There is more than one way to model an elephant. Experiment-driven modeling of the actin cytoskeleton

Model First and Ask Questions Later: Confessions of a Reformed Experimentalist

This paper is an invited perspective written in association with the awarding of the 2018 American Society of Mechanical Engineers Van C. Mow Medal. Inspired by Professor Mow's collaboration with Professor Michael Lai and the role mathematical modeling played in their work on cartilage biomechanics, this article uses our group's work on myocardial infarct healing as an example of the potential value of models in modern experimental biomechanics. Focusing more on the thought process and lessons learned from our studies on infarct mechanics than on the details of the science, this article argues that the complexity of current research questions and the wealth of information already available about almost any cell, tissue, or organ should change how we approach problems and design experiments. In particular, this paper proposes that constructing a mathematical or computational model is now in many cases a critical prerequisite to designing scientifically useful, informative experiments.

From membrane receptors to protein synthesis and actin cytoskeleton: Mechanisms underlying long lasting forms of synaptic plasticity

Synaptic plasticity, the activity dependent change in synaptic strength, forms the molecular foundation of learning and memory. Synaptic plasticity includes structural changes, with spines changing their size to accomodate insertion and removal of postynaptic receptors, which are correlated with functional changes. Of particular relevance for memory storage are the long lasting forms of synaptic plasticity which are protein synthesis dependent. Due to the importance of spine structural plasticity and protein synthesis, this review focuses on the signaling pathways that connect synaptic stimulation with regulation of protein synthesis and remodeling of the actin cytoskeleton. We also review computational models that implement novel aspects of molecular signaling in synaptic plasticity, such as the role of neuromodulators and spatial microdomains, as well as highlight the need for computational models that connect activation of memory kinases with spine actin dynamics.

Probing the Occurrence of Soluble Oligomers through Amyloid Aggregation Scaling Laws

Drug discovery frequently relies on the kinetic analysis of physicochemical reactions that are at the origin of the disease state. Amyloid fibril formation has been extensively investigated in relation to prevalent and rare neurodegenerative diseases, but thus far no therapeutic solution has directly arisen from this knowledge. Other aggregation pathways producing smaller, hard-to-detect soluble oligomers are increasingly appointed as the main reason for cell toxicity and cell-to-cell transmissibility. Here we show that amyloid fibrillation kinetics can be used to unveil the protein oligomerization state. This is illustrated for human insulin and ataxin-3, two model proteins for which the amyloidogenic and oligomeric pathways are well characterized. Aggregation curves measured by the standard thioflavin-T (ThT) fluorescence assay are shown to reflect the relative composition of protein monomers and soluble oligomers measured by nuclear magnetic resonance (NMR) for human insulin, and by dynamic light scattering (DLS) for ataxin-3. Unconventional scaling laws of kinetic measurables were explained using a single set of model parameters consisting of two rate constants, and in the case of ataxin-3, an additional order-of-reaction. The same fitted parameters were used in a discretized population balance that adequately describes time-course measurements of fibril size distributions. Our results provide the opportunity to study oligomeric targets using simple, high-throughput compatible, biophysical assays.

Reconsidering an active role for G-actin in cytoskeletal regulation

Globular (G)-actin, the actin monomer, assembles into polarized filaments that form networks that can provide structural support, generate force and organize the cell. Many of these structures are highly dynamic and to maintain them, the cell relies on a large reserve of monomers. Classically, the G-actin pool has been thought of as homogenous. However, recent work has shown that actin monomers can exist in distinct groups that can be targeted to specific networks, where they drive and modify filament assembly in ways that can have profound effects on cellular behavior. This Review focuses on the potential factors that could create functionally distinct pools of actin monomers in the cell, including differences between the actin isoforms and the regulation of G-actin by monomer binding proteins, such as profilin and thymosin β4. Owing to difficulties in studying and visualizing G-actin, our knowledge over the precise role that specific actin monomer pools play in regulating cellular actin dynamics remains incomplete. Here, we discuss some of these unanswered questions and also provide a summary of the methodologies currently available for the imaging of G-actin.

