Jacek Miękisz

Evolutionary and asymptotic stability in symmetric multi-player games

Abstract.We provide a classification of symmetric three-player games with two strategies and investigate evolutionary and asymptotic stability (in the replicator dynamics) of their Nash equilibria. We discuss similarities and differences between two-player and multi-player games. In particular, we construct examples which exhibit a novel behavior not found in two-player games.

Evolutionary Game Theory and Population Dynamics

Many socio-economic and biological processes can be modeled as systems of interacting individuals. The behaviour of such systems can be often described within game-theoretic models. In these lecture notes, we introduce fundamental concepts of evolutionary game theory and review basic properties of deterministic replicator dynamics and stochastic dynamics of finite populations. We discuss stability of equilibria in deterministic dynamics with migration, time-delay, and in stochastic dynamics of well-mixed populations and spatial games with local interactions. We analyze the dependence of the long-run behaviour of a population on various parameters such as the time delay, the noise level, and the size of the population.

Translational Repression Contributes Greater Noise to Gene Expression than Transcriptional Repression

Stochastic effects in gene expression may result in different physiological states of individual cells, with consequences for pathogen survival and artificial gene network design. We studied the contributions of a regulatory factor to gene expression noise in four basic mechanisms of negative gene expression control: 1), transcriptional regulation by a protein repressor, 2), translational repression by a protein; 3), transcriptional repression by RNA; and 4), RNA interference with the translation. We investigated a general model of a two-gene network, using the chemical master equation and a moment generating function approach. We compared the expression noise of genes with the same effective transcription and translation initiation rates resulting from the action of different repressors, whereas previous studies compared the noise of genes with the same mean expression level but different initiation rates. Our results show that translational repression results in a higher noise than repression on the promoter level, and that this relationship does not depend on quantitative parameter values. We also show that regulation of protein degradation contributes more noise than regulated degradation of mRNA. These are unexpected results, because previous investigations suggested that translational regulation is more accurate. The relative magnitude of the noise introduced by protein and RNA repressors depends on the protein and mRNA degradation rates, and we derived expressions for the threshold below which the noise introduced by a protein repressor is higher than the noise introduced by an RNA repressor.

Decomposing Noise in Biochemical Signaling Systems Highlights the Role of Protein Degradation

Stochasticity is an essential aspect of biochemical processes at the cellular level. We now know that living cells take advantage of stochasticity in some cases and counteract stochastic effects in others. Here we propose a method that allows us to calculate contributions of individual reactions to the total variability of a systems output. We demonstrate that reactions differ significantly in their relative impact on the total noise and we illustrate the importance of protein degradation on the overall variability for a range of molecular processes and signaling systems. With our flexible and generally applicable noise decomposition method, we are able to shed new, to our knowledge, light on the sources and propagation of noise in biochemical reaction networks; in particular, we are able to show how regulated protein degradation can be employed to reduce the noise in biochemical systems.

Nonperiodic long-range order for fast-decaying interactions at positive temperatures

We present the first example of an exponentially decaying interaction which gives rise to nonperiodic long-range order at positive temperatures.

Stable Quasicrystalline Ground States

We give strong evidence that noncrystalline materials such as quasicrystals or incommensurate solids are not exceptions, but rather are generic in some regions of phase space. We show this by constructing classical lattice-gas models with translation-invariant finite-range interactions and with a unique quasiperiodic ground state which is stable against small perturbations of two-body potentials. More generally, we provide a criterion for stability of nonperiodic ground states.

Solving ramified optimal transport problem in the bayesian influence diagram framework

The goal of the ramified optimal transport is to find an optimal transport path between two given probability measures. One measure can be identified with a source while the other one with a target. The problem is well known to be NP---hard. We develop an algorithm for solving a ramified optimal transport problem within the framework of Bayesian networks. It is based on the decision strategy optimisation technique that utilises self---annealing ideas of Chen---style stochastic optimisation. Resulting transport paths are represented in the form of tree---shaped structures. The effectiveness of the algorithm has been tested on computer---generated examples.

A symmetry group of a Thue - Morse quasicrystal

We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue - Morse quasicrystal, i.e. of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction.

Effective reaction rates for diffusion-limited reaction cycles

Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.

Influence of a Topology of a Spring Network on its Ability to Learn Mechanical Behaviour

We discuss how the topology of the spring system/network affects its ability to learn a desired mechanical behaviour. To ensure such a behaviour, physical parameters of springs of the system are adjusted by an appropriate gradient descent learning algorithm. We find the betweenness centrality measure particularly convenient to describe topology of the spring system structure with the best mechanical properties. We apply our results to refine an algorithm generating the structure of a spring network. We also present numerical results confirming our statements.