A.4. Specific Aim 2: Integration of the Approaches Developed for Different Scales

A.4.1. Breadth of molecular and mathematical approaches. The development of physically realistic but mathematically tractable models has been a major goal in the uppermost (mathematical) and lowermost (molecular) approaches, in particular. Extensive progresses have been made in these two groups of studies in the last three decades (see Table A.2). We can refer to two reviews from the pre-NPEBC leading PIs at these two levels (103;106), which provide an overview of the models (discrete and continuous) and methods (analytical and numerical) that have been developed by molecular (106) and mathematical (103) modelers for studying macromolecular and cellular dynamics, respectively. A recent paper by Bahar and coworkers also typifies the efforts (of molecular modelers) for bridging between low resolution analytical approaches and full atomic MD simulations, capturing commonalities, assessing what is being neglected at the higher scale (114). Given the extensive work already accomplished by molecular and mathematical modelers, we will focus here on the approaches for filling the intermediate gap, mainly bridging (i) between mathematical models of cellular networks and stochastic (MC) simulations of microphysiological events, on the one hand, and (ii) between the latter MC simulations and the molecular simulations and analytical methods, on the other. In a broad sense, a two-level approach is adopted, in which the complex signaling networks are modeled in terms of effective control motifs or “circuit” modules, and these modules are modeled using the molecular biochemical and biophysical properties of their elements, as suggested by Asthagiri et al. (115).

A.4.2. Modular Control Theory Analysis of Biochemical Networks. Even relatively small metabolic pathways such as glycolysis can be shown to exhibit chaotic behavior in some regions of parameter space (116). Dynamical systems analysis of simplified network modules (e.g., positive and negative feedback loops) has shown that important properties like adaptation, oscillation, and bistability can emerge from such simple cross-interactions of a small number of signaling molecules (117-119). Bifurcation analysis has been used with remarkable success to show how interactions between cyclin-dependent kinases and their associated proteins control the timing of eukaryotic cell division during early embryogenesis (120). Such studies have shown how positive and negative feedback loops involving Cdc25, Wee1, APC (anaphase-promoting complex) and MPF (M-phase-promoting factor) regulate MPF activation and degradation (121;122), and in more general terms, how dynamic systems theory applied to small modules or subnetworks (or control motifs) that are amenable to theoretical analysis (119), can establish the connection between molecular regulatory mechanisms and cell physiology (115;118;123). These modules are ideally suited for simulations with the adapted forms of MCell (see below). We will explore the dynamics of such modules, or control motifs, both by simulations and analytical methods, and combine these into the large scale network.

The synchronization of the subsystems or the timing of the signals they transmit can, however, have synergistic effects far more complex than those expected from simple additive rules. For example networks have properties –extended signal duration, feedback loops, stimulation thresholds, or multiple signal outputs - that individual pathways do not have (124). Likewise groups of coupled control motifs in complex networks can exhibit distinctive features. We anticipate several cycles of refinement between simulations and customized assays so as to build integrated models that include such coupling effects.

A.4.3. Counter-intuitive Effects of Realistic Spatial Organization (Figure A.3). Two counter-intuitive examples taken from our MC simulations of miniature endplate currents (mEPCs) illustrate the importance of considering space-dependence. We simulated the reversible activation of muscle acetylcholine receptors (AChRs) by the binding of two molecules of diffusing ACh, with subsequent conformational change from the closed state (A₂R³) to an open-channel state (A₂R⁴) that produces the mEPC. The observations are: (i) Adding a JF to an otherwise flat synaptic cleft effectively doubles the concentration of AChRs close to the mouth of the fold, and increases the local cleft volume by a factor of ~1.5. Intuition predicts an increase in mEPC amplitude. However, simulations show a decrease, due to non-linear dynamic effects in the mEPC rising phase. (ii) One might expect that spatially realistic MC simulations would lead to greater noise (variance, s²) in the time course of the reactant amount in a given state, compared to matching simulations in which there is no explicit space. However, Fig. A.3 shows the opposite can be true in a realistic subcellular environment, and that the discrepancy can be substantial. Such counter-intuitive results may have dramatic implications for qualitative and quantitative predictions of steady state, adaptive, oscillatory, and chaotic behaviors of biochemical networks.

