Cell polarity: quantitative modeling as a tool in cell biology

Mathematical model with spatially uniform regulation explains long-range bidirectional transport of early endosomes in fungal hyphae

Cellular cargo transported bidirectionally along microtubules by dynein and kinesin can be organized by spatially nonuniform upstream regulation or can self-organize. A mathematical model of early endosome transport in fungal hyphae demonstrates that spatiotemporally uniform regulation results in cargo dynamics consistent with experiment.

In many cellular contexts, cargo is transported bidirectionally along microtubule bundles by dynein and kinesin-family motors. Upstream factors influence how individual cargoes are locally regulated, as well as how long-range transport is regulated at the whole-cell scale. Although the details of local, single-cargo bidirectional switching have been extensively studied, it remains to be elucidated how this results in cell-scale spatial organization. Here we develop a mathematical model of early endosome transport in Ustilago maydis. We demonstrate that spatiotemporally uniform regulation, with constant transition rates, results in cargo dynamics that is consistent with experimental data, including data from motor mutants. We find that microtubule arrays can be symmetric in plus-end distribution but asymmetric in binding-site distribution in a manner that affects cargo dynamics and that cargo can travel past microtubule ends in microtubule bundles. Our model makes several testable predictions, including secondary features of dynein and cargo distributions.

Establishment of a robust single axis of cell polarity by coupling multiple positive feedback loops

Establishment of cell polarity—or symmetry breaking—relies on local accumulation of polarity regulators. Although simple positive feedback is sufficient to drive symmetry breaking, it is highly sensitive to stochastic fluctuations typical for living cells. Here, by integrating mathematical modelling with quantitative experimental validations, we show that in the yeast Saccharomyces cerevisiae a combination of actin- and guanine nucleotide dissociation inhibitor-dependent recycling of the central polarity regulator Cdc42 is needed to establish robust cell polarity at a single site during yeast budding. The guanine nucleotide dissociation inhibitor pathway consistently generates a single-polarization site, but requires Cdc42 to cycle rapidly between its active and inactive form, and is therefore sensitive to perturbations of the GTPase cycle. Conversely, actin-mediated recycling of Cdc42 induces robust symmetry breaking but cannot restrict polarization to a single site. Our results demonstrate how cells optimize symmetry breaking through coupling between multiple feedback loops.

A positive feedback loop which results in localized accumulation of the small GTPase Cdc42 generates cell polarity in budding yeast; however, such loops are inherently susceptible to noise. Here the authors demonstrate how two pathways that mediate Cdc42 recycling work together to ensure the robustness of symmetry breaking.

How B cells capture, process and present antigens: a crucial role for cell polarity

B cells are key components of the adaptive immune response. Their differentiation into either specific memory B cells or antibody-secreting plasma cells is a consequence of activation steps that involve the processing and presentation of antigens. The engagement of B cell receptors by surface-tethered antigens leads to the formation of an immunological synapse that coordinates cell signalling events and that promotes antigen uptake for presentation on MHC class II molecules. In this Review, we discuss membrane trafficking and the associated molecular mechanisms that are involved in antigen extraction and processing at the B cell synapse, and we highlight how B cells use cell polarity to coordinate the complex events that ultimately lead to efficient humoral responses.

Systems biology and the integration of mechanistic explanation and mathematical explanation

The paper discusses how systems biology is working toward complex accounts that integrate explanation in terms of mechanisms and explanation by mathematical models-which some philosophers have viewed as rival models of explanation. Systems biology is an integrative approach, and it strongly relies on mathematical modeling. Philosophical accounts of mechanisms capture integrative in the sense of multilevel and multifield explanations, yet accounts of mechanistic explanation (as the analysis of a whole in terms of its structural parts and their qualitative interactions) have failed to address how a mathematical model could contribute to such explanations. I discuss how mathematical equations can be explanatorily relevant. Several cases from systems biology are discussed to illustrate the interplay between mechanistic research and mathematical modeling, and I point to questions about qualitative phenomena (rather than the explanation of quantitative details), where quantitative models are still indispensable to the explanation. Systems biology shows that a broader philosophical conception of mechanisms is needed, which takes into account functional-dynamical aspects, interaction in complex networks with feedback loops, system-wide functional properties such as distributed functionality and robustness, and a mechanism's ability to respond to perturbations (beyond its actual operation). I offer general conclusions for philosophical accounts of explanation.

