structure shown in spaces, and capillaries. An idealization of this structure is (B). Arrows show flux of autocrine EGFR ligands out of the depicted in Fig. 2(B). We assume that the cardiomyocytes cells into the intercellular gaps and then into the capillaries. 1322 MALY et al. are arranged in a regular square lattice with the capillar- the intercellular spaces. Because of the slow spontaneous ies positioned where the corners of four neighboring cells shedding of the ligand, it accumulates in the intercellular meet. We assume in addition, that the capillaries obstruct spaces, until the shedding is balanced by loss of ligand into the intercellular spaces completely at these locations, where the bloodstream. The achieved steady state concentration they serve as sinks for the autocrine ligands that reach them is relatively low, and causes modest activation of the re- in the course of diffusion in the intercellular space. Finally, ceptors, and only a negligible activation of the downstream we assume that the tissue structure and the distributions kinases due to the nonlinear signal transduction through of signaling molecules are uniform along the long axes of the intracellular cascade. After this basal steady state is the cardiomyofibrils. In reality, cardiomyocyte orientation established, we introduce tissue compression by beginning spirals relative to this axis depending on location in the to decrease the width of all intercellular gaps very slowly heart, and capillaries will also change orientation. How- (adiabatically), allowing the system to equilibrate at each ever, this final assumption is made for simplicity and to incremental compression level. The quasi-stationary con- demonstrate the important effects of capillaries. centration of the ligand increases, as a function of the de- Figure 2(C) illustrates qualitative expectations for fluxes creasing gap width, only very slowly in the beginning, caus- of ligands released by the cells that the assumed tissue con- ing a negligible increase of the kinase activity downstream straints imply. The ligands are released from the side sur- [Figs. 3(A) and 3(B)]. faces of the cardiomyocytes, diffuse in the narrow spaces As a certain threshold compression is achieved, however, between the cells, and are lost by convection in the blood- the activation level surges to much higher levels [Figs. 3(A) stream once they reach the capillaries. Diffusion of au- and 3(B)]. The new steady state ligand concentration is tocrine ligands in the intercellular spaces toward a sink, well above its equilibrium dissociation constant from the an unsaturable volume, is a similar situation to the one receptor and ensures saturation of the receptors. As much as we found in an epithelial cell layer, particularly from the one-eighth of the total ERK kinase at the end of the down- lung airways. In the latter system,32 the regulation of the stream cascade is now activated. This is a near-saturation ligand flux by the width of the intercellular gap on the way activation level in the present spatially distributed model to the basal opening into the underlying tissue may result where the activation propagates from the cell membrane in an increased steady state concentration of the ligand to with its receptors, but inactivation resides in the cytoplasm which the receptors on the side membrane of the cells are with its phosphatases.13 When the positive feedback from exposed, and so translate the mechanical deformation of the ERK activation to ligand release is blocked or neglected, cell layer into a cell signaling modulation. In the present achieving a high steady state level of ERK activation is idealization of the cardiac muscle, the ligand is diffusing not prevented, but requires a much higher degree of com- in each intercellular gap of the four that surround a car- pression [Fig. 3(C)]. Moreover, in the absence of this feed- diomyocyte toward the two capillaries at the two ends of back loop activation of ERK by the increasing extracellular the gap. Importantly, the perfect-sink, gap-obstructing cap- concentration of the ligand does not display a threshold illaries in the model effectively isolate every gap between behavior, despite the strong nonlinearity of the signal prop- any two cells from all other gaps in the tissue, as far as the agation in the same kinetic model for the kinase cascade autocrine ligand distribution is concerned. The two sides of under the well-mixed conditions.12 The threshold behavior two cells that face each other in the assumed square tissue of the complete model should therefore be attributed to the lattice are exposed to the same concentration of ligand in positive feedback itself. the space that separates them, because we consider the gaps Importantly, when the positive feedback is operative, (∼0.1 µm) too narrow for a gradient of the fast-diffusing the transition to the high signaling activity cannot be re- (∼1 µm2/s) ligand to be established across the distance be- verted by decreasing the intercellular gap width in the tween the cells. Nonetheless, the gap width is treated in the model back across its threshold value. Both the ligand con- model as an important control parameter and its influence centration and the kinase activation vary nearly linearly at on the signaling distribution in the tissue is explored in the the higher values, whether the gap width increases or de- computational analysis. The mathematical formulation of creases [Figs. 3(A) and 3(B)]. The elevated activity of the the model and its numerical treatment are described in the signaling system is relatively insensitive to the regulation Appendix. because it is locked in the positive feedback. The threshold compression that triggers the self-sustained signaling is a decreasing function of the rate of the spontaneous lig- RESULTS and shedding [Fig. 3(D)]. The plot shows that above some threshold shedding rate, no compression is required, as the Tissue Compression Can Induce Autocrine Feedback spontaneous shedding is sufficient to cause accumulation We begin our simulation from an initial state of the of ligand in a noncompressed intercellular space that will system that does not contain any activated forms of the lead to a suprathreshold receptor occupancy and trigger the intracellular signaling proteins, receptors, or any ligand in feedback. Mechanotransduction in Cardiac Muscle 1323

FIGURE 3. Tissue compression triggers autocrine feedback. Autocrine ligand concentration (A) and ERK activation level (B) exhibit a highly nonlinear dependence on the degree of compression of intercellular spaces, and the course of their variation depends on whether the stress is increasing or decreasing. (C) Autocrine ligand concentration (solid curve) and ERK-activation level (dashed curve) as functions of the intercellular space width, in the absence of the positive feedback from ERK activation to ligand release. (D) The critical compression that triggers signaling, as a function of the rate of spontaneous ligand shedding. ([Sos]total = 5nM, −7 −2 −1 kps = 0, G0 = 10 nmol m s .)

