Inferring local molecular dynamics from the global actin network structure: A case study of 2D synthetic branching actin networks.

Cells rely on their cytoskeleton for key processes including division and directed motility. Actin filaments are a primary constituent of the cytoskeleton. Although actin filaments can create a variety of network architectures linked to distinct cell functions, the microscale molecular interactions that give rise to these macroscale structures are not well understood. In this work, we investigate the microscale mechanisms that produce different branched actin network structures using an iterative classification approach. First, we employ a simple yet comprehensive agent-based model that produces synthetic actin networks with precise control over the microscale dynamics. Then we apply machine learning techniques to classify actin networks based on measurable network density and geometry, identifying key mechanistic processes that lead to particular branched actin network architectures. Extensive computational experiments reveal that the most accurate method uses a combination of supervised learning based on network density and unsupervised learning based on network symmetry. This framework can potentially serve as a powerful tool to discover the molecular interactions that produce the wide variety of actin network configurations associated with normal development as well as pathological conditions such as cancer.

Efficient parameter generation for constrained models using MCMC.

Mathematical models of complex systems rely on parameter values to produce a desired behavior. As mathematical and computational models increase in complexity, it becomes correspondingly difficult to find parameter values that satisfy system constraints. We propose a Markov Chain Monte Carlo (MCMC) approach for the problem of constrained model parameter generation by designing a Markov chain that efficiently explores a model's parameter space. We demonstrate the use of our proposed methodology to analyze responses of a newly constructed bistability-constrained model of protein phosphorylation to perturbations in the underlying protein network. Our results suggest that parameter generation for constrained models using MCMC provides powerful tools for modeling-aided analysis of complex natural processes.

Scalable Gromov-Wasserstein Based Comparison of Biological Time Series.

A time series is an extremely abundant data type arising in many areas of scientific research, including the biological sciences. Any method that compares time series data relies on a pairwise distance between trajectories, and the choice of distance measure determines the accuracy and speed of the time series comparison. This paper introduces an optimal transport type distance for comparing time series trajectories that are allowed to lie in spaces of different dimensions and/or with differing numbers of points possibly unequally spaced along each trajectory. The construction is based on a modified Gromov-Wasserstein distance optimization program, reducing the problem to a Wasserstein distance on the real line. The resulting program has a closed-form solution and can be computed quickly due to the scalability of the one-dimensional Wasserstein distance. We discuss theoretical properties of this distance measure, and empirically demonstrate the performance of the proposed distance on several datasets with a range of characteristics commonly found in biologically relevant data. We also use our proposed distance to demonstrate that averaging oscillatory time series trajectories using the recently proposed Fused Gromov-Wasserstein barycenter retains more characteristics in the averaged trajectory when compared to traditional averaging, which demonstrates the applicability of Fused Gromov-Wasserstein barycenters for biological time series. Fast and user friendly software for computing the proposed distance and related applications is provided. The proposed distance allows fast and meaningful comparison of biological time series and can be efficiently used in a wide range of applications.

Dynein localization and pronuclear movement in the C. elegans zygote.

Centrosomes serve as a site for microtubule nucleation and these microtubules will grow and interact with the motor protein dynein at the cortex. The position of the centrosomes determines where the mitotic spindle will develop across all cell types. Centrosome positioning is achieved through dynein and microtubule-mediated force generation. The mechanism and regulation of force generation during centrosome positioning are not fully understood. Centrosome and pronuclear movement in the first cell cycle of the Caenorhabditis elegans early embryo undergoes both centration and rotation prior to cell division. The proteins LET-99 and GPB-1 have been postulated to have a role in force generation associated with pronuclear centration and rotation dynamics. When the expression of these proteins is perturbed, pronuclear positioning exhibits a movement defect characterized by oscillatory ("wobble") behavior of the pronuclear complex (PNC). To determine if this movement defect is due to an effect on cortical dynein distribution, we utilize RNAi-mediated knockdown of LET-99 and GPB-1 to induce wobble and assay for any effects on GFP-tagged dynein localization in the early C. elegans embryo. To compare and quantify the movement defect produced by the knockdown of LET-99 and GPB-1, we devised a quantification method that measures the strength of wobble ("wobble metric") observed under these experimental conditions. Our quantification of pronuclear complex dynamics and dynein localization shows that loss of LET-99 and GPB-1 induces a similar movement defect which is independent of cortical dynein localization in the early C. elegans embryo.

