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7 pages, 5 figures.-- This article is licensed under a Creative Commons Attribution 4.0 International License, Cell fate determination, the process through which cells commit to differentiated states is commonly mediated by gene regulatory motifs with mutually exclusive expression states. The classical deterministic picture for cell fate determination includes bistability and hysteresis, which enables the persistence of the acquired cellular state after withdrawal of the stimulus, ensuring a robust cellular response. However, stochasticity inherent to gene expression dynamics is not compatible with hysteresis, since the stationary solution of the governing Chemical Master Equation does not depend on the initial conditions. We provide a quantitative description of a transient hysteresis phenomenon reconciling experimental evidence of hysteretic behaviour in gene regulatory networks with inherent stochasticity: under sufficiently slow dynamics hysteresis is transient. We quantify this with an estimate of the convergence rate to the equilibrium and introduce a natural landscape capturing system’s evolution that, unlike traditional cell fate potential landscapes, is compatible with coexistence at the microscopic level, M.P. and A.A.A. acknowledge funding from grant PIE201870E041; I.O.M. acknowledges funding from Spanish MINECO (and the European Regional Development Fund) project SYNBIOCONTROL (grant number DPI2017-82896-C2-2-R). C.V. has been partially funded by the spanish MINECO project MTM2016-76497-R and Xunta de Galicia grant ED431C2018/033

3 pages, 1 figure.-- This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, Motivation: Gene regulation is inherently stochastic. In many applications concerning Systems and Synthetic Biology such as the reverse engineering and the de novo design of genetic circuits, stochastic effects (yet potentially crucial) are often neglected due to the high computational cost of stochastic simulations. With advances in these fields there is an increasing need of tools providing accurate approximations of the stochastic dynamics of gene regulatory networks (GRNs) with reduced computational effort. Results: This work presents SELANSI (SEmi-LAgrangian SImulation of GRNs), a software toolbox for the simulation of stochastic multidimensional gene regulatory networks. SELANSI exploits intrinsic structural properties of gene regulatory networks to accurately approximate the corresponding Chemical Master Equation with a partial integral differential equation that is solved by a semi-lagrangian method with high efficiency. Networks under consideration might involve multiple genes with self and cross regulations, in which genes can be regulated by different transcription factors. Moreover, the validity of the method is not restricted to a particular type of kinetics. The tool offers total flexibility regarding network topology, kinetics and parameterization, as well as simulation options. Availability and implementation: SELANSI runs under the MATLAB environment, and is available under GPLv3 license at https://sites.google.com/view/selansi, This work has been supported by Spanish MINECO grants AGL2015-67504- C3-2-R, PIE201230E0M2 (AAA), BES-2013-063112 (MP), MTM2016- 76497-R, MTM2013-47800-C2-1-P (CV) and MINECO and European Regional Development Fund DPI2014-55276-C5-2-R (IOM)

13 pages, 11 figures.--Thiis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited., In this work we explore two potential mechanisms inducing multiple equilibria for weakly reversible networks with mass-action kinetics. The study is performed on a class of polynomial dynamic systems that, under some mild assumptions, are able to accommodate in their state-space form weakly reversible mass-action kinetic networks. The contribution is twofold. We provide an explicit representation of the set of all positive equilibria attained by the system class in terms of a set of (positive parameter dependent) algebraic relations. With this in hand, we prove that deficiency-one networks can only admit multiple equilibria via folding of the equilibrium manifold, whereas a bifurcation leading to multiple branches is only possible in networks with deficiencies larger than one. Interestingly, some kinetic networks within this latter class are capable of sustaining multiple equilibria for any reaction simplex, as we illustrate with one example., Manuel Pájaro acknowledges support from Spanish MINECO