Chemical reaction systems are dynamical systems that arise in chemical engineering and systems biology. In this work, we consider the question of whether the minimal (in a precise sense) multistationary chemical reaction networks, which we propose to call 'atoms of multistationarity,' characterize the entire set of multistationary networks. Our main result states that the answer to this question is 'yes' in the context of fully open continuous-flow stirred-tank reactors (CFSTRs), which are networks in which all chemical species take part in the inflow and outflow. In order to prove this result, we show that if a subnetwork admits multiple steady states, then these steady states can be lifted to a larger network, provided that the two networks share the same stoichiometric subspace. We also prove an analogous result when a smaller network is obtained from a larger network by 'removing species.' Our results provide the mathematical foundation for a technique used by SiegalGaskins et al. of establishing bistability by way of 'network ancestry.' Additionally, our work provides sufficient conditions for establishing multistationarity by way of atoms and moreover reduces the problem of classifying multistationary CFSTRs to that of cataloging atoms of multistationarity. As an application, we enumerate and classify all 386 bimolecular and reversible two-reaction networks. Of these, exactly 35 admit multiple positive steady states. Moreover, each admits a unique minimal multistationary subnetwork, and these subnetworks form a poset (with respect to the relation of 'removing species') which has 11 minimal elements (the atoms of multistationarity).
Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network N that is embedded in a larger network G, which means that N is obtained from G by removing some subsets of chemical species and reactions. This embedding relation is a significant generalization of the subnetwork relation. For arbitrary networks, it is not true that if N is embedded in G, then the steady states of N lift to G. Nonetheless, this does hold for certain classes of networks; one such class is that of fully open networks. This motivates the search for embedding-minimal multistationary networks: those networks which admit multiple steady states but no proper, embedded networks admit multiple steady states. We present results about such minimal networks, including several new constructions of infinite families of these networks.
Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. Bistable reaction networks are posited to underlie biochemical switches, which motivates the following question: which reaction networks have the capacity for multiple steady states? Mathematically, this asks: among certain parametrized families of polynomial systems, which admit multiple positive roots? No complete answer is known. This work analyzes the smallest networks, those with only a few chemical species or reactions. For these "smallest" networks, we completely answer the question of multistationarity and, in some cases, multistability too, thereby extending related work of Boros. Our results highlight the role played by the Newton polytope of a network (the convex hull of the reactant vectors). Also, our work is motivated by recent results that explain how a given network's capacity for multistationarity arises from that of certain related networks which are typically smaller. Hence, we are interested in classifying small multistationary networks, and our work forms a first step in this direction.
In this paper we develop a new mathematical model of immunotherapy and cancer vaccination, focusing on the role of antigen presentation and co-stimulatory signaling pathways in cancer immunology. We investigate the effect of different cancer vaccination protocols on the well-documented phenomena of cancer dormancy and recurrence, and we provide a possible explanation of why adoptive (i.e. passive) immunotherapy protocols can sometimes actually promote tumour growth instead of inhibiting it (a phenomenon called immunostimulation), as opposed to active vaccination protocols based on tumour-antigen pulsed dendritic cells. Significantly, the results of our computational simulations suggest that elevated numbers of professional antigen presenting cells correlate well with prolonged time periods of cancer dormancy.
Certain chemical reaction networks (CRNs) when modeled as a deterministic dynamical system taken with mass-action kinetics have the property of reaction network detailed balance (RNDB) which is achieved by imposing network-related constraints on the reaction rate constants. Markov chains (whether arising as models of CRNs or otherwise) have their own notion of detailed balance, imposed by the network structure of the graph of the transition matrix of the Markov chain. When considering Markov chains arising from chemical reaction networks with mass-action kinetics, we will refer to this property as Markov chain detailed balance (MCDB). Finally, we refer to the stochastic analog of RNDB as Whittle stochastic detailed balance (WSDB). It is known that RNDB and WSDB are equivalent. We prove that WSDB and MCDB are also intimately related but are not equivalent. While RNDB implies MCDB, the converse is not true. The conditions on rate constants that result in networks with MCDB but without RNDB are stringent, and thus examples of this phenomenon are rare, a notable exception is a network whose Markov chain is a birth and death process. We give a new algorithm to find conditions on the rate constants that are required for MCDB.
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