Simulated actin reorganization mediated by motor proteins.

Cortical actin networks are highly dynamic and play critical roles in shaping the mechanical properties of cells. The actin cytoskeleton undergoes significant reorganization in many different contexts, including during directed cell migration and over the course of the cell cycle, when cortical actin can transition between different configurations such as open patched meshworks, homogeneous distributions, and aligned bundles. Several types of myosin motor proteins, characterized by different kinetic parameters, have been involved in this reorganization of actin filaments. Given the limitations in studying the interactions of actin with myosin in vivo, we propose stochastic agent-based models and develop a set of data analysis measures to assess how myosin motor proteins mediate various actin organizations. In particular, we identify individual motor parameters, such as motor binding rate and step size, that generate actin networks with different levels of contractility and different patterns of myosin motor localization, which have previously been observed experimentally. In simulations where two motor populations with distinct kinetic parameters interact with the same actin network, we find that motors may act in a complementary way, by tuning the actin network organization, or in an antagonistic way, where one motor emerges as dominant. This modeling and data analysis framework also uncovers parameter regimes where spatial segregation between motor populations is achieved. By allowing for changes in kinetic rates during the actin-myosin dynamic simulations, our work suggests that certain actin-myosin organizations may require additional regulation beyond mediation by motor proteins in order to reconfigure the cytoskeleton network on experimentally-observed timescales.

Identification of approximate symmetries in biological development.

Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However, precise measures of symmetry are often difficult to formulate and apply in a meaningful way to biological systems, where symmetries and asymmetries can be dynamic and transient, or be visually apparent but not reliably quantifiable using standard measures from mathematics and physics. Here, we present and illustrate a novel measure that draws on concepts from information theory to quantify the degree of symmetry, enabling the identification of approximate symmetries that may be present in a pattern or a biological image. We apply the measure to rotation, reflection and translation symmetries in patterns produced by a Turing model, as well as natural objects (algae, flowers and leaves). This method of symmetry quantification is unbiased and rigorous, and requires minimal manual processing compared to alternative measures. The proposed method is therefore a useful tool for comparison and identification of symmetries in biological systems, with potential future applications to symmetries that arise during development, as observed in vivo or as produced by mathematical models. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks.

In developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

Coordination of local and long range signaling modulates developmental patterning.

The development of multicellular organisms relies on correct patterns of cell fates to produce functional tissues in the mature organism. A commonly observed developmental pattern consists of alternating cell fates, where neighboring cells take on distinct cell fates characterized by contrasting gene and protein expression levels, and this cell fate pattern repeats over two or more cells. The patterns produced by these fate decisions are regulated by a small number of highly conserved signaling networks, some of which are mediated by long range diffusible signals and others mediated by local contact-dependent signals. However, it is not completely understood how local and long range signals associated with these networks interact to produce fate patterns that are both robust and flexible. Here we analyze mathematical models to investigate the patterning of cell fates in an array of cells, focusing on a two cell repeating pattern. Bifurcation analysis of a multicellular ODE model, where we consider the cells as discrete compartments, suggests that cells must balance sensitivity to external signals with robustness to perturbations. To focus on the patterning dynamics close to the bifurcation point, we derive a continuum PDE model that integrates local and long range signaling. For those cells with dynamics close to the bifurcation point, sensitivity to long range signals determines how far a pattern extends in space, while the number of local signaling connections determines the type of pattern produced. This investigation provides a general framework for understanding developmental patterning, and how both long range and local signals play a role in generating features observed across biology, such as species differences in nematode vulval development and insect bristle patterning, as well as medically relevant processes such as control of stem cell fate in the intestinal crypt.

CDC-42 Interactions with Par Proteins Are Critical for Proper Patterning in Polarization.