A.4.4. Integration of Cellular and Microphysiological Models. The aim of these studies will be to characterize the effects of stochastic and spatial parameters on the dynamics of abstract network motifs, and to develop heuristic rules and functions that can be added to cellular models based on differential equations. By abstract network motifs we mean generic positive and negative feedback loops (and combinations thereof) that appear in virtually all biochemical networks of interest, including the three developmental projects DP1-3. Our general method will involve the formulation of several models for the same biological questions, defined at different resolutions, for studying the limitation at each level as well as their relations. For example, consider four levels of modeling: discrete particles with space dependence (e.g. MC simulations); discrete particles/probabilities, no space dependence (e.g. master equation formalism); continuum models with PDEs (a limit in the first model), and continuum with ODEs (a limit of the second model). A specific study may be:

1) Identify or postulate a control motif in the network of interest, and build its block diagram based on experimental information.

2) Analyze the predictions of (1) using continuum models and phase plane and/or bifurcation analysis as allowed by the problem's dimensionality.

1) Examine the dynamics of the same motif with discrete modeling using MCell and "identical" initial conditions: (a) well-mixed reactants (uniform concentration) in a symmetrical unit volume (Fig. A.4), (b) well-mixed reactants in different volumes, i.e. varying the number of reactants, (c) varying the mobility of the species, while maintaining average binding rates (16), (d) holding the total number of reactants fixed, but varying their spatial distributions, and (e) repeating the simulations with the above mentioned stochastic Markov model.

2) Study the passage from discrete or hybrid models to continuous models inasmuch as possible, and develop heuristic rules that relate spatial parameters, reactant numbers, and reactant mobilities to network behavior, so that the mathematical models can be adapted to make similar qualitative and quantitative predictions. Hybrid modeling methods that combine the classical approach of differential equations with stochastic and MC methods (125) will also be considered for this aim.

3) Tune model parameters with experimental data using inverse problem formulation via optimization techniques. This allows efficient sampling of the parameter space and saves the need to reiterate similar models many times. Such approaches have been done successfully in material science, aerodynamics shape design and more. Our techniques for optimization and inverse problems (126-128) will be used for that part.

4) If agreement with experimental data is not satisfactory after optimal values have been selected in step 5, reevaluate the model system assumptions, and reiterate steps 1-5.

MCell is presently capable of simulating arbitrary reactions between diffusing small molecules and static binding sites (receptors) on reflective, transparent, or absorptive surfaces. Step 3 above can be begun with present capabilities, but will also require simulation of interactions between diffusing molecules themselves. This extension of MCell is presently being pursued. Step 3 will also ultimately lead to large-scale parameter sweeps, and the resources and support of PSC will be a major factor that will ensure high throughput. In addition, MCell use is now being extended to Grid-based computation, with NSF support (ITR0086092). This is a collaborative project between MCell developers, M. Ellisman (Neuroscience, UCSD), and grid-based distributed computation experts (F. Berman (Project Director), H. Casanova, R. Wolski, UCSD; J. Dongarra, Univ. Tennessee Knoxville). The Computational Grid is an emerging platform for deployment of large-scale scientific and engineering projects. MCell is the chosen prototype application for this project, because large sets of simulations can generally be run in parallel with different ranges of spatial and kinetic parameter sweeps and sensitivity analyses.

Developing large scale models from a microscopic one is a hard question in general. The applications we are considering here may benefit from experience obtained over many years in other fields. A classical example is the emergence of the Navier-Stokes (NS) equations from molecular motion modeled by Newton’s equations of motion. An interesting lesson from this example is that macroscopic variables may have different forms than their microscopic counterparts (e.g. temperature in NS equations, accounted for by velocity distributions on the particle level). Thus, the set of proper variables to use on each level is a most difficult question. Additional lessons can be learned from the coarse-graining processes that lead to macroscopic equations. The passage to larger scales uses probability distributions and their dynamical equations, the Boltzmann equation being the most obvious example. This supports a description of the system in terms of probability functions, in addition to the other approaches mentioned. Averaging techniques applied to such models result in continuum models.