Spatial Organization of Autocrine Signaling in Tissue bution of autocrine signaling in the model cardiomyocytes as a function of the Sos expression level and the strength of The near-saturation signaling activity in the preceding the ERK-Sos negative feedback. simulation has been induced by the compression under a The computation showed that in a range of Sos lev- sufficiently high level of expression (i.e., total cellular con- els, autocrine signaling is sustained in the above four-sided centration) of Sos, the activator of signaling downstream configuration. In particular, all four sides of the model car- of the receptors in our model. Spatially, this active state diomyocyte have a substantial level of receptor occupancy is characterized by equally high concentrations of ligand, by the ligand. Interestingly, a decrease of the Sos level does levels of receptor engagement, and intracellular kinase acti- not cause the signaling to fade away uniformly. As shown vation next to all four sides of our idealized cardiomyocyte. in Fig. 4(B), signaling turns off sequentially at the sides of Recently, we found that decreasing the signal transduction the cell instead. Accordingly, there are four qualitatively capacity of the EGFR autocrine circuit, either by lowering different steady state distributions of self-sustained signal- the expression level of Sos or by increasing the strength of ing in the model cardiomyocyte. An equivalent succession the ERK-Sos negative feedback, did not result in a mono- of four-, three-, two-, and zero-sided signaling configura- tonic and homogeneous decrease of the signaling activity in tions is obtained when the negative feedback is enhanced the cell. Rather, both the ligand release and the intracellular in the simulation [Fig. 4(C)], starting from the same initial activation concentrated on one (arbitrarily oriented) side of condition. a spherical isolated cell when the signal transduction capac- The corresponding distributions of the intracellular sig- ity did not allow for the uniformly high autocrine activity naling are shown in Fig. 5 and reveal the reason why there all over the cell.18 We decided to explore the spatial distri- is no state in which the receptors would be activated on 1324 MALY et al.

FIGURE 4. Spatial distribution of autocrine signaling at different levels of expression of Sos. The diagram in part A introduces the coordinate axis of the position along the cell perimeter, which is used in part B and other figures. (B) Color-coded receptor occupancy as a function of the position on the cell surface and Sos expression level. The color bar in part B also pertains to all the other color-coded plots. (C) Receptor occupancy distribution at varied strength of negative feedback. The feedback strength is measured as the rate constant of ERK-dependent Sos phosphorylation. (w = 0.25 µm, kps = 0(B),[Sos]total = 5nM(C),G0 = 0.) but one side of the myocyte. Although in the proximity of was considered by Postma and Van Haastert20). Under these the capillaries, which are functioning as sinks for the ex- conditions of local activation and global inhibition, the ac- tracellular ligand at the corners of the myocytes, the ligand tivation spills over into regions of the cell (like other cor- concentration and receptor occupancy are lower than near ners), where the inactivation has taken hold, and therefore the middle of the cell sides, the intracellular activation of is unable to turn these regions into activated ones. ERK is favored at the cell corners. This could be explained For a spatially distributed kinetic system of this com- by a higher effective concentration of the membrane and plexity ascertaining how many stable states exist for any receptors at the corners, where more membrane is exposed given set of parameters does not appear to be an easy propo- to a given volume of cytoplasm where the downstream sition. For instance, the three-corner state can be achieved kinases reside. Therefore, the minimal signaling configura- by starting from the four-corner state and then diminishing tion is activation concentrated at one corner, with two cell slowly (adiabatically) the cellular pool of Sos beyond some sides emitting ligands at an appreciable rate. That the effect threshold. But is the three-corner state the only stable state of the surface-to-volume ratio dominates over the effect of this system that would be characterized by a signaling of the extracellular ligand distribution should be attributed activity above the low basal level, at a specific Sos level to the fact that signal-response nonlinearity in our model where it is achieved by the adiabatic sweep? It would be resides primarily downstream of the receptors. an interesting possibility if the number of active corners The distribution of the signaling to just some of the went down stepwise as the overall signaling in the cell was corners and not the other can be stable because this sig- gradually inhibited by lowering the Sos levels because the naling system displays the properties of local activation system was only losing the signaling activity spontaneously and global inhibition, the conditions defined by Gierer and at the bifurcation points, jumping each time to the lower of , Meinhardt for Turing-type spatial self-organization.8 33 The the stable branches of solutions. That another four-corner spatial range of activation of signaling is restricted because solution exists as a higher-activation, stable solution branch the kinases of the cascade become inactivated by dephos- alongside the three-corner branch, may not be excluded phorylation before they can diffuse more than a fraction apriori. We searched for such solutions numerically. The of the cell size away from the point of activation, and the state of this signaling system can be perturbed efficiently autocrine ligands are likely to get lost into the environment by changing the relative amounts of the different receptor (bloodstream) than bind to a receptor after diffusing farther complexes, because the rates of their mutual transformation , , from the point of their release.2 18 30 The signal-induced are much lower than the rates of the transformations in the inhibition of signaling is global because the cellular pools kinase cascade and of diffusion. It was found that a simu- of Grb2 and Sos, the intracellular components of the signal- lated binding of ligand to all of the free receptors always ing complex, are depleted as these proteins bind to locally led, in the course of the relaxation of the perturbation, to activated receptors, and the intracellular diffusion assures the same steady state that existed before the perturbation. that the lowered concentrations of Grb2 and Sos become Specifically, the three-corner state relaxed back to the three- uniform throughout the cell18 (an analogous mechanism corner state, and the two-corner state relaxed back to the Mechanotransduction in Cardiac Muscle 1325

