Intrinsic catalytic properties of histone H3 lysine-9 methyltransferases preserve monomethylation levels under low S-adenosylmethionine.

S-adenosylmethionine (SAM) is the methyl donor for site-specific methylation reactions on histone proteins, imparting key epigenetic information. During SAM-depleted conditions that can arise from dietary methionine restriction, lysine di- and tri-methylation are reduced while sites such as Histone-3 lysine-9 (H3K9) are actively maintained, allowing cells to restore higher-state methylation upon metabolic recovery. Here, we investigated if the intrinsic catalytic properties of H3K9 histone methyltransferases (HMTs) contribute to this epigenetic persistence. We employed systematic kinetic analyses and substrate binding assays using four recombinant H3K9 HMTs (i.e., EHMT1, EHMT2, SUV39H1, and SUV39H2). At both high and low (i.e., sub-saturating) SAM, all HMTs displayed the highest catalytic efficiency (kcat/KM) for monomethylation compared to di- and trimethylation on H3 peptide substrates. The favored monomethylation reaction was also reflected in kcat values, apart from SUV39H2 which displayed a similar kcat regardless of substrate methylation state. Using differentially methylated nucleosomes as substrates, kinetic analyses of EHMT1 and EHMT2 revealed similar catalytic preferences. Orthogonal binding assays revealed only small differences in substrate affinity across methylation states, suggesting that catalytic steps dictate the monomethylation preferences of EHMT1, EHMT2, and SUV39H1. To link in vitro catalytic rates with nuclear methylation dynamics, we built a mathematical model incorporating measured kinetic parameters and a time course of mass spectrometry-based H3K9 methylation measurements following cellular SAM depletion. The model revealed that the intrinsic kinetic constants of the catalytic domains could recapitulate in vivo observations. Together, these results suggest catalytic discrimination by H3K9 HMTs maintains nuclear H3K9me1, ensuring epigenetic persistence after metabolic stress.

Homeostasis and injectivity: a reaction network perspective.

Homeostasis represents the idea that a feature may remain invariant despite changes in some external parameters. We establish a connection between homeostasis and injectivity for reaction network models. In particular, we show that a reaction network cannot exhibit homeostasis if a modified version of the network (which we call homeostasis-associated network) is injective. We provide examples of reaction networks which can or cannot exhibit homeostasis by analyzing the injectivity of their homeostasis-associated networks.

Foundations of static and dynamic absolute concentration robustness.

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness-the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (Science 327:1389-1391, 2010) (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.

Multistationarity in Cyclic Sequestration-Transmutation Networks.

We consider a natural class of reaction networks which consist of reactions where either two species can inactivate each other (i.e., sequestration), or some species can be transformed into another (i.e., transmutation), in a way that gives rise to a feedback cycle. We completely characterize the capacity of multistationarity of these networks. This is especially interesting because such networks provide simple examples of "atoms of multistationarity", i.e., minimal networks that can give rise to multiple positive steady states.

Autocatalytic recombination systems: A reaction network perspective.

Autocatalytic systems called hypercycles are very often incorporated in "origin of life" models. We investigate the dynamics of certain related models called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that it can be analyzed using well-chosen autonomous polynomial dynamical systems. Moreover, we use results from reaction network theory to prove persistence and permanence of several families of bimolecular autocatalytic systems called autocatalytic recombination systems.

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems.

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversible deficiency zero (WR0) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, WR0 systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

Delay stability of reaction systems.

Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.

A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems.

Mass-action kinetics and its generalizations appear in mathematical models of (bio)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.

Realizations of kinetic differential equations.

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.

Conditions for extinction events in chemical reaction networks with discrete state spaces.

We study chemical reaction networks with discrete state spaces and present sufficient conditions on the structure of the network that guarantee the system exhibits an extinction event. The conditions we derive involve creating a modified chemical reaction network called a domination-expanded reaction network and then checking properties of this network. Unlike previous results, our analysis allows algorithmic implementation via systems of equalities and inequalities and suggests sequences of reactions which may lead to extinction events. We apply the results to several networks including an EnvZ-OmpR signaling pathway in Escherichia coli.

High Throughput Analyses of Budding Yeast ARSs Reveal New DNA Elements Capable of Conferring Centromere-Independent Plasmid Propagation.

The ability of plasmids to propagate in Saccharomyces cerevisiae has been instrumental in defining eukaryotic chromosomal control elements. Stable propagation demands both plasmid replication, which requires a chromosomal replication origin (i.e., an ARS), and plasmid distribution to dividing cells, which requires either a chromosomal centromere for segregation or a plasmid-partitioning element. While our knowledge of yeast ARSs and centromeres is relatively advanced, we know less about chromosomal regions that can function as plasmid partitioning elements. The Rap1 protein-binding site (RAP1) present in transcriptional silencers and telomeres of budding yeast is a known plasmid-partitioning element that functions to anchor a plasmid to the inner nuclear membrane (INM), which in turn facilitates plasmid distribution to daughter cells. This Rap1-dependent INM-anchoring also has an important chromosomal role in higher-order chromosomal structures that enhance transcriptional silencing and telomere stability. Thus, plasmid partitioning can reflect fundamental features of chromosome structure and biology, yet a systematic screen for plasmid partitioning elements has not been reported. Here, we couple deep sequencing with competitive growth experiments of a plasmid library containing thousands of short ARS fragments to identify new plasmid partitioning elements. Competitive growth experiments were performed with libraries that differed only in terms of the presence or absence of a centromere. Comparisons of the behavior of ARS fragments in the two experiments allowed us to identify sequences that were likely to drive plasmid partitioning. In addition to the silencer RAP1 site, we identified 74 new putative plasmid-partitioning motifs predicted to act as binding sites for DNA binding proteins enriched for roles in negative regulation of gene expression and G2/M-phase associated biology. These data expand our knowledge of chromosomal elements that may function in plasmid partitioning and suggest underlying biological roles shared by such elements.