Multi-agent model of hepatitis C virus infection

OBJECTIVES

The objective of this study is to design a method for modeling hepatitis C virus (HCV) infection using multi-agent simulation and to verify it in practice.

METHODS AND MATERIALS

In this paper, first, the modeling of HCV infection using a multi-agent system is compared with the most commonly used model type, which is based on differential equations. Then, the implementation and results of the model using a multi-agent simulation is presented. To find the values of the parameters used in the model, a method using inverted simulation flow and genetic algorithm is proposed. All of the data regarding HCV infection are taken from the paper describing the model based on the differential equation to which the proposed method is compared.

RESULTS

Important advantages of the proposed method are noted and demonstrated: these include flexibility, clarity, re-usability and the possibility to model more complex dependencies. Then, the simulation framework that uses the proposed approach is successfully implemented in C++ and is verified by comparing it to the approach based on differential equations. The verification proves that an objective function that performs the best is the function that minimizes the maximal differences in the data. Finally, an analysis of one of the already known models is performed, and it is proved that it incorrectly models a decay in the hepatocytes number by 40%.

CONCLUSIONS

The proposed method has many advantages in comparison to the currently used model types and can be used successfully for analyzing HCV infection. With almost no modifications, it can also be used for other types of viral infections.

The PAR network: redundancy and robustness in a symmetry-breaking system

To become polarized, cells must first 'break symmetry'. Symmetry breaking is the process by which an unpolarized, symmetric cell develops a singularity, often at the cell periphery, that is used to develop a polarity axis. The Caenorhabditis elegans zygote breaks symmetry under the influence of the sperm-donated centrosome, which causes the PAR polarity regulators to sort into distinct anterior and posterior cortical domains. Modelling analyses have shown that cortical flows induced by the centrosome combined with antagonism between anterior and posterior PARs (mutual exclusion) are sufficient, in principle, to break symmetry, provided that anterior and posterior PAR activities are precisely balanced. Experimental evidence indicates, however, that the system is surprisingly robust to changes in cortical flows, mutual exclusion and PAR balance. We suggest that this robustness derives from redundant symmetry-breaking inputs that engage two positive feedback loops mediated by the anterior and posterior PAR proteins. In particular, the PAR-2 feedback loop stabilizes the polarized state by creating a domain where posterior PARs are immune to exclusion by anterior PARs. The two feedback loops in the PAR network share characteristics with the two feedback loops in the Cdc42 polarization network of Saccharomyces cerevisiae.

Contraction-driven cell motility

We propose a mechanism for the initiation of cell motility that is based on myosin-induced contraction and does not require actin polymerization. The translocation of a cell is induced by symmetry breaking of the motor-driven flow, and the ensuing asymmetry gives rise to a steady motion of the center of mass of a cell. The predictions of the model are consistent with observations on keratocytes.

Cellular polarization: interaction between extrinsic bounded noises and the wave-pinning mechanism

Cell polarization (cued or uncued) is a fundamental mechanism in cell biology. As an alternative to the classical Turing bifurcation, it has been proposed that the onset of cell polarity might arise by means of the well-known phenomenon of wave-pinning [Gamba et al., Proc. Natl. Acad. Sci. USA 102, 16927 (2005)]. A particularly simple and elegant deterministic model of cell polarization based on the wave-pinning mechanism has been proposed by Edelstein-Keshet and coworkers [Biophys. J. 94, 3684 (2008)]. This model consists of a small biomolecular network where an active membrane-bound factor interconverts into its inactive form that freely diffuses in the cell cytosol. However, biomolecular networks do communicate with other networks as well as with the external world. Thus, their dynamics must be considered as perturbed by extrinsic noises. These noises may have both a spatial and a temporal correlation, and in any case they must be bounded to preserve the biological meaningfulness of the perturbed parameters. Here we numerically show that the inclusion of external spatiotemporal bounded parametric perturbations in the above wave-pinning-based model of cellular polarization may sometimes destroy the polarized state. The polarization loss depends on both the extent of temporal and spatial correlations and on the kind of noise employed. For example, an increase of the spatial correlation of the noise induces an increase of the probability of cell polarization. However, if the noise is spatially homogeneous then the polarization is lost in the majority of cases. These phenomena are independent of the type of noise. Conversely, an increase of the temporal autocorrelation of the noise induces an effect that depends on the model of noise.