FIGURE 5. Distributions of activated, double-phosphorylated ERK in the tissue cells at 5, 4, 3, and 1 nM of cellular Sos are shown in parts (A)–(D), respectively. The cell boundaries are shown in black (the intercellular spaces are too narrow to appear on this plot). Color coding of the fractional activation of ERK as defined by the color bar in Fig. 4(B). Compare these distributions of intracellular signaling with the four patterns of receptor occupancy distributions at these values of Sos in Fig. 4(B), and also with the four similar patterns in Fig. 4(C). (w = 0.25 µm, kps = 0, G0 = 0.) two-corner state. Therefore, it seems likely that the high- the most active side of each cell faces its counterpart in activation states are unique, symmetry variants not being the neighboring cell. As a result, parallel planes (shown considered, across the range of the Sos levels. It is our per- in cross-section in the figure) of elevated signaling activity ception that the model does not possess multiple, diverse span the tissue. The orientation of the planes [horizontal stable solutions. rather than vertical in Fig. 5(C)] is arbitrary in this case Among the four types of the qualitatively different sig- because of the symmetry of the square lattice tissue geom- naling distributions (Fig. 5), the one with two active cor- etry. In the simulation, one or the other orientation will be ners stands out. This signaling configuration [Fig. 5(C)] achieved depending on the practically random inaccuracies is anisotropic on the tissue level. In general, a stable ar- of the numerical solution that proceeds from a symmetric rangement of cells in a given state of signaling is achieved initial signaling distribution. For comparison, starting from when a side in one cell faces its counterpart, in terms of the the same initial condition, a different simulation, in which cellular signaling distribution, of the neighboring cell, in the negative feedback was enhanced, lead to establishment the symmetric arrangement. In the case shown in Fig. 5(C), of the most active signaling at another, vertical side of the 1326 MALY et al. cell [compare Figs. 4(B) and 4(C)], which is of course achieved [Fig. 6(B), right], all four corners of each cell be- again only a random choice. A different, systematic origin come active again. However, the intracellular signaling now of anisotropy is considered in the next section. spreads from the corners along the vertical, but not horizontal, sides of the cell [Fig. 7(B), bottom]. Thus, the signaling distribution at high stress is different from both distribu- Induction of Signaling Anisotropy tions that were found at a lower stress. Most importantly, by Anisotropic Compression the planes of signaling at the tissue level are now orthogonal We next explored the effect of anisotropic, unidirectional to the direction of the compression [Figs. 7(A) and 7(B), compression of the tissue. Specifically, horizontal compres- bottom]. sion was modeled by very slowly decreasing the width of The ultimate redistribution of signaling into the planes the intercellular gaps along the vertical sides of the cell, orthogonal to the direction of tissue compression should and correspondingly widening the horizontal gaps to keep be attributed to the fact that the intercellular gaps that run the total volume of all the intercellular spaces constant. orthogonally to the direction of compression become nar- The simplification that the cell itself was not deformed was rower. This would allow for a build-up of higher concen- motivated by the fact that cell aspect ratio only changed by trations of autocrine ligands that are trapped more effec- about 10% when the gaps were compressed about eightfold tively in a narrower gap on their way to the capillary.32 in the previous experiments.32 We found that the signaling The intracellular distribution of signaling should follow the distribution, as evidenced by the distribution of the receptor extracellular one as the kinase cascade is more efficiently occupancy along the cell perimeter (Fig. 6), undergoes two activated next to the membrane exposed to the higher ligand abrupt transitions as the stress is gradually increased. This concentration. two-threshold behavior was observed whether the simula- The two-step response of the tissue in the basal state to tion began in the basal state of signaling in which the posi- the increasing compression, the sequential activation of sig- tive feedback had not had been triggered, or when it began naling at the two cell sides that face the intercellular spaces from the state of with the activated, four-sided signaling being compressed, is comparatively easy to understand. As configuration. the receptor engagement is accompanied by inhibition of The diagram in Fig. 7 shows the characteristic steady the signal propagation downstream due to depletion of Grb state arrangements of intracellular signaling in the tissue at and Sos that bind to the occupied receptors, there can be a the three stages that the stress-induced transitions separate. degree of compression that is sufficient to trigger the posi- Starting from the basal state of signaling, a certain threshold tive feedback, but insufficient to overcome the global intra- compression is necessary to trigger the positive feedback cellular signaling inhibition, so the signaling is restricted [Fig. 7(A)]. After this threshold is achieved, the signaling to just one of the two sides that are favored by compres- is triggered primarily along one side of the cell, one of sion. Still, the existence of such an intermediate stage of the two facing the compressed intercellular gaps. Beyond activation should depend on the exact nonlinearity of the the second threshold, both sides facing the compressed gaps system. become active. Starting from the symmetric, four-sided sig- Explanation of why the planes of signaling are aligned naling configuration [Fig. 7(B)], and before some threshold with the direction of tissue compression, when the com- horizontal compression is achieved, the signaling distribu- pression is not as strong and the initial state is the uni- tion retains its original symmetry to a large extent: all four formly activated one, appears more problematic. At the corners in a cell have a high level of ERK activation, and all lower load, the “corner effect” appears to play the major four sides emit sufficiently large amounts of ligand to oc- role that favors concentration of intracellular signaling in cupy the receptors [compare Figs. 6(B) and 7(B)]. Beyond the cell corners where the surface–volume ratio is higher, the first threshold level of the compression in the horizontal as discussed above. The compression itself favors signal- direction, two of the cell corners lose the high ERK acti- ing along planes orthogonal to it, but it takes a stronger vation. The ERK activation redistributes to the two other compression to enable signaling away from the corners corners, without changing appreciably on a per cell basis. than it takes to destabilize the four-corner signaling. This Remarkably, the active corners belong to one side of the explanation is not quite satisfactory, however, and this case cell, so the tissue-level steady state arrangement becomes may for now represent an example of nonlinearity defying anisotropic. In this regard, it is similar to the self-organized intuition. anisotropic arrangement [compare Fig. 7(B), middle, with Fig. 5(C)]. However, the direction of the high-signaling Hysteresis of Anisotropy planes is now systematic, aligned with the direction of compression. As in the case shown in Fig. 7(B) the compression When the Sos expression levels are moderate and in the is horizontal, so is the direction in which the stripes of high absence of deformation only two corners of a cell have a signaling level run in the shown cross-section of the tis- high ERK activity, application of a significant directional sue. After the second threshold of the gap deformation is compression drives the system to a similar state to the one Mechanotransduction in Cardiac Muscle 1327

