A family tree of Markov models in systems biology

Motivated by applications in systems biology, we seek a probabilistic framework based on Markov processes to represent intracellular processes. We review the formal relationships between different stochastic models referred to in the systems biology literature. As part of this review, we present a novel derivation of the differential Chapman-Kolmogorov equation for a general multidimensional Markov process made up of both continuous and jump processes. We start with the definition of a time-derivative for a probability density but place no restrictions on the probability distribution, in particular, we do not assume it to be confined to a region that has a surface (on which the probability is zero). In our derivation, the master equation gives the jump part of the Markov process while the Fokker-Planck equation gives the continuous part. We thereby sketch a {}``family tree'' for stochastic models in systems biology, providing explicit derivations of their formal relationship and clarifying assumptions involved.Comment: 18 pages, 2 figure

Stochasticity in the miR-9/Hes1 oscillatory network can account for clonal heterogeneity in the timing of differentiation.

Recent studies suggest that cells make stochastic choices with respect to differentiation or division. However, the molecular mechanism underlying such stochasticity is unknown. We previously proposed that the timing of vertebrate neuronal differentiation is regulated by molecular oscillations of a transcriptional repressor, HES1, tuned by a post-transcriptional repressor, miR-9. Here, we computationally model the effects of intrinsic noise on the Hes1/miR-9 oscillator as a consequence of low molecular numbers of interacting species, determined experimentally. We report that increased stochasticity spreads the timing of differentiation in a population, such that initially equivalent cells differentiate over a period of time. Surprisingly, inherent stochasticity also increases the robustness of the progenitor state and lessens the impact of unequal, random distribution of molecules at cell division on the temporal spread of differentiation at the population level. This advantageous use of biological noise contrasts with the view that noise needs to be counteracted

Tourette’s disorder and other tic disorders in DSM-5: a comment

Classification of tic disorders will be revised in the forthcoming edition of the Diagnostic and Statistical Manual of Mental Disorders (5th ed.; DSM-5). We do not support the suggestion to move tic disorders to “Anxiety and Obsessive–Compulsive Disorders”, if the section “Disorders Usually First Diagnosed in Infancy, Childhood, or Adolescence” is not retained. Other than that, most proposed changes of the criteria for tic disorders contain a number of welcome improvements, e.g., the more unified definition of tics including the removal of the term “stereotyped” and the better capture of the temporal pattern of tics (e.g., removal of the maximum 3 months criterion for a tic-free period in chronic tic disorders). But, unfortunately there are some inconsistencies in detail, e.g., the unification of diagnostic criteria for tic disorders had not been consistently pursued in transient tic disorder. In sum, the proposed DSM-5 criteria could be seen as an important step forward particularly in clinical routine. However, continued research is needed to justify the existing and proposed classification of tic disorders as well as to better clarify what other changes should be made in the DSM-5 and beyond

Three-Dimensional Traction Force Microscopy: A New Tool for Quantifying Cell-Matrix Interactions

The interactions between biochemical processes and mechanical signaling play important roles during various cellular processes such as wound healing, embryogenesis, metastasis, and cell migration. While traditional traction force measurements have provided quantitative information about cell matrix interactions in two dimensions, recent studies have shown significant differences in the behavior and morphology of cells when placed in three-dimensional environments. Hence new quantitative experimental techniques are needed to accurately determine cell traction forces in three dimensions. Recently, two approaches both based on laser scanning confocal microscopy have emerged to address this need. This study highlights the details, implementation and advantages of such a three-dimensional imaging methodology with the capability to compute cellular traction forces dynamically during cell migration and locomotion. An application of this newly developed three-dimensional traction force microscopy (3D TFM) technique to single cell migration studies of 3T3 fibroblasts is presented to show that this methodology offers a new quantitative vantage point to investigate the three-dimensional nature of cell-ECM interactions

Matrix Rigidity Induces Osteolytic Gene Expression of Metastatic Breast Cancer Cells

Nearly 70% of breast cancer patients with advanced disease will develop bone metastases. Once established in bone, tumor cells produce factors that cause changes in normal bone remodeling, such as parathyroid hormone-related protein (PTHrP). While enhanced expression of PTHrP is known to stimulate osteoclasts to resorb bone, the environmental factors driving tumor cells to express PTHrP in the early stages of development of metastatic bone disease are unknown. In this study, we have shown that tumor cells known to metastasize to bone respond to 2D substrates with rigidities comparable to that of the bone microenvironment by increasing expression and production of PTHrP. The cellular response is regulated by Rho-dependent actomyosin contractility mediated by TGF-ß signaling. Inhibition of Rho-associated kinase (ROCK) using both pharmacological and genetic approaches decreased PTHrP expression. Furthermore, cells expressing a dominant negative form of the TGF-ß receptor did not respond to substrate rigidity, and inhibition of ROCK decreased PTHrP expression induced by exogenous TGF-ß. These observations suggest a role for the differential rigidity of the mineralized bone microenvironment in early stages of tumor-induced osteolysis, which is especially important in metastatic cancer since many cancers (such as those of the breast and lung) preferentially metastasize to bone

Stochastic Responses May Allow Genetically Diverse Cell Populations to Optimize Performance with Simpler Signaling Networks

Two theories have emerged for the role that stochasticity plays in biological responses: first, that it degrades biological responses, so the performance of biological signaling machinery could be improved by increasing molecular copy numbers of key proteins; second, that it enhances biological performance, by enabling diversification of population-level responses. Using T cell biology as an example, we demonstrate that these roles for stochastic responses are not sufficient to understand experimental observations of stochastic response in complex biological systems that utilize environmental and genetic diversity to make cooperative responses. We propose a new role for stochastic responses in biology: they enable populations to make complex responses with simpler biochemical signaling machinery than would be required in the absence of stochasticity. Thus, the evolution of stochastic responses may be linked to the evolvability of different signaling machineries.National Institutes of Health (U.S.). Pioneer Awar

From Molecular Signal Activation to Locomotion: An Integrated, Multiscale Analysis of Cell Motility on Defined Matrices

The adhesion, mechanics, and motility of eukaryotic cells are highly sensitive to the ligand density and stiffness of the extracellular matrix (ECM). This relationship bears profound implications for stem cell engineering, tumor invasion and metastasis. Yet, our quantitative understanding of how ECM biophysical properties, mechanotransductive signals, and assembly of contractile and adhesive structures collude to control these cell behaviors remains extremely limited. Here we present a novel multiscale model of cell migration on ECMs of defined biophysical properties that integrates local activation of biochemical signals with adhesion and force generation at the cell-ECM interface. We capture the mechanosensitivity of individual cellular components by dynamically coupling ECM properties to the activation of Rho and Rac GTPases in specific portions of the cell with actomyosin contractility, cell-ECM adhesion bond formation and rupture, and process extension and retraction. We show that our framework is capable of recreating key experimentally-observed features of the relationship between cell migration and ECM biophysical properties. In particular, our model predicts for the first time recently reported transitions from filopodial to “stick-slip” to gliding motility on ECMs of increasing stiffness, previously observed dependences of migration speed on ECM stiffness and ligand density, and high-resolution measurements of mechanosensitive protrusion dynamics during cell motility we newly obtained for this study. It also relates the biphasic dependence of cell migration speed on ECM stiffness to the tendency of the cell to polarize. By enabling the investigation of experimentally-inaccessible microscale relationships between mechanotransductive signaling, adhesion, and motility, our model offers new insight into how these factors interact with one another to produce complex migration patterns across a variety of ECM conditions