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

Mathematical modeling has established its value for investigating the interplay of biochemical and mechanical mechanisms underlying actin-based motility. Because of the complex nature of actin dynamics and its regulation, many of these models are phenomenological or conceptual, providing a general understanding of the physics at play. But the wealth of carefully measured kinetic data on the interactions of many of the players in actin biochemistry cries out for the creation of more detailed and accurate models that could permit investigators to dissect interdependent roles of individual molecular components. Moreover, no human mind can assimilate all of the mechanisms underlying complex protein networks; so an additional benefit of a detailed kinetic model is that the numerous binding proteins, signaling mechanisms, and biochemical reactions can be computationally organized in a fully explicit, accessible, visualizable, and reusable structure. In this review, we will focus on how comprehensive and adaptable modeling allows investigators to explain experimental observations and develop testable hypotheses on the intracellular dynamics of the actin cytoskeleton.

The effective application of a discrete transition model to explore cell-cycle regulation in yeast

Background

Bench biologists often do not take part in the development of computational models for their systems, and therefore, they frequently employ them as “black-boxes”. Our aim was to construct and test a model that does not depend on the availability of quantitative data, and can be directly used without a need for intensive computational background.

Results

We present a discrete transition model. We used cell-cycle in budding yeast as a paradigm for a complex network, demonstrating phenomena such as sequential protein expression and activity, and cell-cycle oscillation. The structure of the network was validated by its response to computational perturbations such as mutations, and its response to mating-pheromone or nitrogen depletion. The model has a strong predicative capability, demonstrating how the activity of a specific transcription factor, Hcm1, is regulated, and what determines commitment of cells to enter and complete the cell-cycle.

Conclusion

The model presented herein is intuitive, yet is expressive enough to elucidate the intrinsic structure and qualitative behavior of large and complex regulatory networks. Moreover our model allowed us to examine multiple hypotheses in a simple and intuitive manner, giving rise to testable predictions. This methodology can be easily integrated as a useful approach for the study of networks, enriching experimental biology with computational insights.

Spatial stochastic dynamics enable robust cell polarization

Although cell polarity is an essential feature of living cells, it is far from being well-understood. Using a combination of computational modeling and biological experiments we closely examine an important prototype of cell polarity: the pheromone-induced formation of the yeast polarisome. Focusing on the role of noise and spatial heterogeneity, we develop and investigate two mechanistic spatial models of polarisome formation, one deterministic and the other stochastic, and compare the contrasting predictions of these two models against experimental phenotypes of wild-type and mutant cells. We find that the stochastic model can more robustly reproduce two fundamental characteristics observed in wild-type cells: a highly polarized phenotype via a mechanism that we refer to as spatial stochastic amplification, and the ability of the polarisome to track a moving pheromone input. Moreover, we find that only the stochastic model can simultaneously reproduce these characteristics of the wild-type phenotype and the multi-polarisome phenotype of a deletion mutant of the scaffolding protein Spa2. Significantly, our analysis also demonstrates that higher levels of stochastic noise results in increased robustness of polarization to parameter variation. Furthermore, our work suggests a novel role for a polarisome protein in the stabilization of actin cables. These findings elucidate the intricate role of spatial stochastic effects in cell polarity, giving support to a cellular model where noise and spatial heterogeneity combine to achieve robust biological function.