FIGURE 6. Receptor occupancy as a function of position on the cell surface and deformation of the intercellular spaces. The relative deformation is defined as fractional narrowing of vertical spaces and fractional widening of horizontal spaces. Color-coding as in Fig. 4(B). (A) The undeformed steady state is the state with the basal level of signaling, without a significant activation of the positive feedback, as on the lower-right solution branch in Figs. 3(A) and 3(B). (B) The undeformed steady state is equal signaling −7 −2 −1 on all four sides of a cell, with the positive feedback activated, as in Fig. 5(A). ([Sos]total = 5nM,kps = 0, G0 = 10 nmol m s (A), G0 = 0(B).) shown in Fig. 7, bottom. Under a slow relaxation of the ap- of compression (compare Figs. 8 and 9). This new arrange- plied stress, the system then exhibits an interesting behav- ment remains stable when the stress is completely removed. ior. At first, it jumps from the four-corner to the two-corner Intrinsically, the state of the system is now the same as it anisotropic state, retaining the orientation of the signaling was before the application of stress, which was the same as planes in the tissue that is perpendicular to the direction shown in Fig. 5(C). But the orientation of this anisotropic 1328 MALY et al.

FIGURE 7. Transformations of the distribution of active ERK in the tissue as compression increases. The distributions shown here are representative examples of the qualitatively different signaling states shown in Fig. 6. The solid arrows denote the transitions, the block arrows show the direction and strength of the load; other conventions as in Fig. 5. (A) Effect of deformation on tissue that initially had not activated signaling [see Fig. 6(A)]. (B) Effect of deformation on tissue that initially had activated signaling in all four corners of every cell [see Fig. 6(B)]. distribution is not random now, but set by the preceding tion cycle. Anisotropy with the signaling planes parallel to deformation cycle: still orthogonal to the compression that the direction of compression is maintained until a certain has been relieved. degree of compression is achieved. Then the system jumps A gradual application of compression in the orthogonal into a state with the signaling planes running orthogonally direction can be seen as continuation of the same deforma- to the direction of stress. This state could be achieved by Mechanotransduction in Cardiac Muscle 1329

FIGURE 8. Receptor occupancy as a function of position on the cell surface and deformation of the intercellular spaces, when the undeformed steady state is anisotropic signaling as in Fig. 5(C). The arrows show the direction of change of the deformation; other conventions as in Fig. 6. ([Sos]total = 4.3 nM, kps = 0, G0 = 0.) rotation of the state strongly compressed in the other direc- anisotropy in the absence of stress, exhibits hysteresis of tion, and the two threshold stress levels are also the same the orientation of the signaling anisotropy with respect to for the two directions of compression. What appears to be the direction of the deformation. an important property of the system is that as the stress is relaxed again to complete the cycle, the distribution of DISCUSSION signaling retains the orientation of the signaling planes that it had at the time of the last extreme compression (Fig. 9). Our computational analysis of a spatially distributed ki- Thus, the system that is capable of self-organization of netic model of autocrine EGFR signaling in heart muscle 1330 MALY et al.

