A Logic for Checking the Probabilistic Steady-State Properties of Reaction Networks.

Designing probabilistic reaction models and determining their stochastic kinetic parameters are major issues in systems biology. To assist in the construction of reaction network models, we introduce a logic that allows one to express asymptotic properties about the steady-state stochastic dynamics of a reaction network. Basically, the formulas can express properties on expectancies, variances, and covariances. If a formula encoding for experimental observations on the system is not satisfiable, then the reaction network model can be rejected. We demonstrate that deciding the satisfiability of a formula is NP-hard, but we provide a decision method based on solving systems of polynomial constraints. We illustrate our method on a toy example.

Model of the delayed translation of cyclin B maternal mRNA after sea urchin fertilization.

Sea urchin eggs exhibit a cap-dependent increase in protein synthesis within minutes after fertilization. This rise in protein synthesis occurs at a constant rate for a great number of proteins translated from the different available mRNAs. Surprisingly, we found that cyclin B, a major cell-cycle regulator, follows a synthesis pattern that is distinct from the global protein population, so we developed a mathematical model to analyze this dissimilarity in biosynthesis kinetic patterns. The model includes two pathways for cyclin B mRNA entry into the translational machinery: one from immediately available mRNA (mRNAcyclinB) and one from mRNA activated solely after fertilization (XXmRNAcyclinB). Two coefficients, α and ß, were added to fit the measured scales of global protein and cyclin B synthesis, respectively. The model was simplified to identify the synthesis parameters and to allow its simulation. The calculated parameters for activation of the specific cyclin B synthesis pathway after fertilization included a kinetic constant (ka ) of 0.024 sec-1 , for the activation of XXmRNAcyclinB, and a critical time interval (t2 ) of 42 min. The proportion of XXmRNAcyclinB form was also calculated to be largely dominant over the mRNAcyclinB form. Regulation of cyclin B biosynthesis is an example of a select protein whose translation is controlled by pathways that are distinct from housekeeping proteins, even though both involve the same cap-dependent initiation pathway. Therefore, this model should help provide insight to the signaling utilized for the biosynthesis of cyclin B and other select proteins. Mol. Reprod. Dev. 83: 1070-1082, 2016. © 2016 Wiley Periodicals, Inc.