Identifying Intermolecular Interactions in Single-Molecule Localization Microscopy.

Intermolecular interactions underlie all cellular functions, yet visualizing these interactions at the single-molecule level remains challenging. Single-molecule localization microscopy (SMLM) offers a potential solution. Given a nanoscale map of two putative interaction partners, it should be possible to assign molecules either to the class of coupled pairs or to the class of non-coupled bystanders. Here, we developed a probabilistic algorithm that allows accurate determination of both the absolute number and the proportion of molecules that form coupled pairs. The algorithm calculates interaction probabilities for all possible pairs of localized molecules, selects the most likely interaction set, and corrects for any spurious colocalizations. Benchmarking this approach across a set of simulated molecular localization maps with varying densities (up to ∼ 50 molecules µm - 2 ) and localization precisions (5 to 50 nm) showed typical errors in the identification of correct pairs of only a few percent. At molecular densities of ∼ 5-10 molecules µm - 2 and localization precisions of 20-30 nm, which are typical parameters for SMLM imaging, the recall was ∼ 90%. The algorithm was effective at differentiating between non-interacting and coupled molecules both in simulations and experiments. Finally, it correctly inferred the number of coupled pairs over time in a simulated reaction-diffusion system, enabling determination of the underlying rate constants. The proposed approach promises to enable direct visualization and quantification of intermolecular interactions using SMLM.

Nonmodular oscillator and switch based on RNA decay drive regeneration of multimodal gene expression.

Periodic gene expression dynamics are key to cell and organism physiology. Studies of oscillatory expression have focused on networks with intuitive regulatory negative feedback loops, leaving unknown whether other common biochemical reactions can produce oscillations. Oscillation and noise have been proposed to support mammalian progenitor cells' capacity to restore heterogenous, multimodal expression from extreme subpopulations, but underlying networks and specific roles of noise remained elusive. We use mass-action-based models to show that regulated RNA degradation involving as few as two RNA species-applicable to nearly half of human protein-coding genes-can generate sustained oscillations without explicit feedback. Diverging oscillation periods synergize with noise to robustly restore cell populations' bimodal expression on timescales of days. The global bifurcation organizing this divergence relies on an oscillator and bistable switch which cannot be decomposed into two structural modules. Our work reveals surprisingly rich dynamics of post-transcriptional reactions and a potentially widespread mechanism underlying development, tissue regeneration, and cancer cell heterogeneity.

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.