A.4.5. Integration of Microphysiological and Molecular Models. As mentioned above, there have been numerous studies by the pre-NPEBC investigators, aiming at developing models and tools that can explore the mesoscale dynamics. Two noteworthy examples are, the recent extension of the elastic network models to multimeric complexes of 10⁴ -10⁵ residues (91;92), and the solution of the Poisson-Nernst-Planck PDEs to compute the current-voltage characteristics of ion channels (30). The former is a purely mechanical approach for a collection of discrete interaction sites, while the latter resorts to a continuum description of permeating ions in a self-consistent mean-field of electrostatic interactions. These are therefore quite different, in a sense orthogonal, approaches, and both have approximations, adopted for mathematical simplicity or computational efficiency. The former neglects chemical specificities or energetics, while the latter cannot address the effects of finite size or excluded volume in the case of narrow channels. Yet, both approaches yield results in quantitative agreement with experimental data, and provide insights as to the dominant mechanisms of collective dynamics. And to address the limitations of these approaches, a strategy is to perform simulations. BD simulations of the same ion channel were indeed performed to this aim (129), and likewise, the GNM results were compared with those from MD simulations coupled with essential dynamics analysis (115). This type of cross-examination of a given system by both more detailed and reduced methods has proven useful for assessing the most important ‘details’ that need to be incorporated in the higher level models and methods. An even more synergistic interaction between the two orthogonal approaches would to combine in a hybrid model the substates and fluctuations information.

Our goal is to extend this type of biochemical and biophysical interactions to even higher scales, and merge with MCell-type simulations. There are many possibilities for such integrating approaches, and a major aim of planning studies will be to identify the most needed and feasible ones. Examples are;

1) Cell signaling and regulation dynamics are generally dependent on the assumed molecular substates and rate constants for the passages between them (130). Coarse-grained molecular models coupled with analytical tools such as mode analysis, as developed by the pre-NPEBC investigators (Table A.2), could provide information on the dominant substates, their relative free energies, and their fluctuation or transition rates.

2) Coarse-grained molecular studies could also be used to provide membrane properties or charge distributions, which in turn could be implemented as new empirical features in microphysiological simulations. Eventually, it should be possible to superimpose energetic effects (empirical force fields at the cellular scale) on the presently random displacements of molecules in MCell, based on the data provided by theoretical and experimental studies at the molecular level.

3) Molecular simulations could be used to provide surface topologies and properties of macromolecular complexes, which then would be incorporated into microphysiological simulations as mesh objects.

4) Coarse-grained molecular studies could ultimately provide input to microphysiological simulations based on volume meshes rather than surface meshes, and which thus could combine present Monte Carlo algorithms with finite element methods. It is a challenge, however, to select the structural and/or spatial characteristics at an optimal level of detail, given the trade-off between accuracy and computational efficiency.

Finally, we need to further develop the analytical tools for the interpretation and generalization of the results. The software developed by Ermentrout (7) is an excellent resource for analyzing mathematical models. Interestingly, models and methods used by molecular and mathematical modelers share several common features: For example, MC methods are used both for simulating microphysiological processes (16;17) and macromolecular dynamics (131); (ii) the network connectivity is an important determinant of dynamics both at the cellular level (between interacting molecules) and at the molecular structural model (between residues), (iii) the master equation formalism has been advantageously used for exploring macromolecular dynamics (132;133) (106), protein folding kinetics (134), or biochemical networks (96; 97;105;135;136), and same methods of analysis based on eigenvalue decomposition hold for all cases. These examples suggest that the mathematical tools that we have developed for analyzing the dynamics of intramolecular interactions, can be applied, with suitable redefinition of system parameters, to intermolecular or cellular networks, and vice versa.