Cell polarity is the fundamental process of breaking symmetry to create asymmetric cellular structures. It is an open question how randomness (stochasticity) in the cell hinders or helps cell polarity. In this work, we focus on the ability of yeast cells to sense a spatial gradient of mating pheromone and respond by forming a projection in the direction of the mating partner. A key element is the polarisome, which is at the tip of the mating projection. We introduce the first model of polarisome formation in yeast. The model is well-supported by experimental data. We perform modeling to explore the role of noise in the formation of the polarisome. By running simulations with and without noise, we arrive at the surprising conclusion, that gradient-dependent polarization is enhanced by stochasticity. Both the tight localization (amplification) and the ability to respond to directional change of the input (tracking) are enhanced by stochastic dynamics, resulting in a more robust behavior. Mutants in which key polarisome proteins have been deleted exhibit broader, noisier polarisome than the wild type. The mutant phenotype is accurately captured by our stochastic simulations. These results demonstrate the importance of stochasticity in the study of cell polarity.

An intracellular partitioning-based framework for tissue cell polarity in plants and animals

Tissue cell polarity plays a major role in plant and animal development. We propose that a fundamental building block for tissue cell polarity is the process of intracellular partitioning, which can establish individual cell polarity in the absence of asymmetric cues. Coordination of polarities may then arise through cell-cell coupling, which can operate directly, through membrane-spanning complexes, or indirectly, through diffusible molecules. Polarity is anchored to tissues through organisers located at boundaries. We show how this intracellular partitioning-based framework can be applied to both plant and animal systems, allowing different processes to be placed in a common evolutionary and mechanistic context.

Non-uniform membrane diffusion enables steady-state cell polarization via vesicular trafficking

Actin-based vesicular trafficking of Cdc42, leading to a polarized concentration of the GTPase, has been implicated in cell polarization, but it was recently debated whether this mechanism allows stable maintenance of cell polarity. Here we show that endocytosis and exocytosis are spatially segregated in the polar plasma membrane, with sites of exocytosis correlating with microdomains of higher concentration and slower diffusion of Cdc42 compared with surrounding regions. Numerical simulations using experimentally obtained diffusion coefficients and trafficking geometry revealed that non-uniform membrane diffusion of Cdc42 in fact enables temporally sustained cell polarity. We show further that phosphatidylserine, a phospholipid recently found to be crucial for cell polarity, is enriched in Cdc42 microdomains. Weakening a potential interaction between phosphatidylserine and Cdc42 enhances Cdc42 diffusion in the microdomains but impedes the strength of polarization. These findings demonstrate a critical role for membrane microdomains in vesicular trafficking-mediated cell polarity.

An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks

Discrete dynamic models are a powerful tool for the understanding and modeling of large biological networks. Although a lot of progress has been made in developing analysis tools for these models, there is still a need to find approaches that can directly relate the network structure to its dynamics. Of special interest is identifying the stable patterns of activity, i.e., the attractors of the system. This is a problem for large networks, because the state space of the system increases exponentially with network size. In this work, we present a novel network reduction approach that is based on finding network motifs that stabilize in a fixed state. Notably, we use a topological criterion to identify these motifs. Specifically, we find certain types of strongly connected components in a suitably expanded representation of the network. To test our method, we apply it to a dynamic network model for a type of cytotoxic T cell cancer and to an ensemble of random Boolean networks of size up to 200. Our results show that our method goes beyond reducing the network and in most cases can actually predict the dynamical repertoire of the nodes (fixed states or oscillations) in the attractors of the system.

Envisioning migration: mathematics in both experimental analysis and modeling of cell behavior

The complex nature of cell migration highlights the power and challenges of applying mathematics to biological studies. Mathematics may be used to create model equations that recapitulate migration, which can predict phenomena not easily uncovered by experiments or intuition alone. Alternatively, mathematics may be applied to interpreting complex data sets with better resolution--potentially empowering scientists to discern subtle patterns amid the noise and heterogeneity typical of migrating cells. Iteration between these two methods is necessary in order to reveal connections within the cell migration signaling network, as well as to understand the behavior that arises from those connections. Here, we review recent quantitative analysis and mathematical modeling approaches to the cell migration problem.