FIGURE 9. Transformations of the distribution of active ERK in the tissue as compression changes between horizontal and vertical. The distributions shown here are representative examples of the qualitatively different signaling states shown in Fig. 8. The solid arrows denote the transitions, the block arrows show the direction of the load, where present; other conventions as in Fig. 5.

tissue predicts that mechanical load on the tissue leads to Our results indicate that capillaries may have a crucial triggering and complex modulation of the cellular signal- role in local EGF signaling. In normal myocardium, a cap- ing. The EGFR signaling system plays an important role in illary is adjacent to every myocyte, allowing diffusion of cardiac hypertrophy and is a candidate system for mediat- oxygen from blood to the mitochondria in the center of ing transformation of mechanical forces into hypertrophic the relatively large myocyte. Capillaries are likely to be responses. Expression of HB-EGF is increased in hyperten- effective at removing ligands by convection. In our ideal- sive rats,6 and activation of EGF receptors causes growth ized model, the corner geometry of the myocyte increased of cardiomyocytes.24 In addition, EGF receptor transacti- activation due to surface-to-volume relations. Our results vation plays a central role in angiotensin II-induced cardiac suggest that in normal myocardium, the variable myocyte hypertrophy.26,31 Our results indicate that compression of geometries and capillary locations may regulate EGF re- the extracellular matrix may narrow intercellular gaps and ceptor activation. trigger the signaling through the autocrine feedback. This The spatially distributed and tissue-level nature of our finding can be explained on the basis of our earlier result32 new model allowed us to find in addition the different spa- that narrowing an intercellular gap makes the autocrine lig- tial patterns of the autocrine signaling at the tissue level. ands that have been shed into the gap accumulate to higher The spatial-pattern formation in tissues of autocrine cells concentrations. The present model, however, additionally is a widespread phenomenon that plays a major role in predicts an existence of a threshold compression that causes development of organisms and has been subject of mod- an abrupt and irreversible increase of the level of the sig- eling research, particularly as regards the autocrine EGFR naling activity. This type of behavior with the threshold signaling.22,23,29 Here, we predict that the signaling distri- and bistability was exhibited by a previous model that also bution in cardiac muscle can be anisotropic. The anisotropy, incorporated the positive feedback of receptor stimulation according to our computations, may be a result of a spon- on the ligand release.28 In this earlier model, the triggering taneous self-organization. In this respect, it represents a receptor activation was caused by ligand exogenous to the higher-level self-organization via interactions of autocrine cell. In contrast, in the present model the ligand that triggers cells that possess the self-organizational capacity on their the feedback is also of an autocrine origin, and the increase own, as in our single-cell model,18 but are now coordinated of its concentration to which the cell is exposed is caused through sharing the common pool of autocrine ligands in by narrowing the intercellular gap as a result of the tissue the intercellular spaces that separate them in the tissue. compression. The anisotropy can also be induced in our model by an Mechanotransduction in Cardiac Muscle 1331 anisotropic mechanical stress in the tissue that makes some naling response in the present model, as well as in our lung of the intercellular gaps wider and other smaller, depending mechanotransduction model,32 is easier to understand as a on their orientation with respect to the direction of com- behavior of the whole system. pression. Interestingly, the orientation of the planes of the The present model rests on a number of arguable as- elevated signaling activity in the tissue can be orthogonal to sumptions. So, some of the parameters that characterize the the compression or aligned with it, depending of the level connections between the modules in our model, such as of the stress. The intrinsic anisotropy can also be modulated the rate of Raf activation by the receptor complex and the by the application of stress, in which case the orientation kinetic strengths of the feedback interactions, were given of the signaling planes exhibits hysteresis as the direction ad hoc, if physically and biologically plausible, values.18 of stress is altered. Computational analysis presented here and in the previous It should be emphasized that the current model does paper,18 however, demonstrates that the essential features not predict the orientation of the signaling planes in the of the behavior of the model do not depend on the specific otherwise symmetric, uncompressed model tissue. This ori- choice of the parameter values, but rather some qualitative entation originates from asymmetric errors of the numer- type of behavior is exhibited in a range of the parameter val- ical simulation. Also, the planes are infinite in the ideal ues. On the other hand, the geometry in the model makes a geometry at steady state. In reality, the geometry is far rather strong idealization of the structure of cardiac muscle. from ideal, and the sources of asymmetry that can bias In our assessment, though, the idealization of the geome- the self-organization of anisotropy should be numerous, try should match the crudeness and uncertainty associated including the stochastic nature of signaling events and with the very kinetic description of the signaling network influence from the myocardium environment. The self- involved. Making these kinetic and structural assumptions organized anisotropy under these conditions should be only allows us to explore the spatial behaviors of the system that local, and even the anisotropy biased strongly by compres- can be expected to hold, qualitatively, in the more realistic sion should be confined to finite domains in the tissue. geometry and with better substantiated parameter values. It We believe that these potential capabilities of autocrine is our goal, by demonstrating the implications of the cho- EGFR signaling in the heart muscle, unanticipated without sen assumptions for the spatial organization and behavior of a computational analysis of a spatially distributed model signaling at the tissue level, to draw attention to the potential of the signaling network involved, point to some new di- of a more detailed quantitative experimentation and more rections of inquiry into the mechanisms of physiological realistic modeling of these aspects of tissue organization. adaptation and pathophysiology of the myocardium. So, At this stage, the predictions of our modeling are the poten- the irreversibility and hysteresis found in the model imply tial unanticipated modes of the system dynamics. Whether that the tissue possesses a type of long-term memory of they are realized in the actual heart muscle, and what the its previous mechanical load. Given the involvement of relevant kinetic and structural conditions actually are, are the autocrine EGFR signaling in heart hypertrophy,1 the the questions to be answered in the course of quantitative possible anisotropic distribution of signaling that is related, experimentation that our modeling may help motivate. although not necessarily univocally, to the anisotropy of mechanical stress also indicates the possibility of anisotropic APPENDIX hypertrophic response of the myocardium, attuned physio- logically or pathologically to the directionality of the local The model is described by a set of reaction-diffusion load on the tissue. kinetic equations. The spatial coordinates are x and y; these It is remarkable that the signaling system in our model orthogonal axes are also orthogonal to the long axis of a responds to the mechanical deformation, but no element cardiomyocyte. They cross the cell membrane at x = x1, x2, of it can be appropriately assigned the function of the re- and y = y1, y2. By assumption, the concentrations of the sponse element, or mechanosignaling transducer. Thus, no chemical species that are studied here do not depend on the signaling protein in this model changes its conformation coordinate along the cell axis. The extracellular concentra- as it experiences the mechanical load, which would change tion of EGFR ligand is denoted as L, the surface density of its properties with regard to signaling interactions. Instead, EGFR as R, and the surface density of ligand–receptor com- a narrower intercellular gap, as a geometrical property of plexes as C. Volume concentrations and surface densities the tissue subject to change with the load applied, means a of other species are denoted as [species], using the symbol higher concentration of ligands being released into it, which C for the extracellular ligand–receptor component of the could at some point exceed a threshold of the nonlinear transmembrane complexes, and Grb for Grb2. Nomencla- signal transduction downstream of the receptors,5 and trig- ture and values of parameters are given in Table A1. ger the autocrine feedback. Although in general the dis- All of the equations that pertain to signaling in the cell tinction between an intrinsic property of an element in a membrane and inside the cell are identical to the ones in system and an emergent, system-level behavior is difficult our model for a single cell18 and are listed below for con- to delineate unambiguously, we notice that the mechanosig- venience. The boundary conditions are different insofar as 1332 MALY et al.

