Identifying Markov Chain Models from Time-to-Event Data: An Algebraic Approach.

Many biological and medical questions can be modeled using time-to-event data in finite-state Markov chains, with the phase-type distribution describing intervals between events. We solve the inverse problem: given a phase-type distribution, can we identify the transition rate parameters of the underlying Markov chain? For a specific class of solvable Markov models, we show this problem has a unique solution up to finite symmetry transformations, and we outline a recursive method for computing symbolic solutions for these models across any number of states. Using the Thomas decomposition technique from computer algebra, we further provide symbolic solutions for any model.

Evolution of biological cooperation: an algorithmic approach.

This manuscript presents an algorithmic approach to cooperation in biological systems, drawing on fundamental ideas from statistical mechanics and probability theory. Fisher's geometric model of adaptation suggests that the evolution of organisms well adapted to multiple constraints comes at a significant complexity cost. By utilizing combinatorial models of fitness, we demonstrate that the probability of adapting to all constraints decreases exponentially with the number of constraints, thereby generalizing Fisher's result.

A Simple 3-Parameter Model for Cancer Incidences.

We propose a simple 3-parameter model that provides very good fits for incidence curves of 18 common solid cancers even when variations due to different locations, races, or periods are taken into account. From a data perspective, we use model selection (Akaike information criterion) to show that this model, which is based on the Weibull distribution, outperforms other simple models like the Gamma distribution. From a modeling perspective, the Weibull distribution can be justified as modeling the accumulation of driver events, which establishes a link to stem cell division based cancer development models and a connection to a recursion formula for intrinsic cancer risk published by Wu et al.

Geometric analysis of pathways dynamics: Application to versatility of TGF-β receptors.

We propose a new geometric approach to describe the qualitative dynamics of chemical reactions networks. By this method we identify metastable regimes, defined as low dimensional regions of the phase space close to which the dynamics is much slower compared to the rest of the phase space. These metastable regimes depend on the network topology and on the orders of magnitude of the kinetic parameters.

A Geometric Method for Model Reduction of Biochemical Networks with Polynomial Rate Functions.

Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds.