Reactive Vortexes in a Naturally Activated Process: Non-Diffusive Rotational Fluxes at Transition State Uncovered by Persistent Homology.

The dynamics of reaction coordinates during barrier-crossing are key to understanding activated processes in complex systems such as proteins. The default assumption from Kramers' physical intuition is that of a diffusion process. However, the dynamics of barrier-crossing in natural complex molecules are largely unexplored. Here we investigate the transition dynamics of alanine dipeptide isomerization, the simplest complex system with a large number of non-reaction coordinates that can serve as an adequate thermal bath feeding energy into the reaction coordinates. We separate conformations along the time axis and construct the dynamic probability surface of reaction. We quantify its topological structure and rotational flux using persistent homology and differential form. Our results uncovered a region with a strong reactive vortex in the configuration-time space, where the highest probability peak and the transition state ensemble are located. This reactive region contains strong rotational fluxes: Most reactive trajectories swirl multiple times around this region in the subspace of the two most important reaction coordinates. Furthermore, the rotational fluxes result from cooperative movement along the isocommitter surfaces and orthogonal barrier-crossing. Overall, our findings offer a first glimpse into the reactive vortex regions that characterize the non-diffusive dynamics of barrier-crossing of a naturally occurring activation process.

Identifying Transient Cells During Reprogramming via Persistent Homology.

Single-cell RNA sequencing is a powerful method that helps delineate the regulatory mechanisms shaping the diverse cellular populations. Heterogeneous cell populations consist of individual cells, and the expression of distinct sets of genes can differentiate one sub-population of cells from another, as they are responsible for the emergence of distinct cellular phenotypes. Of particular importance are cells at transition states that bridge these different cellular phenotypes. In this study, we develop a method to identify the cells at transition states bridging different cellular phenotypes. Our approach is based on persistent homology, which enabled us to identify the group of cells located on the boundaries between different sub-populations of cells. We applied this method to study the reprogramming of human fibroblasts toward induced pluripotent stem cells using single-cell time-course data. Even though only the data that is representative of the early stages of the reprogramming process are analyzed, we are able to uncover transient cells bridging different cell sub-populations. The most prominent group of transient cells are found to be enriched for NANOG, which is a known stem cell transcription factor that takes part in the maintenance of pluripotency and other stem cell marker genes. Overall, our method can identify cells in transient states bridging major cellular phenotypes, even though they are only a small fraction of the overall cell population. We also discuss how this approach can link the topology of the surface of cellular transcripts and bring order to the transition between cellular states and how it automatically uncovers the underlying time process.

Exact Probability Landscapes of Stochastic Phenotype Switching in Feed-Forward Loops: Phase Diagrams of Multimodality.

Feed-forward loops (FFLs) are among the most ubiquitously found motifs of reaction networks in nature. However, little is known about their stochastic behavior and the variety of network phenotypes they can exhibit. In this study, we provide full characterizations of the properties of stochastic multimodality of FFLs, and how switching between different network phenotypes are controlled. We have computed the exact steady-state probability landscapes of all eight types of coherent and incoherent FFLs using the finite-butter Accurate Chemical Master Equation (ACME) algorithm, and quantified the exact topological features of their high-dimensional probability landscapes using persistent homology. Through analysis of the degree of multimodality for each of a set of 10,812 probability landscapes, where each landscape resides over 105-106 microstates, we have constructed comprehensive phase diagrams of all relevant behavior of FFL multimodality over broad ranges of input and regulation intensities, as well as different regimes of promoter binding dynamics. In addition, we have quantified the topological sensitivity of the multimodality of the landscapes to regulation intensities. Our results show that with slow binding and unbinding dynamics of transcription factor to promoter, FFLs exhibit strong stochastic behavior that is very different from what would be inferred from deterministic models. In addition, input intensity play major roles in the phenotypes of FFLs: At weak input intensity, FFL exhibit monomodality, but strong input intensity may result in up to 6 stable phenotypes. Furthermore, we found that gene duplication can enlarge stable regions of specific multimodalities and enrich the phenotypic diversity of FFL networks, providing means for cells toward better adaptation to changing environment. Our results are directly applicable to analysis of behavior of FFLs in biological processes such as stem cell differentiation and for design of synthetic networks when certain phenotypic behavior is desired.

Exact Topology of the Dynamic Probability Surface of an Activated Process by Persistent Homology.

To gain insight into the reaction mechanism of activated processes, we introduce an exact approach for quantifying the topology of high-dimensional probability surfaces of the underlying dynamic processes. Instead of Morse indexes, we study the homology groups of a sequence of superlevel sets of the probability surface over high-dimensional configuration spaces using persistent homology. For alanine-dipeptide isomerization, a prototype of activated processes, we identify locations of probability peaks and connecting ridges, along with measures of their global prominence. Instead of a saddle point, the transition state ensemble (TSE) of conformations is at the most prominent probability peak after reactants/products, when proper reaction coordinates are included. Intuition-based models, even those exhibiting a double-well, fail to capture the dynamics of the activated process. Peak occurrence, prominence, and locations can be distorted upon subspace projection. While principal component analysis accounts for conformational variance, it inflates the complexity of the surface topology and destroys the dynamic properties of the topological features. In contrast, TSE emerges naturally as the most prominent peak beyond the reactant/product basins, when projected to a subspace of minimum dimension containing the reaction coordinates. Our approach is general and can be applied to investigate the topology of high-dimensional probability surfaces of other activated processes.