Optically triggering spatiotemporally confined GPCR activity in a cell and programming neurite initiation and extension

G-protein-coupled receptor (GPCR) activity gradients evoke important cell behavior but there is a dearth of methods to induce such asymmetric signaling in a cell. Here we achieved reversible, rapidly switchable patterns of spatiotemporally restricted GPCR activity in a single cell. We recruited properties of nonrhodopsin opsins--rapid deactivation, distinct spectral tuning, and resistance to bleaching--to activate native Gi, Gq, or Gs signaling in selected regions of a cell. Optical inputs were designed to spatiotemporally control levels of second messengers, IP3, phosphatidylinositol (3,4,5)-triphosphate, and cAMP in a cell. Spectrally selective imaging was accomplished to simultaneously monitor optically evoked molecular and cellular response dynamics. We show that localized optical activation of an opsin-based trigger can induce neurite initiation, phosphatidylinositol (3,4,5)-triphosphate increase, and actin remodeling. Serial optical inputs to neurite tips can refashion early neuron differentiation. Methods here can be widely applied to program GPCR-mediated cell behaviors.

Crawling of a driven adherent membrane

We discuss motions of an elastic N × M membrane model whose constituents can bind reversibly with strength ε to adhesive sites of a flat substrate. One of the edges of the membrane ("front") is driven in one direction at rate constant p by N stochastically treadmilling short parallel lines ("cortex"). The main conclusions derived from Monte Carlo studies of this model are the following: (a) Since the polymerizing cortex pushes only the leading edge of the membrane, the major part of the membranes is dragged behind. Therefore, the locomotion of the membrane can be described by frictional sliding processes which are asymmetrically distributed between front and rear of the membrane. A signature of this asymmetry is the difference between the life times of adhesion bonds at front and rear, τ(1) and τ(M), respectively, where τ(1) ≫ τ(M). (b) There are four characteristic times for the membrane motion: The first time, T(0) ~ τ(M) ~ e(aε), is the resting time where the displacement of the membrane is practically zero. The second time, T(p) ~ τ(1) ~ M, is the friction time which characterizes the time between two consecutive ruptures of adhesion bonds at the front, and which signalizes the onset of drift ("protrusion") at the leading edge. The third time, T(r) ~ M(γ(ε)) (γ > 1), characterizes the "retraction" of the trailing edge, which is the retarded response to the pulling leading edge. The fourth time, T(L) ~ M(2), is the growth time for fluctuation of the end-to-end distance. (c) The separation of time scales, T(r)/T(p) ~ M(γ(ε) - 1), leads to stretched fluctuations of the end-to-end distance, which are considered as stochastic cycles of protrusion and retraction on the time scale of T(L). (d) The drift velocity v obeys anomalous scaling, v/p~f(p(1/γ(ε))M), where f(z) ~ const. for small drag pM ≪ 1, and f(z) ~ z(-γ(ε)) for pM ≫ 1, which implies v~M(-γ(ε)). These results may also turn out to be useful for the (more difficult) problem of understanding the protrusion-retraction cycle of crawling biological cells. We compare our model and our results to previous two-particle theories for membrane protrusion and to known stochastic friction models.

Cell polarity: models and mechanisms from yeast, worms and flies

Determinants of cell polarity orient the behaviour of many cell types during development. Pioneering genetic screens in yeast, worms and flies have identified key polarity determinants that are evolutionarily conserved across the animal kingdom. Recent work in these three model organisms has combined computer modelling with experimental analysis to reveal the molecular mechanisms that drive the polarisation of determinants. Two key principles have emerged: the first is the requirement for a positive-feedback loop to drive self-recruitment of determinants to the plasma membrane; the second is the requirement for mutual antagonism between determinants that localise to opposite ends of the cell.