TABLE A1. Kinetic and geometric parameters.

Symbol Parameter Value

w Intercellular gap width 0.02–0.75 × 10−6 m s Cell size 2.5 × 10−5 m −12 2 −1 DL Diffusion constant of ligand 10 m s a −3 −1 koff Rate constant of dissociation of ligand from EGFR 1.67 × 10 s a 6 −1 −1 kon Rate constant of association of ligand with EGFR 1.67 × 10 M s −5 −2 −1 G1 Maximal rate density of induced ligand shedding 10 nmol m s −7 −2 −1 G0 Rate density of spontaneous ligand shedding 0–10 nmol m s D Diffusion constant of proteins in cytoplasm 10−12 m2 s−1 b 6 −1 −1 kagc Association rate constant of Grb2 with receptor 3 × 10 M s b −2 −1 kdgc Dissociation rate constant of Grb2 from receptor 5 × 10 s b 6 −1 −1 kagsc Association rate constant of Grb2-Sos with receptor 4.5 × 10 M s b −2 −1 kdgsc Dissociation rate constant of Grb2-Sos from receptor 3 × 10 s b 7 −1 −1 kasc Association rate constant of Sos with receptor-Grb2 1 × 10 M s b −2 −1 kdsc Dissociation rate constant of Sos from receptor-Grb2 6 × 10 s b 5 −1 −1 kasg Association rate constant of Sos with Grb2 1 × 10 M s b −3 −1 kdsg Dissociation rate constant of Sos from Grb2 1.5 × 10 s 9 −1 −1 kps Phosphorylation rate constant of Sos 0–1 × 10 M s −3 −1 kdps Dephosphorylation rate constant of pSos 1 × 10 s −7 [Raf]tot Total concentration of all forms of Raf 1 × 10 M −7 [MEK]tot Total concentration of all forms of MEK 3 × 10 M −7 [ERK]tot Total concentration of all forms of ERK 3 × 10 M −1 k1 Rate constant of Raf phosphorylation 6.17 s −8 K1 Michaelis constant of Raf phosphorylation 1 × 10 M −9 −1 V2 Maximal rate of pRaf dephosphorylation 7.5 × 10 Ms −9 K2 Michaelis constant of pRaf dephosphorylation 8 × 10 M −1 k3 Rate constant of MEK phosphorylation 0.75 s −8 K3 Michaelis constant of MEK phosphorylation 1.5 × 10 M −1 k4 Rate constant of pMEK phosphorylation 0.75 s −8 K4 Michaelis constant of pMEK phosphorylation 1.5 × 10 M −8 −1 V5 Maximal rate of ppMEK dephosphorylation 2.25 × 10 Ms −8 K5 Michaelis constant of ppMEK dephosphorylation 1.5 × 10 M −8 −1 V6 Maximal rate of pMEK dephosphorylation 2.25 × 10 Ms −8 K6 Michaelis constant of pMEK dephosphorylation 1.5 × 10 M −1 k7 Rate constant of ERK phosphorylation 0.75 s −8 K7 Michaelis constant of ERK phosphorylation 1.5 × 10 M −1 k8 Rate constant of pERK phosphorylation 0.75 s −8 K8 Michaelis constant of pERK phosphorylation 1.5 × 10 M −8 −1 V9 Maximal rate of ppERK dephosphorylation 1.5 × 10 Ms −8 K9 Michaelis constant of ppERK dephosphorylation 1.5 × 10 M −8 −1 V10 Maximal rate of pERK dephosphorylation 1.5 × 10 Ms −8 K10 Michaelis constant of pERK dephosphorylation 1.5 × 10 M 15 −2 Rtotal Total surface density of receptors and complexes 1.75 × 10 m −9 [Grb]total Total cellular concentration of Grb2 and complexes 1.8 × 10 M −9 [Sos]total Total cellular concentration of Sos and complexes 0–5 × 10 M