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.

Modeling the endocrine control of vitellogenin production in female rainbow trout.

The rainbow trout endocrine system is sensitive to changes in annual day length, which is likely the principal environmental cue controlling its reproductive cycle. This study focuses on the endocrine regulation of vitellogenin (Vg) protein synthesis, which is the major egg yolk precursor in this fish species. We present a model of Vg production in female rainbow trout which incorporates a biological pathway beginning with sex steroid estradiol-17ß levels in the plasma and concluding with Vg secretion by the liver and sequestration in the oocytes. Numerical simulation results based on this model are compared with experimental data for estrogen receptor mRNA, Vg mRNA, and Vg in the plasma from female rainbow trout over a normal annual reproductive cycle. We also analyze the response of the model to parameter changes. The model is subsequently tested against experimental data from female trout under a compressed photoperiod regime. Comparison of numerical and experimental results suggests the possibility of a time-dependent change in oocyte Vg uptake rate. This model is part of a larger effort that is developing a mathematical description of the endocrine control of reproduction in female rainbow trout. We anticipate that these mathematical and computational models will play an important role in future regulatory toxicity assessments and in the prediction of ecological risk.

Dynamical properties of Discrete Reaction Networks.

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: irreducibility, i.e., the ability to reach any discrete state from any other state; and recurrence, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.

Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks.

We describe a necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive, for general chemical reaction networks modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph), which is a bipartite graph with different nodes representing species and reactions. If the SR graph satisfies certain conditions, similar to the conditions for ruling out multiple equilibria in spatially homogeneous differential equations systems, then the corresponding mass-action reaction-diffusion system cannot exhibit zero-eigenvalue Turing instability for any parameter values. On the other hand, if the graph-theoretic condition for ruling out zero-eigenvalue Turing instability is not satisfied, then the corresponding model may display zero-eigenvalue Turing instability for some parameter values. The technique is illustrated with a model of a bifunctional enzyme.

MOST HOMEOMORPHISMS WITH A FIXED POINT HAVE A CANTOR SET OF FIXED POINTS.

We show that, for any n ≠ 2, most orientation preserving homeomorphisms of the sphere S2n have a Cantor set of fixed points. In other words, the set of such homeomorphisms that do not have a Cantor set of fixed points is of the first Baire category within the set of all homeomorphisms. Similarly, most orientation reversing homeomorphisms of the sphere S2n+1 have a Cantor set of fixed points for any n ≠ 0. More generally, suppose that M is a compact manifold of dimension > 1 and ≠ 4 and ℋ is an open set of homeomorphisms h : M → M such that all elements of ℋ have at least one fixed point. Then we show that most elements of ℋ have a Cantor set of fixed points.

Statistical Model for Biochemical Network Inference.

We describe a statistical method for predicting most likely reactions in a biochemical reaction network from the longitudinal data on species concentrations. Such data is relatively easily available in biochemical laboratories, for instance, via the popular RT-PCR technology. Under the assumed kinetics of the law of mass action, we also propose the data-based algorithms for estimating the prediction errors and for network dimension reduction. The second algorithm allows in particular for the application of the original algebraic inferential procedure described in [4] without the unnecessary restrictions on the dimension of the network stoichiometric space. Simulated examples of biochemical networks are analyzed, in order to assess the proposed methods' performance.

Mass Distributions of Linear Chain Polymers.

Biochemistry has many examples of linear chain polymers, i.e., molecules formed from a sequence of units from a finite set of possibilities; examples include proteins, RNA, single-stranded DNA, and paired DNA. In the field of mass spectrometry, it is useful to consider the idea of weighted alphabets, with a word inheriting weight from its letters. We describe the distribution of the mass of these words in terms of a simple recurrence relation, the general solution to that relation, and a canonical form that explicitly describes both the exponential form of this distribution and its periodic features, thus explaining a wave pattern that has been observed in protein mass databases. Further, we show that a pure exponential term dominates the distribution and that there is exactly one such purely exponential term. Finally, we illustrate the use of this theorem by describing a formula for the integer mass distribution of peptides and we compare our theoretical results with mass distributions of human and yeast peptides.

Periodic patterns in distributions of peptide masses.

We are investigating the distribution of the number of peptides for given masses, and especially the observation that peptide density reaches a local maximum approximately every 14Da. This wave pattern exists across species (e.g. human or yeast) and enzyme digestion techniques. To analyze this phenomenon we have developed a mathematical method for computing the mass distributions of peptides, and we present both theoretical and empirical evidence that this 14-Da periodicity does not arise from species selection of peptides but from the number- theoretic properties of the masses of amino acid residues. We also describe other, more subtle periodic patterns in the distribution of peptide masses. We also show that these periodic patterns are robust under a variety of conditions, including the addition of amino acid modifications and selection of mass accuracy scale. The method used here is also applicable to any family of sequential molecules, such as linear hydrocarbons, RNA, single- and double-stranded DNA.

Product-form stationary distributions for deficiency zero chemical reaction networks.

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.