Regulation of actin dynamics and protein trafficking during spermatogenesis--insights into a complex process

In the mammalian testis, extensive restructuring takes place across the seminiferous epithelium at the Sertoli-Sertoli and Sertoli-germ cell interface during the epithelial cycle of spermatogenesis, which is important to facilitate changes in the cell shape and morphology of developing germ cells. However, precise communications also take place at the cell junctions to coordinate the discrete events pertinent to spermatogenesis, namely spermatogonial renewal via mitosis, cell cycle progression and meiosis, spermiogenesis and spermiation. It is obvious that these cellular events are intimately related to the underlying actin-based cytoskeleton which is being used by different cell junctions for their attachment. However, little is known on the biology and regulation of this cytoskeleton, in particular its possible involvement in endocytic vesicle-mediated trafficking during spermatogenesis, which in turn affects cell adhesive function and communication at the cell-cell interface. Studies in other epithelia in recent years have shed insightful information on the intimate involvement of actin dynamics and protein trafficking in regulating cell adhesion and communications. The goal of this critical review is to provide an updated assessment of the latest findings in the field on how these complex processes are being regulated during spermatogenesis. We also provide a working model based on the latest findings in the field including our laboratory to provide our thoughts on an apparent complicated subject, which also serves as the framework for investigators in the field. It is obvious that this model will be rapidly updated when more data are available in future years.

Synergistic action of gravity and temperature on the motor system within the lifespan: a "Baby Astronaut" hypothesis

Here we describe GATO (gravity, age, thermoregulation, and oxygenation) hypothesis (or a "Baby Astronaut" hypothesis) which we suggest to explain synergistic effect of these factors on the motor system. Taken separately, microgravity (in spaceflight, G~0), the early age, heat and hypoxia exert identical effect on the motor system. We posit that synergy of these factors originate from their synchronicity during intrauterine immersion (analog microgravity) of the fetus in warm hypoxic condition. We further postulate three successive motor adaptive strategies, driven lifelong by gravity as the key factor. The first by age, fetal/microgravity (FM)-strategy, induced by the intrauterine immersion of the fetus, is based on domination of fast type muscle fibers. After birth, thought to be analog for landing from orbit, newborn is subjected to combined influence of cooler ambient temperature, normoxia, and 1G Earth gravity, which cooperatively form a slower GE-strategy. Eventually, healthy ageing results in further domination of slow type muscle fibers that forms the slowest (SL)-strategy. Our hypothesis implies that specific sensory conditions may substitute for each other owing to their synergistic action on the motor system. According to GATO hypothesis heating and hypoxia may be considered as "pro-microgravity" factors, while cold and hyperoxia - as "pro-gravity" ones. As such, cold may act as a partial "surrogate" for gravity, estimated as ~0.2G. That may have potential to elaborate countermeasures for muscle atrophy in astronauts either on-board in long-term spaceflight or for post-flight rehabilitation. Based on GATO hypothesis, predictions on muscle remodeling caused by illumination, sound/noise, and gravidity are discussed.

Designing synthetic regulatory networks capable of self-organizing cell polarization

How cells form global, self-organized structures using genetically encoded molecular rules remains elusive. Here, we take a synthetic biology approach to investigate the design principles governing cell polarization. First, using a coarse-grained computational model, we searched for all possible simple networks that can achieve polarization. All solutions contained one of three minimal motifs: positive feedback, mutual inhibition, or inhibitor with positive feedback. These minimal motifs alone could achieve polarization under limited conditions; circuits that combined two or more of these motifs were significantly more robust. With these design principles as a blueprint, we experimentally constructed artificial polarization networks in yeast, using a toolkit of chimeric signaling proteins that spatially direct the synthesis and degradation of phosphatidylinositol (3,4,5)-trisphosphate (PIP(3)). Circuits with combinatorial motifs yielded clear foci of synthetic PIP(3) that can persist for nearly an hour. Thus, by harnessing localization-regulated signaling molecules, we can engineer simple molecular circuits that reliably execute spatial self-organized programs.

Traveling waves in actin dynamics and cell motility

Much of current understanding of cell motility arose from studying steady treadmilling of actin arrays. Recently, there have been a growing number of observations of a more complex, non-steady, actin behavior, including self-organized waves. It is becoming clear that these waves result from activation and inhibition feedbacks in actin dynamics acting on different scales, but the exact molecular nature of these feedbacks and the respective roles of biomechanics and biochemistry are still unclear. Here, we review recent advances achieved in experimental and theoretical studies of actin waves and discuss mechanisms and physiological significance of wavy protrusions.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.