Cell-substrate mechanics guide collective cell migration through intercellular adhesion: a dynamic finite element cellular model.

During the process of tissue formation and regeneration, cells migrate collectively while remaining connected through intercellular adhesions. However, the roles of cell-substrate and cell-cell mechanical interactions in regulating collective cell migration are still unclear. In this study, we employ a newly developed finite element cellular model to study collective cell migration by exploring the effects of mechanical feedback between cell and substrate and mechanical signal transmission between adjacent cells. Our viscoelastic model of cells consists many triangular elements and is of high resolution. Cadherin adhesion between cells is modeled explicitly as linear springs at subcellular level. In addition, we incorporate a mechano-chemical feedback loop between cell-substrate mechanics and Rac-mediated cell protrusion. Our model can reproduce a number of experimentally observed patterns of collective cell migration during wound healing, including cell migration persistence, separation distance between cell pairs and migration direction. Moreover, we demonstrate that cell protrusion determined by the cell-substrate mechanics plays an important role in guiding persistent and oriented collective cell migration. Furthermore, this guidance cue can be maintained and transmitted to submarginal cells of long distance through intercellular adhesions. Our study illustrates that our finite element cellular model can be employed to study broad problems of complex tissue in dynamic changes at subcellular level.

Evolution of Coagulation-Fragmentation Stochastic Processes Using Accurate Chemical Master Equation Approach.

Coagulation and fragmentation (CF) is a fundamental process in which smaller particles attach to each other to form larger clusters while existing clusters break up into smaller particles . It is a ubiquitous process that plays important roles in many physical and biological phenomena. CF is typically a stochastic process that often occurs in confined spaces with a limited number of available particles . Here, we study the CF process formulated with the discrete Chemical Master Equation (dCME). Using the newly developed Accurate Chemical Master Equation (ACME) method, we examine the time-dependent behavior of the CF system. We investigate the effects of a number of important factors that influence the overall behavior of the system, including the dimensionality, the ratio of attachment to detachment rates among clusters, and the initial conditions. By comparing CF in one and three dimensions, we conclude that systems in three dimensions are more likely to form large clusters. We also demonstrate how the ratio of the attachment to detachment rates affects the dynamics and the steady-state of the system. Finally, we demonstrate the relationship between the formation of large clusters and the initial condition.

A viscoelastic model for axonal microtubule rupture.

Axon is an important part of the neuronal cells and axonal microtubules are bundles in axons. In axons, microtubules are coated with microtubule-associated protein tau, a natively unfolded filamentous protein in the central nervous system. These proteins are responsible for cross-linking axonal microtubule bundles. Through complimentary dimerization with other tau proteins, bridges are formed between nearby microtubules creating bundles. Formation of bundles of microtubules causes their transverse reinforcement and has been shown to enhance their ability to bear compressive loads. Though microtubules are conventionally regarded as bearing compressive loads, in certain circumstances during traumatic brain injuries, they are placed in tension. In our model, microtubule bundles were formed from a large number of discrete masses. We employed Standard Linear Solid model (SLS), a viscoelastic model, to computationally simulate microtubules. In this study, we investigated the dynamic responses of two dimensional axonal microtubules under suddenly applied end forces by implementing discrete masses connected to their neighboring masses with a Standard Linear Solid unit. We also investigated the effect of the applied force rate and magnitude on the deformation of bundles. Under tension, a microtubule fiber may rupture as a result of a sudden force. Using the developed model, we could predict the critical regions of the axonal microtubule bundles in the presence of varying end forces. We finally analyzed the nature of microtubular failure under varying mechanical stresses.

Dynamic response of axonal microtubules under suddenly applied end forces.

Axon is a filament in neuronal system and axonal microtubules are bundles in axons. In axons, microtubules are coated with microtubule-associated protein tau, a natively unfolded profuse filamentous protein in the central nervous system. These proteins are responsible for the cross-linked structure of the axonal microtubule bundles. Through complimentary dimerization with other tau proteins, bridges are formed to nearby microtubules to create bundles. The transverse reinforcement of microtubules by cross-linking to the cytoskeleton has been shown to enhance their ability to bear compressive loads. Though microtubules are conventionally regarded as bearing compressive loads, in certain circumstances such as in traumatic stretch injury, they are placed in tension. We employ Standard Linear Solid, a viscoelastic model, to computationally simulate microtubules. This study investigates the dynamic response of two dimensional axonal microtubules under suddenly applied end forces. We obtain the results for steady state behavior of axonal microtubule for different forces.