Elimination of Intermediate Species in Multiscale Stochastic Reaction Networks

We study networks of biochemical reactions modelled by continuous-time Markov processes. Such networks typically contain many molecular species and reactions and are hard to study analytically as well as by simulation. Particularly, we are interested in reaction networks with intermediate species such as the substrate-enzyme complex in the Michaelis-Menten mechanism. These species are virtually in all real-world networks, they are typically short-lived, degraded at a fast rate and hard to observe experimentally. We provide conditions under which the Markov process of a multiscale reaction network with intermediate species is approximated in finite dimensional distribution by the Markov process of a simpler reduced reaction network without intermediate species. We do so by embedding the Markov processes into a one-parameter family of processes, where reaction rates and species abundances are scaled in the parameter. Further, we show that there are close links between these stochastic models and deterministic ODE models of the same networks

Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States

We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady states, called node balanced steady states, is naturally associated with such a reaction graph. We show that complex and detailed balanced steady states are special cases of node balanced steady states by choosing appropriate reaction graphs. Further, we show that node balanced steady states have properties analogous to complex balanced steady states, such as uniqueness and asymptotical stability in each stoichiometric compatibility class. Moreover, we associate an integer, called the deficiency, to a reaction graph that gives the number of independent relations in the reaction rate constants that need to be satisfied for a positive node balanced steady state to exist. The set of reaction graphs (modulo isomorphism) is equipped with a partial order that has the complex balanced reaction graph as minimal element. We relate this order to the deficiency and to the set of reaction rate constants for which a positive node balanced steady state exists

Limits for Stochastic Reaction Networks

Reaction systems have been introduced in the 70s to model biochemical systems. Nowadays their range of applications has increased and they are fruitfully used in different fields. The concept is simple: some chemical species react, the set of chemical reactions form a graph and a rate function is associated with each reaction. Such functions describe the speed of the different reactions, or their propensities. Two modelling regimes are then available: the evolution of the different species concentrations can be deterministically modelled through a system of ODE, while the counts of the different species at a certain time are stochastically modelled by means of a continuous-time Markov chain. Our work concerns primarily stochastic reaction systems, and their asymptotic properties. In Paper I, we consider a reaction system with intermediate species, i.e. species that are produced and fast degraded along a path of reactions. Let the rates of degradation of the intermediate species be functions of a parameter N that tends to infinity. We consider a reduced system where the intermediate species have been eliminated, and find conditions on the degradation rate of the intermediates such that the behaviour of the reduced network tends to that of the original one. In particular, we prove a uniform punctual convergence in distribution and weak convergence of the integrals of continuous functions along the paths of the two models. Under some extra conditions, we also prove weak convergence of the two processes. The result is stated in the setting of multiscale reaction systems: the amounts of all the species and the rates of all the reactions of the original model can scale as powers of N. A similar result also holds for the deterministic case, as shown in Appendix IA. In Paper II, we focus on the stationary distributions of the stochastic reaction systems. Specifically, we build a theory for stochastic reaction systems that is parallel to the deficiency zero theory for deterministic systems, which dates back to the 70s. A deficiency theory for stochastic reaction systems was missing, and few results connecting deficiency and stochastic reaction systems were known. The theory we build connects special form of product-form stationary distributions with structural properties of the reaction graph of the system. In Paper III, a special class of reaction systems is considered, namely systems exhibiting absolute concentration robust species. Such species, in the deterministic modelling regime, assume always the same value at any positive steady state. In the stochastic setting, we prove that, if the initial condition is a point in the basin of attraction of a positive steady state of the corresponding deterministic model and tends to infinity, then up to a fixed time T the counts of the species exhibiting absolute concentration robustness are, on average, near to their equilibrium value. The result is not obvious because when the counts of some species tend to infinity, so do some rate functions, and the study of the system may become hard. Moreover, the result states a substantial concordance between the paths of the stochastic and the deterministic models

Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness

Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an "absolute concentration robustness" (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Followup mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later mathematical work focused on the behavior of this same class of reaction networks, though under the assumption that the dynamics were stochastic. Under the stochastic assumption, it was proven that the system will undergo an extinction event with a probability of one so long as the system is conservative, showing starkly different long-time behavior than in the deterministic setting. Here we consider a general class of stochastic models that intersects with the class of ACR systems studied previously. We consider a specific system scaling over compact time intervals and prove that in a limit of this scaling the distribution of the abundances of the ACR species converges to a certain product-form Poisson distribution whose mean is the ACR value of the deterministic model. This result is in agreement with recent conjectures pertaining to the behavior of ACR networks endowed with stochastic kinetics, and helps to resolve the conflicting theoretical results pertaining to deterministic and stochastic models in this setting

Between external and internal space : an urban transition

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Architecture, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Page 283 blank. Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 278-282).The aim of this dissertation is to explore the evolution of both architecture and urban space, in terms of mutual relationship between solids and voids, with particular attention to two transitional moments of ancient and modern history: the Hellenistic and Baroque periods. This study is the result of the consideration that in certain periods, at least in western history, there is a clear predominance of either interior or exterior space in relation to architecture. If on one hand external space seems to predominate in Greek and modern architecture, interior space is prevalent between the Roman and the Renaissance periods. The hypothesis is that both the Hellenistic and Baroque periods represent intermediate phases in the historical transition between interior and exterior space and that this transition is manifested, through the transformations of the urban fabric, in the enclosed civic spaces of forums and squares. The methodological approach can be more easily described defining what this analysis is not meant to be: this examination is neither intended to be an urban theory nor a historical study. The intention is to interrelate theory and history, remaining distant from the necessary abstraction of urban design theory and, at the same time, avoiding the indispensable specificity and attention to details required by architecture history.by Daniele Cappelletti.S.M