Many cells rearrange proteins and other components into spatially distinct domains in a process called polarization. This asymmetric patterning is required for a number of biological processes including asymmetric division, cell migration, and embryonic development. Proteins involved in polarization are highly conserved and include members of the Par and Rho protein families. Despite the importance of these proteins in polarization, it is not yet known how they interact and regulate each other to produce the protein localization patterns associated with polarization. In this study, we develop and analyse a biologically based mathematical model of polarization that incorporates interactions between Par and Rho proteins that are consistent with experimental observations of CDC-42. Using minimal network and eFAST sensitivity analyses, we demonstrate that CDC-42 is predicted to reinforce maintenance of anterior PAR protein polarity which in turn feedbacks to maintain CDC-42 polarization, as well as supporting posterior PAR protein polarization maintenance. The mechanisms for polarity maintenance identified by these methods are not sufficient for the generation of polarization in the absence of cortical flow. Additional inhibitory interactions mediated by the posterior Par proteins are predicted to play a role in the generation of Par protein polarity. More generally, these results provide new insights into the role of CDC-42 in polarization and the mutual regulation of key polarity determinants, in addition to providing a foundation for further investigations.

Evolution of Transcriptional Repressors Impacts Caenorhabditis Vulval Development.

Comparative genomic sequence analysis has found that the genes for many chromatin-associated proteins are poorly conserved, but the biological consequences of these sequence changes are not understood. Here, we show that four genes identified for an Inappropriate Vulval cell Proliferation (ivp) phenotype in the nematode Caenorhabditis briggsae exhibit distinct functions and genetic interactions when compared with their orthologs in C. elegans. Specifically, we show that the four C. briggsae ivp genes encode the noncanonical histone HTZ-1/H2A.z and three nematode-specific proteins predicted to function in the nucleus. The mutants exhibit ectopic vulval precursor cell proliferation (the multivulva [Muv] phenotype) due to inappropriate expression of the lin-3/EGF gene, and RNAseq analysis suggests a broad role for these ivp genes in transcriptional repression. Importantly, although the C. briggsae phenotypes have parallels with those seen in the C. elegans synMuv system, except for the highly conserved HTZ-1/H2A.z, comparable mutations in C. elegans ivp orthologs do not exhibit synMuv gene interactions or phenotypes. These results demonstrate the evolutionary changes that can underlie conserved biological outputs and argue that proteins critical to repress inappropriate expression from the genome participate in a rapidly evolving functional landscape.

Changes in morphogen kinetics and pollen grain size are potential mechanisms of aberrant pollen aperture patterning in previously observed and novel mutants of Arabidopsis thaliana.

Pollen provides an excellent system to study pattern formation at the single-cell level. Pollen surface is covered by the pollen wall exine, whose deposition is excluded from certain surface areas, the apertures, which vary between the species in their numbers, positions, and morphology. What determines aperture patterns is not understood. Arabidopsis thaliana normally develops three apertures, equally spaced along the pollen equator. However, Arabidopsis mutants whose pollen has higher ploidy and larger volume develop four or more apertures. To explore possible mechanisms responsible for aperture patterning, we developed a mathematical model based on the Gierer-Meinhardt system of equations. This model was able to recapitulate aperture patterns observed in the wild-type and higher-ploidy pollen. We then used this model to further explore geometric and kinetic factors that may influence aperture patterns and found that pollen size, as well as certain kinetic parameters, like diffusion and decay of morphogens, could play a role in formation of aperture patterns. In conjunction with mathematical modeling, we also performed a forward genetic screen in Arabidopsis and discovered two mutants with aperture patterns that had not been previously observed in this species but were predicted by our model. The macaron mutant develops a single ring-like aperture, matching the unusual ring-like pattern produced by the model. The doughnut mutant forms two pore-like apertures at the poles of the pollen grain. Further tests on these novel mutants, motivated by the modeling results, suggested the existence of an area of inhibition around apertures that prevents formation of additional apertures in their vicinity. This work demonstrates the ability of the theoretical model to help focus experimental efforts and to provide fundamental insights into an important biological process.

Centriole Biogenesis: Symmetry Breaking and Site Selection.

Centrioles must duplicate as cells progress through the cell cycle but it is unclear how the site of duplication is selected. A recent computational study demonstrates that two separate but interacting feedback mechanisms (autocatalytic activation and substrate depletion) are capable of selecting a single site for centriole biogenesis.

Unite to divide - how models and biological experimentation have come together to reveal mechanisms of cytokinesis.

