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.

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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 are modeled explicitly as linear springs at subcellular level. In addition, we incorporate a mechanochemical 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 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

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.

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.