Fragility of foot process morphology in kidney podocytes arises from chaotic spatial propagation of cytoskeletal instability

Kidney podocytes' function depends on fingerlike projections (foot processes) that interdigitate with those from neighboring cells to form the glomerular filtration barrier. The integrity of the barrier depends on spatial control of dynamics of actin cytoskeleton in the foot processes. We determined how imbalances in regulation of actin cytoskeletal dynamics could result in pathological morphology. We obtained 3-D electron microscopy images of podocytes and used quantitative features to build dynamical models to investigate how regulation of actin dynamics within foot processes controls local morphology. We find that imbalances in regulation of actin bundling lead to chaotic spatial patterns that could impair the foot process morphology. Simulation results are consistent with experimental observations for cytoskeletal reconfiguration through dysregulated RhoA or Rac1, and they predict compensatory mechanisms for biochemical stability. We conclude that podocyte morphology, optimized for filtration, is intrinsically fragile, whereby local transient biochemical imbalances may lead to permanent morphological changes associated with pathophysiology.

Multivalent Molecules as Modulators of RNA Granule Size and Composition

RNA granules are ensembles of specific RNA and protein molecules that mediate localized translation in eukaryotic cells. The mechanisms for formation and selectivity of RNA granules are unknown. Here we present a model for assembly of one type of RNA granule based on experimentally measured binding interactions among three core multivalent molecular components necessary for such assembly: specific RNA molecules that contain a cis-acting sequence called the A2 response element (A2RE), hnRNP A2 proteins that bind specifically (with high affinity) to A2RE sequences or nonspecifically (with lower affinity) to other RNA sequences, and heptavalent protein cytoskeleton-associated protein 5 (CKAP5, an alternative name for TOG protein) that binds both hnRNP A2 molecules and RNA. Non-A2RE RNA molecules (RNA without the A2RE sequence) that may be recruited to the granules through nonspecific interactions are also considered in the model. Modeling multivalent molecular interactions in granules is challenging because of combinatorial complexity in the number of potential molecular complexes among these core components and dynamic changes in granule composition and structure in response to changes in local intracellular environment. We use a hybrid modeling approach (deterministic-stochastic-statistical) that is appropriate when the overall compositions of multimolecular ensembles are of greater importance than the specific interactions among individual molecular components. Modeling studies titrating the concentrations of various granule components and varying effective site pair affinities and RNA valency demonstrate that interactions between multivalent components (TOG and RNA) are modulated by a bivalent adaptor molecule (hnRNP A2). Formation and disruption of granules, as well as RNA selectivity in granule composition are regulated by distinct concentration regimes of A2. Our results suggest that granule assembly is tightly controlled by multivalent molecular interactions among RNA molecules, adaptor proteins, and scaffold proteins.

Strong Inference in Mathematical Modeling: A Method for Robust Science in the Twenty-First Century

While there are many opinions on what mathematical modeling in biology is, in essence, modeling is a mathematical tool, like a microscope, which allows consequences to logically follow from a set of assumptions. Only when this tool is applied appropriately, as microscope is used to look at small items, it may allow to understand importance of specific mechanisms/assumptions in biological processes. Mathematical modeling can be less useful or even misleading if used inappropriately, for example, when a microscope is used to study stars. According to some philosophers (Oreskes et al., 1994), the best use of mathematical models is not when a model is used to confirm a hypothesis but rather when a model shows inconsistency of the model (defined by a specific set of assumptions) and data. Following the principle of strong inference for experimental sciences proposed by Platt (1964), I suggest “strong inference in mathematical modeling” as an effective and robust way of using mathematical modeling to understand mechanisms driving dynamics of biological systems. The major steps of strong inference in mathematical modeling are (1) to develop multiple alternative models for the phenomenon in question; (2) to compare the models with available experimental data and to determine which of the models are not consistent with the data; (3) to determine reasons why rejected models failed to explain the data, and (4) to suggest experiments which would allow to discriminate between remaining alternative models. The use of strong inference is likely to provide better robustness of predictions of mathematical models and it should be strongly encouraged in mathematical modeling-based publications in the Twenty-First century.

Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer

The ability to rapidly assess the efficacy of therapeutic strategies for incurable bone metastatic prostate cancer is an urgent need. Pre-clinical in vivo models are limited in their ability to define the temporal effects of therapies on simultaneous multicellular interactions in the cancer-bone microenvironment. Integrating biological and computational modeling approaches can overcome this limitation. Here, we generated a biologically driven discrete hybrid cellular automaton (HCA) model of bone metastatic prostate cancer to identify the optimal therapeutic window for putative targeted therapies. As proof of principle, we focused on TGFβ because of its known pleiotropic cellular effects. HCA simulations predict an optimal effect for TGFβ inhibition in a pre-metastatic setting with quantitative outputs indicating a significant impact on prostate cancer cell viability, osteoclast formation and osteoblast differentiation. In silico predictions were validated in vivo with models of bone metastatic prostate cancer (PAIII and C4-2B). Analysis of human bone metastatic prostate cancer specimens reveals heterogeneous cancer cell use of TGFβ. Patient specific information was seeded into the HCA model to predict the effect of TGFβ inhibitor treatment on disease evolution. Collectively, we demonstrate how an integrated computational/biological approach can rapidly optimize the efficacy of potential targeted therapies on bone metastatic prostate cancer.