Cytokinesis is the fundamental and ancient cellular process by which one cell physically divides into two. Cytokinesis in animal and fungal cells is achieved by contraction of an actomyosin cytoskeletal ring assembled in the cell cortex, typically at the cell equator. Cytokinesis is essential for the development of fertilized eggs into multicellular organisms and for homeostatic replenishment of cells. Correct execution of cytokinesis is also necessary for genome stability and the evasion of diseases including cancer. Cytokinesis has fascinated scientists for well over a century, but its speed and dynamics make experiments challenging to perform and interpret. The presence of redundant mechanisms is also a challenge to understand cytokinesis, leaving many fundamental questions unresolved. For example, how does a disordered cytoskeletal network transform into a coherent ring? What are the long-distance effects of localized contractility? Here, we provide a general introduction to 'modeling for biologists', and review how agent-based modeling and continuum mechanics modeling have helped to address these questions.

FACT complex gene duplicates exhibit redundant and non-redundant functions in C. elegans.

FACT (facilitates chromatin transcription) is a histone chaperone complex important in genomic processes including transcription, DNA replication, and DNA repair. FACT is composed of two proteins, SSRP1 and SPT16, which are highly conserved across eukaryotes. While the mechanisms for FACT in nucleosome reorganization and its relationship to DNA processes is well established, how these roles impact coordination in multicellular animal development are less well understood. Here we characterize the genes encoding FACT complex proteins in the nematode C. elegans. We show that whereas C. elegans includes one SPT16 gene (spt-16), two genes (hmg-3 and hmg-4) encode SSRP1 proteins. Depletion of FACT complex genes interferes with embryonic cell division and cell cycle timing generally, with anterior pharynx development especially sensitive to these defects. hmg-3 and hmg-4 exhibit redundancy for these maternally-provided embryonic functions, but are each uniquely required zygotically for normal germline development. This work provides a framework to study FACT gene function in developmental processes, and identifies that distinct functional requirements for gene duplicates can be manifest within a single tissue.

The importance of mechanical constraints for proper polarization and psuedo-cleavage furrow generation in the early Caenorhabditis elegans embryo.

Intracellular polarization, where a cell specifies a spatial axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode worm Caenorhabditis elegans (C. elegans), polarization is often accompanied by deformations of the cortex, a highly contractile structure consisting of actin filaments cross-linked by the motor protein myosin (actomyosin). It has been suggested that the eggshell surrounding the early embryo plays a role in polarization although its function is not understood. Here we develop a mathematical model which couples a reaction-diffusion model of actomyosin dynamics with a phase field model of the cell cortex to implicitly track cell shape changes in the early C. elegans embryo. We investigate the potential rigidity effect of the geometric constraint imposed by the presence and size of the eggshell on polarization dynamics. Our model suggests that the geometric constraint of the eggshell is essential for proper polarization and the size of the eggshell also affects the dynamics of polarization. Therefore, we conclude that geometric constraint on a cell might affect the dynamics of a biochemical process.

PAR proteins regulate maintenance-phase myosin dynamics during Caenorhabditis elegans zygote polarization.

Establishment of anterior-posterior polarity in the Caenorhabditis elegans zygote requires two different processes: mechanical activity of the actin-myosin cortex and biochemical activity of partitioning-defective (PAR) proteins. Here we analyze how PARs regulate the behavior of the cortical motor protein nonmuscle myosin (NMY-2) to complement recent efforts that investigate how PARs regulate the Rho GTPase CDC-42, which in turn regulates the actin-myosin cortex. We find that PAR-3 and PAR-6 concentrate CDC-42-dependent NMY-2 in the anterior cortex, whereas PAR-2 inhibits CDC-42-dependent NMY-2 in the posterior domain by inhibiting PAR-3 and PAR-6. In addition, we find that PAR-1 and PAR-3 are necessary for inhibiting movement of NMY-2 across the cortex. PAR-1 protects NMY-2 from being moved across the cortex by forces likely originating in the cytoplasm. Meanwhile, PAR-3 stabilizes NMY-2 against PAR-2 and PAR-6 dynamics on the cortex. We find that PAR signaling fulfills two roles: localizing NMY-2 to the anterior cortex and preventing displacement of the polarized cortical actin-myosin network.

Existence and uniqueness for a coupled PDE model for motor-induced microtubule organization.

