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Discrete state Markov chains in discrete or continuous time are widely used to model phenomena in the social, physical and life sciences. In many cases, the model can feature a large state space, with extreme differences between the fastest and slowest transition timescales. Analysis of such ill-conditioned models is often intractable with finite precision linear algebra techniques. In this contribution, we propose a solution to this problem, namely partial graph transformation, to iteratively eliminate and renormalize states, producing a low-rank Markov chain from an ill-conditioned initial model. We show that the error induced by this procedure can be minimized by retaining both the renormalized nodes that represent metastable superbasins, and those through which reactive pathways concentrate, i.e. the dividing surface in the discrete state space. This procedure typically returns a much lower rank model, where trajectories can be efficiently generated with kinetic path sampling. We apply this approach to an ill-conditioned Markov chain for a model multi-community system, measuring the accuracy by direct comparison with trajectories and transition statistics. This article is part of a discussion meeting issue ‘Supercomputing simulations of advanced materials’.

From proteins to chromosomes, polymers fold into specific conformations that control their biological function. Polymer folding has long been studied with equilibrium thermodynamics, yet intracellular organization and regulation involve energy-consuming, active processes. Signatures of activity have been measured in the context of chromatin motion, which shows spatial correlations and enhanced subdiffusion only in the presence of adenosine triphosphate (ATP). Moreover, chromatin motion varies with genomic coordinate, pointing towards a heterogeneous pattern of active processes along the sequence. How do such patterns of activity affect the conformation of a polymer such as chromatin? We address this question by combining analytical theory and simulations to study a polymer subjected to sequence-dependent correlated active forces. Our analysis shows that a local increase in activity (larger active forces) can cause the polymer backbone to bend and expand, while less active segments straighten out and condense. Our simulations further predict that modest activity differences can drive compartmentalization of the polymer consistent with the patterns observed in chromosome conformation capture experiments. Moreover, segments of the polymer that show correlated active (sub)diffusion attract each other through effective long-ranged harmonic interactions, whereas anticorrelations lead to effective repulsions. Thus, our theory offers non-equilibrium mechanisms for forming genomic compartments, which cannot be distinguished from affinity-based folding using structural data alone. As a first step toward disentangling active and passive mechanisms of folding, we discuss a data-driven approach to discern if and how active processes affect genome organization.

We present a theoretical model that demonstrates the integral role chromosome organization and structural mechanics play in the spreading of histone modifications involved in epigenetic regulation. Our model shows that heterogeneous nucleosome positioning, and the resulting position-dependent mechanical properties, must be included to reproduce several qualitative features of experimental data of histone methylation spreading around an artificially induced “nucleation site.” We show that our model recreates both the extent of spreading and the presence of a subdominant peak upstream of the transcription start site. Our model indicates that the spreading of epigenetic modifications is sensitive to heterogeneity in chromatin organization and the resulting variability in the chromatin’s mechanical properties, suggesting that nucleosome spacing can directly control the conferral of epigenetic marks by modifying the structural mechanics of the chromosome. It further illustrates how the physical organization of the DNA polymer may play a significant role in re-establishing the epigenetic code upon cell division.

International audience; Markov chains can accurately model the state-to-state dynamics of a wide range of complex systems, but the underlying transition matrix is ill-conditioned when the dynamics feature a separation of timescales. Graph transformation (GT) provides a numerically stable method to compute exact mean first passage times (MF-PTs) between states, which are the usual dynamical observables, in continuous-time Markov chains (CTMCs). Here, we generalize the GT algorithm to discrete-time Markov chains (DTMCs), which are commonly estimated from simulation data, for example in the Markov State Model approach. We then consider the dimensionality reduction of CTMCs and DTMCs, which aids model interpretation and facilitates more expensive computations, including sampling of pathways. We perform a detailed numerical analysis of existing methods to compute the optimal reduced CTMC, given a partitioning of the network into metastable communities (macrostates) of nodes (microstates). We show that approaches based on linear algebra encounter numerical problems that arise from the requisite metastability. We propose an alternative approach, using GT to compute the matrix of intermicrostate MFPTs in the original Markov chain, from which a matrix of weighted intermacrostate MFPTs can be obtained. We also propose an approximation to the weighted-MFPT matrix in the strongly metastable limit. Inversion of the weighted-MFPT matrix, which is better conditioned than the matrices that must be inverted in alternative dimensionality reduction schemes, then yields the optimal reduced Markov chain. The superior numerical stability of the GT approach therefore enables us to realize optimal Markovian coarse-graining of systems with rare event dynamics.