aFollowing Shvartsman and colleagues.30 bFollowing Kholodenko and colleagues.14 they represent a different cell and tissue geometry. The [ppERK] equation for the ligand dynamics is also different because × G0 + G1 − konRL + koffC [ERK] the intercellular spaces are much thinner than a cell, and tot we assume that diffusion across the intercellular distance ∂ R = k C − k RL is always at equilibrium. The capillaries as sinks for the ∂t off on ligand at the corners of the cell are modeled by the impos- ∂C ing L = 0 at those positions.The dynamics at each of the = k RL − k C − k [Grb]C + k [CGrb] ∂t on off agc dgc four sides of the cell are described by the following set of − + equations: kagsc[GrbSos]C kdgsc[CGrbSos] ∂ L 1 ∂[CGrb] = D L + = kagc[Grb]C − kdgc[CGrb] ∂t L w ∂t Mechanotransduction in Cardiac Muscle 1333 − + kasc[Sos][CGrb] kdsc[CGrbSos] − V5[ppMEK] , ∂[CGrbSos] K5 + [ppMEK] = kagsc[GrbSos]C − kdgsc[CGrbSos] ∂t ∂[pERK] = D[pERK] + kasc[Sos][CGrb] − kdsc[CGrbSos]. ∂t − − In the diffusion term of the equation for the ligand concen- + k7[ppMEK]([ERK]tot [pERK] [ppERK]) tration, denotes the Laplacian operator in one dimension, K7 + ([ERK]tot − [pERK] − [ppERK]) which is either x or y, depending on which side of the cell V9[ppERK] k8[ppMEK][pERK] the intercellular space is adjacent to: + − K9 + [ppERK] K8 + [pERK] ∂2 L = L, − V10[pERK] , ∂x2 K10 + [pERK] for intercellular spaces that run “horizontally,” and ∂[ppERK] k [ppMEK][pERK] = D[ppERK] + 8 ∂2 ∂t K + [pERK] L = L, 8 ∂y2 − V9[ppERK] , for intercellular spaces that run “vertically.” K9 + [ppERK] The fluxes of the cytoplasmic species off the cell mem- where denotes the Laplacian operator in the two dimen- brane are defined as sions, x and y, = − , IGrb kdgc[CGrb] kagc[Grb]C ∂2 ∂2 = + . 2 2 ISos = kdsc[CGrbSos] − kasc[Sos][CGrb], ∂x ∂y µ IGrbSos = kdgsc[CGrbSos] − kagsc[GrbSos]C, The model was discretized in space with the step size 2.5 m for x and y, using the finite difference approximations for k [CGrbSos]([Raf] − [pRaf]) I = 1 tot , the spatial derivatives of variables. This transforms, for pRaf + − K1 ([Raf]tot [pRaf]) example, an equation for a variable f that represents the and zero for pSos and all forms of MEK and ERK. The concentration of a cytoplasmic protein, such as phospho- dynamics of the cytoplasmic species is described as follows: ERK, into Nx by Ny ordinary differential equations (ODEs), ∂[Grb] one for each node on the x–y space grid. Using central differ- = D[Grb] − k [Sos][Grb] + k [GrbSos], ences, the spatial (diffusion) term in the partial differential ∂t asg dsg equation for f is approximated, in the ordinary differential ∂ Sos [ ] equation for f , , the value of the variable f at the grid node = D[Sos] − kasg[Sos][Grb] + kdsg[GrbSos] i j ∂t that has number i along the x dimension and number j along − kps[ppERK][Sos] + kdps[pSos], the y dimension, by