Quantifying Limits on Replication, Death, and Quiescence of Mycobacterium tuberculosis in Mice

When an individual is exposed to Mycobacterium tuberculosis (Mtb) three outcomes are possible: bacterial clearance, active disease, or latent infection. It is generally believed that most individuals exposed to Mtb become latently infected and carry the mycobacteria for life. How Mtb is maintained during this latent infection remains largely unknown. During an Mtb infection in mice, there is a phase of rapid increase in bacterial numbers in the murine lungs within the first 3 weeks, and then bacterial numbers either stabilize or increase slowly over the period of many months. It has been debated whether the relatively constant numbers of bacteria in the chronic infection result from latent (dormant, quiescent), non-replicating bacteria, or whether the observed Mtb cell numbers are due to balance between rapid replication and death. A recent study of mice, infected with a Mtb strain carrying an unstable plasmid, showed that during the chronic phase, Mtb was replicating at significant rates. Using experimental data from this study and mathematical modeling we investigated the limits of the rates of bacterial replication, death, and quiescence during Mtb infection of mice. First, we found that to explain the data the rates of bacterial replication and death could not be constant and had to decrease with time since infection unless there were large changes in plasmid segregation probability over time. While a decrease in the rate of Mtb replication with time since infection was expected due to depletion of host's resources, a decrease in the Mtb death rate was counterintuitive since Mtb-specific immune response, appearing in the lungs 3–4 weeks after infection, should increase removal of bacteria. Interestingly, we found no significant correlation between estimated rates of Mtb replication and death suggesting the decline in these rates was driven by independent mechanisms. Second, we found that the data could not be explained by assuming that bacteria do not die, suggesting that some removal of bacteria from lungs of these mice had to occur even though the total bacterial counts in these mice always increased over time. Third and finally, we showed that to explain the data the majority of bacterial cells (at least ~60%) must be replicating in the chronic phase of infection further challenging widespread belief of nonreplicating Mtb in latency. Our predictions were robust to some changes in the structure of the model, for example, when the loss of plasmid-bearing cells was mainly due to high fitness cost of the plasmid. Further studies should determine if more mechanistic models for Mtb dynamics are also able to accurately explain these data.

Actin-myosin spatial patterns from a simplified isotropic viscoelastic model

F-actin networks are involved in cell mechanical processes ranging from motility to endocytosis. The mesoscale architecture of assemblies of individual F-actin polymers that gives rise to micrometer-scale rheological properties is poorly understood, despite numerous in vivo and vitro studies. In vitro networks have been shown to organize into spatial patterns when spatially confined, including dense spherical shells inside spherical emulsion droplets. Here we develop a simplified model of an isotropic, compressible, viscoelastic material continually assembling and disassembling. We demonstrate that spherical shells emerge naturally when the strain relaxation rate (corresponding to internal network reorganization) is slower than the disassembly rate (corresponding to F-actin depolymerization). These patterns are consistent with recent experiments, including a collapse of shells to a central high-density focus of F-actin when either assembly or disassembly is reduced with drugs. Our results demonstrate how complex spatio-temporal patterns can emerge without spatially distributed force generation, polar alignment of F-actin polymers, or spatially nonuniform regulation of F-actin by upstream biochemical networks.

The value of mechanistic biophysical information for systems-level understanding of complex biological processes such as cytokinesis

This review illustrates the value of quantitative information including concentrations, kinetic constants and equilibrium constants in modeling and simulating complex biological processes. Although much has been learned about some biological systems without these parameter values, they greatly strengthen mechanistic accounts of dynamical systems. The analysis of muscle contraction is a classic example of the value of combining an inventory of the molecules, atomic structures of the molecules, kinetic constants for the reactions, reconstitutions with purified proteins and theoretical modeling to account for the contraction of whole muscles. A similar strategy is now being used to understand the mechanism of cytokinesis using fission yeast as a favorable model system.