Dysregulated activity of the protease matriptase is a key contributor to aggressive tumor growth, cancer metastasis, and osteoarthritis. Methods for the detection and quantification of matriptase activity and inhibition would be useful tools. To address this need, we developed a matriptase-sensitive protein biosensor based on a dimerization-dependent red fluorescent protein (ddRFP) reporter system. In this platform, two adjoining protein domains, connected by a protease-labile linker, produce fluorescence when assembled and are nonfluorescent when the linker is cleaved by matriptase. A panel of ddRFP-based matriptase biosensor designs was created that contained different linker lengths between the protein domains. These constructs were characterized for linker-specific cleavage, matriptase activity, and matriptase selectivity; a biosensor containing a RSKLRVGGH linker (termed B4) was expressed at high yields and displayed both high catalytic efficiency and matriptase specificity. This biosensor detects matriptase inhibition by soluble and yeast cell surface expressed inhibitor domains with up to a 5-fold dynamic range and also detects matriptase activity expressed by human cancer cell lines. In addition to matriptase, we highlight a strategy that can be used to create effective biosensors for quantifying activity and inhibition of other proteases of interest. [Image: see text]

Within a living cell, the myriad of proteins that bind DNA introduce heterogeneously spaced kinks into an otherwise semiflexible DNA double helix. To investigate the effects of heterogeneous nucleosome binding on chromatin organization, we extend the wormlike chain (WLC) model to include statistically spaced, rigid kinks. On time scales where nucleosome positions are fixed, we find that the probability of chromatin loop formation can differ by up to six orders of magnitude between two sets of nucleosome positions drawn from the same distribution. On longer time scales, we show that continuous re-randomization due to nucleosome turnover results in chromatin tracing out an effective WLC with a dramatically smaller Kuhn length than bare DNA. Together, these observations demonstrate that heterogeneity in nucleosome spacing acts as the dominant source of chromatin elasticity and governs both local and global chromatin organization.

Anti-cancer immunotherapy is encountering its own checkpoint. Responses are dramatic and long lasting but occur in a subset of tumors and are largely dependent upon the pre-existing immune contexture of individual cancers. Available data suggest that three landscapes best define the cancer microenvironment: immune-active, immune-deserted and immune-excluded. This trichotomy is observable across most solid tumors (although the frequency of each landscape varies depending on tumor tissue of origin) and is associated with cancer prognosis and response to checkpoint inhibitor therapy (CIT). Various gene signatures (e.g. Immunological Constant of Rejection - ICR and Tumor Inflammation Signature - TIS) that delineate these landscapes have been described by different groups. In an effort to explain the mechanisms of cancer immune responsiveness or resistance to CIT, several models have been proposed that are loosely associated with the three landscapes. Here, we propose a strategy to integrate compelling data from various paradigms into a “Theory of Everything”. Founded upon this unified theory, we also propose the creation of a task force led by the Society for Immunotherapy of Cancer (SITC) aimed at systematically addressing salient questions relevant to cancer immune responsiveness and immune evasion. This multidisciplinary effort will encompass aspects of genetics, tumor cell biology, and immunology that are pertinent to the understanding of this multifaceted problem.

Dysregulated matriptase activity has been established as a key contributor to cancer progression through its activation of growth factors, including the hepatocyte growth factor (HGF). Despite its critical role and prevalence in many human cancers, limitations to developing an effective matriptase inhibitor include weak binding affinity, poor selectivity, and short circulating half-life. We applied rational and combinatorial approaches to engineer a potent inhibitor based on the hepatocyte growth factor activator inhibitor type-1 (HAI-1), a natural matriptase inhibitor. The first Kunitz domain (KD1) of HAI-1 has been well established as a minimal matriptase-binding and inhibition domain, whereas the second Kunitz domain (KD2) is inactive and involved in negative regulation. Here, we replaced the inactive KD2 domain of HAI-1 with an engineered chimeric variant of KD2/KD1 domains and fused the resulting construct to an antibody Fc domain to increase valency and circulating serum half-life. The final protein variant contains four stoichiometric binding sites that we showed were needed to effectively inhibit matriptase with a K(i) of 70 ± 5 pm, an increase of 120-fold compared with the natural HAI-1 inhibitor, to our knowledge making it one of the most potent matriptase inhibitors identified to date. Furthermore, the engineered inhibitor demonstrates a protease selectivity profile similar to that of wildtype KD1 but distinct from that of HAI-1. It also inhibits activation of the natural pro-HGF substrate and matriptase expressed on cancer cells with at least an order of magnitude greater efficacy than KD1.