WASSERSTEIN DISTANCE ON SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS

In this paper, we shall focus on the Wasserstein distance between two jump processes determined by stochastic differential equations in Rd or the Riemannian manifold M. As an application, the study on the Wasserstein distance implies that the law of the subordinated Brownian motion on M is different from the one of the canonical projected process of the Marcus-type equation with jumps valued in the bundle of orthonormal frames O(M)

Horizontal transfer between loose compartments stabilizes replication of fragmented ribozymes

The emergence of replicases that can replicate themselves is a central issue in the origin of life. Recent experiments suggest that such replicases can be realized if an RNA polymerase ribozyme is divided into fragments short enough to be replicable by the ribozyme and if these fragments self-assemble into a functional ribozyme. However, the continued self-replication of such replicases requires that the production of every essential fragment be balanced and sustained. Here, we use mathematical modeling to investigate whether and under what conditions fragmented replicases achieve continued self-replication. We first show that under a simple batch condition, the replicases fail to display continued self-replication owing to positive feedback inherent in these replicases. This positive feedback inevitably biases replication toward a subset of fragments, so that the replicases eventually fail to sustain the production of all essential fragments. We then show that this inherent instability can be resolved by small rates of random content exchange between loose compartments (i.e., horizontal transfer). In this case, the balanced production of all fragments is achieved through negative frequency-dependent selection operating in the population dynamics of compartments. This selection mechanism arises from an interaction mediated by horizontal transfer between intracellular and intercellular symmetry breaking. The horizontal transfer also ensures the presence of all essential fragments in each compartment, sustaining self-replication. Taken together, our results underline compartmentalization and horizontal transfer in the origin of the first self-replicating replicases.Comment: 14 pages, 4 figures, and supplemental materia