Microtubules (MTs) are protein filaments that provide structure to the cytoskeleton of cells and a platform for the movement of intracellular substances. The spatial organization of MTs is crucial for a cell's form and function. MTs interact with a class of proteins called motor proteins that can transport and position individual filaments, thus contributing to overall organization. In this paper, we study the mathematical properties of a coupled partial differential equation (PDE) model, introduced by White et al. in 2015, that describes the motor-induced organization of MTs. The model consists of a nonlinear coupling of a hyperbolic PDE for bound motor proteins, a parabolic PDE for unbound motor proteins, and a transport equation for MT dynamics. We locally smooth the motor drift velocity in the equation for bound motor proteins. The mollification is not only critical for the analysis of the model, but also adds biological realism. We then use a Banach Fixed Point argument to show local existence and uniqueness of mild solutions. We highlight the applicability of the model by showing numerical simulations that are consistent with in vitro experiments.

Differing roles for sur-2/MED23 in C. elegans and C. briggsae vulval development.

Normal vulval development in the nematode Caenorhabditis briggsae is identical to that in the related Caenorhabditis elegans. However, several experiments suggest that there are differences between the two species with respect to the contribution of EGF/Ras signaling. To investigate these differences genetically, we have characterized a C. briggsae mutant strain that phenocopies the effect observed when C. briggsae animals are treated with U0126, an inhibitor of the EGF pathway component MEK. We identify that the gene affected in the mutant strain is Cbr-sur-2, which encodes a MED23 mediator complex protein that acts downstream of EGF signaling in C. elegans and other organisms, such as mammals. When Cbr-sur-2 and Cel-sur-2 mutants are compared, we find that the production of additional vulval cells from P5.p and P7.p in C. elegans is dependent on proper development of P6.p, while C. briggsae does not have a similar requirement. Combined chemical and genetic interference with the EGF pathway completely eliminates vulval development in C. elegans but not in C. briggsae. Our results provide genetic evidence for the differing requirements for EGF signaling in the two species.

A computational model predicts genetic nodes that allow switching between species-specific responses in a conserved signaling network.

Cell signaling networks regulate a variety of developmental and physiological processes, and changes in their response to external stimuli are often implicated in disease initiation and progression. To elucidate how different responses can arise from conserved signaling networks, we have developed a mathematical model of the well-characterized Caenorhabditis vulval development network involving EGF, Wnt and Notch signaling that recapitulates biologically observed behaviors. We experimentally block a specific element of the EGF pathway (MEK), and find different behaviors in vulval development in two Caenorhabditis species, C. elegans and C. briggsae. When we separate our parameters into subsets that correspond to these two responses, they yield model behaviors that are consistent with observed experimental results, despite the initial parameter grouping based on perturbation in a single node of the EGF pathway. Finally, our analysis predicts specific parameters that may be critical for the theoretically and experimentally observed differences, suggesting modifications that might allow intentional switching between the two species' responses. Our results indicate that all manipulations within a signal transduction pathway do not yield the same outcome, and provide a framework to identify the specific genetic perturbations within a conserved network that will confer unique behaviors on the network.

Antagonistic Behaviors of NMY-1 and NMY-2 Maintain Ring Channels in the C. elegans Gonad.

Contractile rings play critical roles in a number of biological processes, including oogenesis, wound healing, and cytokinesis. In many cases, the activity of motor proteins such as nonmuscle myosins is required for appropriate constriction of these contractile rings. In the gonad of the nematode worm Caenorhabditis elegans, ring channels are a specialized form of contractile ring that are maintained at a constant diameter before oogenesis. We propose a model of ring channel maintenance that explicitly incorporates force generation by motor proteins that can act normally or tangentially to the ring channel opening. We find that both modes of force generation are needed to maintain the ring channels. We demonstrate experimentally that the type II myosins NMY-1 and NMY-2 antagonize each other in the ring channels by producing force in perpendicular directions: the experimental depletion of NMY-1/theoretical decrease in orthogonal force allows premature ring constriction and cellularization, whereas the experimental depletion of NMY-2/theoretical decrease in tangential force opens the ring channels and prevents cellularization. Together, our experimental and theoretical results show that both forces, mediated by NMY-1 and NMY-2, are crucial for maintaining the appropriate ring channel diameter and dynamics throughout the gonad.