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.

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.

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.

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.

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.

Stronger net posterior cortical forces and asymmetric microtubule arrays produce simultaneous centration and rotation of the pronuclear complex in the early Caenorhabditis elegans embryo.

Positioning of microtubule-organizing centers (MTOCs) incorporates biochemical and mechanical cues for proper alignment of the mitotic spindle and cell division site. Current experimental and theoretical studies in the early Caenorhabditis elegans embryo assume remarkable changes in the origin and polarity of forces acting on the MTOCs. These changes must occur over a few minutes, between initial centration and rotation of the pronuclear complex and entry into mitosis, and the models do not replicate in vivo timing of centration and rotation. Here we propose a model that incorporates asymmetry in the microtubule arrays generated by each MTOC, which we demonstrate with in vivo measurements, and a similar asymmetric force profile to that required for posterior-directed spindle displacement during mitosis. We find that these asymmetries are capable of and important for recapitulating the simultaneous centration and rotation of the pronuclear complex observed in vivo. The combination of theoretical and experimental evidence provided here offers a unified framework for the spatial organization and forces needed for pronuclear centration, rotation, and spindle displacement in the early C. elegans embryo.

Protein abundance may regulate sensitivity to external cues in polarized cells.

Cell polarization is a ubiquitous process which results in cellular constituents being organized into discrete intracellular spatial domains. It occurs in a variety of cell types, including epithelial cells, immune system cells and neurons. A key player in this process is the Par protein family whose asymmetric localization to anterior and posterior parts of the cell is crucial for proper division and cell fate specification. In this paper, we explore a stochastic analogue of the temporal model of Par protein interactions first developed in Dawes & Munro (Dawes and Munro 2011 Biophys. J. 101, 1412-1422. (doi:10.1016/j.bpj.2011.07.030)). We focus on how protein abundance influences the behaviour of both the deterministic and stochastic versions of the model. In Dawes & Munro (2011), it was found that bistable behaviour in the temporal model of Par protein led to the existence of complementary domains in the corresponding spatio-temporal model. Here, we find that the corresponding temporal stochastic model permits switching behaviour (the model solution 'jumps' between steady states) for lower protein abundances, whereas for higher protein abundances the stochastic and deterministic models are in good agreement (the model solution evolves to one of two steady states). This led us to the testable hypothesis that cells with lower abundances of Par protein may be more sensitive to external cues, whereas cells with higher abundances of Par protein may be less sensitive to external cues. In order to gain more control over the precise abundance of Par protein, we proposed and explored a second model (again, examining both deterministic and stochastic versions) in which the total number of Par molecules is conserved. We found that this model required an additional dimerization reaction in the cytoplasm in order for bistable and switching behaviour to be found. Once this additional reaction was included, we found that both the first and second models gave qualitatively similar results but in different regions of the parameter space, suggesting a further regulatory mechanism that cells could potentially use to modulate their response to external signals.

Actomyosin regulation and symmetry breaking in a model of polarization in the early Caenorhabditis elegans embryo : symmetry breaking in cell polarization.

Polarization, whereby a cell defines a spatial axis by segregating specific determinants to distinct regions, is an essential and highly conserved biological process. The process of polarization is initiated by a cue that breaks an initially symmetric distribution of determinants, allowing for a spatially asymmetric redistribution. The nature of this cue is currently not well understood. Utilizing the conservation of polarization process and its determinants, we theoretically investigate the nature of the cue and the regulation of contractility that enables the establishment of polarity in early embryos of the nematode worm Caenorhabditis elegans (C. elegans). Our biologically based model, which consists of coupled partial differential equations, suggests that a biochemical but not mechanical cue is sufficient for symmetry breaking, and inhibition of contractile elements by specific determinants is needed for sustained spatial redistribution.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos.

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.