∂[GrbSos] fi−1, j − 2 fi, j + fi+1, j fi, j−1 − 2 fi, j + fi, j+1 = D[GrbSos] + k [Sos][Grb] + , ∂t asg (x)2 (y)2

− kdsg[GrbSos], where x and y are the intervals between the nodes along ∂[pSos] the x and y dimensions on the uniform grid. = D[pSos] + k [ppERK][Sos] − k [pSos], The resultant large, sparse ODE problem was solved ∂t ps dps with the MATLAB (The MathWorks Inc., Natick, MA) ∂[pRaf] V [pRaf] = D[pRaf] − 2 , routine ode15s, a variable-order solver for stiff problems ∂t K2 + [pRaf] that is based on the numerical differentiation formulas and ∂[pMEK] uses an adaptive time step size. The relative error tolerance = D[pMEK] ∂t was set to 0.001. In the initial condition, all variables were set to zero, k [pRaf]([MEK] − [pMEK] − [ppMEK]) + 3 tot except [Grb] = [Grb] ,[Sos] = [Sos] , R = R .Re- + − − total total total K3 ([MEK]tot [pMEK] [ppMEK]) laxation of an initial condition was followed by an adiabatic + V5[ppMEK] − k4[pRaf][pMEK] change of the control parameters, one at a time. The adia- K5 + [ppMEK] K4 + [pMEK] batic change of the total concentration of Sos in the numerical procedure is equivalent to adding a very small constant V [pMEK] − 6 , term to the equation for the dynamics of this species. + K6 [pMEK] These simulations were done for one cell only and as- ∂[ppMEK] k [pRaf][pMEK] suming w = w/2 to represent a symmetric ligand release = D[ppMEK] + 4 ∂t K4 + [pMEK] from the other cell facing the same intercellular space. To 1334 MALY et al. obtain the distribution of signaling in the tissue, the one- cell distributions were combined in the square lattice in such a way that the symmetry at the interfaces was preserved, and a simulation was run for the 4 by 4 cell lattice with the same symmetry conditions at its boundaries. All the proper tissue arrangements of the stable single-cell solutions proved numerically stable. We note parenthetically that, first, unstable single-cell solutions and combinations that violated the symmetry were unstable in the tissue simulations and evolved into the same configurations that could be obtained by relaxing the single cells first and then ar- ranging them properly in the contiguous tissue, and second, that the procedure chosen was found efficient in terms of the computation time. The algorithm behind the ode15s routine27 is based on numerical differentiation formulae (NDF) and is fully implicit. In the framework of the von Neumann stability analysis21 the first-order NDF27 with the central-difference discretization in two spatial dimensions leads to the following eigenmode amplification factor: √ 1 − 2κ ± 1 + 4κα Dt kx ξ = ,α= 8 sin2 , 2(1 − κ + α) (x)2 2

where D is the diffusion constant, t and x are the time and space intervals (equal intervals in both x and y spatial dimensions), κ<0 is an NDF parameter,27 and k is the wave number of the eigenmode. It can be seen from this expression that at the minimal α = 0, the larger value of ξ equals 1, and that this value decreases remaining positive as α increases to −1/(4κ). For still larger values of α,|ξ|2 becomes −κ/(1−κ+α), which is again a decreasing function of α. Thus, the stability criterion |ξ| ≤ 1issimplyα> FIGURE 10. Benchmarking of the numerical procedure. (A) 0, which is true for any t and x, demonstrating that this Analytical (solid curve) and numerical (crosses) solution to the mimic model, with 11 spatial nodes and relaxation time numerical scheme is unconditionally stable. 105. (B) Square space interval vs. maximal absolute value of For benchmarking the numerical procedure, we com- the relative error in the numerical solution. pared the numerical results with the analytical solution to the following mimic model: ACKNOWLEDGMENTS ∂y ∂2 y = D − ky, y| = = y , y| = = y . ∂t ∂x2 x 0 0 x L L This work was partially supported by NIH grants R01- GM62575 to DAL and P01-HL64858 to DAL and RTL. With D = 1, k = 1, L = 10, it has the same characteristic times of “reaction” and “diffusion” as our model for intra- REFERENCES cellular signaling has. The analytical steady state solution to this model is 1Asakura, M., M. Kitakaze, S. Takashima, Y. Liao, F. Ishikura, √ √ T. Yoshinaka, H. Ohmoto, K. Node, K. Yoshino, H. Ishiguro, − k √ k √ H. Asanuma, S. Sanada, Y. Matsumura, H. Takeda, S. Beppu, − D L D L − yL y0 e k x y0 e yL − k x y = √ √ e D + √ √ e D M. Tada, M. Hori, and S. Higashiyama. Cardiac hypertrophy is k L − k L k L − k L e D − e D e D − e D inhibited by antagonism of ADAM12 processing of HB-EGF: Metalloproteinase inhibitors as a new therapy. Nat. Med. 8:35– = = | 40, 2002. The numerical solution with y0 yL 1, y t=0,0

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