Membrane related dynamics and the formation of actin in cells growing on micro-topographies: a spatial computational model

Background

Intra-cellular processes of cells at the interface to an implant surface are influenced significantly by their extra-cellular surrounding. Specifically, when growing osteoblasts on titanium surfaces with regular micro-ranged geometry, filaments are shorter, less aligned and they concentrate at the top of the geometric structures. Changes to the cytoskeleton network, i. e., its localization, alignment, orientation, and lengths of the filaments, as well as the overall concentration and distribution of key-actors are induced. For example, integrin is distributed homogeneously, whereas integrin in activated state and vinculin, both components of focal adhesions, have been found clustered on the micro-ranged geometries. Also, the concentration of Rho, an intracellular signaling protein related to focal adhesion regulation, was significantly lower.

Results

To explore whether regulations associated with the focal adhesion complex can be responsible for the changed actin filament patterns, a spatial computational model has been developed using ML-Space, a rule-based model description language, and its associated Brownian-motion-based simulator. The focus has been on the deactivation of cofilin in the vicinity of the focal adhesion complex. The results underline the importance of sensing mechanisms to support a clustering of actin filament nucleations on the micro-ranged geometries, and of intracellular diffusion processes, which lead to spatially heterogeneous distributions of active (dephosphorylated) cofilin, which in turn influences the organization of the actin network. We find, for example, that the spatial heterogeneity of key molecular actors can explain the difference in filament lengths in cells on different micro-geometries partly, but to explain the full extent, further model assumptions need to be added and experimentally validated. In particular, our findings and hypothesis referring to the role, distribution, and amount of active cofilin have still to be verified in wet-lab experiments.

Conclusion

Letting cells grow on surface structures is a possibility to shed new light on the intricate mechanisms that relate membrane and actin related dynamics in the cell. Our results demonstrate the need for declarative expressive spatial modeling approaches that allow probing different hypotheses, and the central role of the focal adhesion complex not only for nucleating actin filaments, but also for regulating possible severing agents locally.

Computational modeling of subcellular transport and signaling

Numerous signaling processes in the cell are controlled in microdomains that are defined by cellular structures ranging from nm to μm in size. Recent improvements in microscopy enable the resolution and reconstruction of these micro domains, while new computational methods provide the means to elucidate their functional roles. Collectively these tools allow for a biophysical understanding of the cellular environment and its pathological progression in disease. Here we review recent advancements in microscopy, and subcellular modeling on the basis of reconstructed geometries, with a special focus on signaling microdomains that are important for the excitation contraction coupling in cardiac myocytes.

Regulation of actin by ion-linked equilibria

Actin assembly, filament mechanical properties, and interactions with regulatory proteins depend on the types and concentrations of salts in solution. Salts modulate actin through both nonspecific electrostatic effects and specific binding to discrete sites. Multiple cation-binding site classes spanning a broad range of affinities (nanomolar to millimolar) have been identified on actin monomers and filaments. This review focuses on discrete, low-affinity cation-binding interactions that drive polymerization, regulate filament-bending mechanics, and modulate interactions with regulatory proteins. Cation binding may be perturbed by actin post-translational modifications and linked equilibria. Partial cation occupancy under physiological and commonly used in vitro solution conditions likely contribute to filament mechanical heterogeneity and structural polymorphism. Site-specific cation-binding residues are conserved in Arp2 and Arp3, and may play a role in Arp2/3 complex activation and actin-filament branching activity. Actin-salt interactions demonstrate the relevance of ion-linked equilibria in the operation and regulation of complex biological systems.

Molecular origins of binding affinity: seeking the Archimedean point

Connecting three dimensional structure and affinity is analogous to seeking the 'Archimedean point', a vantage point from where any observer can quantitatively perceive the subject of inquiry. Here we review current knowledge and challenges that lie ahead of us in the quest for this Archimedean point. We argue that current models are limited in reproducing measured data because molecular description of binding affinity must expand beyond the interfacial contribution and also incorporate effects stemming from conformational changes/dynamics and long-range interactions. Fortunately, explicit modeling of various kinetic schemes underlying biomolecular recognition and confined systems that reflect in vivo interactions are coming within reach. This quest will hopefully lead to an accurate biophysical interpretation of binding affinity that would allow unprecedented understanding of the molecular basis of life through unraveling the why's of interaction networks.