Spectral properties of non-local uniformly-elliptic operators

In this paper we consider the spectral properties of a class of non-local uniformly elliptic operators, which arise from the study of non-local uniformly elliptic partial differential equations. Such equations arise naturally in the study of a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators, and the non-local perturbation is in the form of an integral term. We study the eigenvalues, the multiplicities of these eigenvalues, and the existence of corresponding positive eigenfunctions. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. However, we show that under suitable hypotheses, there still exists a principal eigenvalue of these operators

Self-adjoint boundary-value problems on time-scales

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form $$L u := -[p u^{ abla}]^{Delta} + qu,$$ on an arbitrary, bounded time-scale $mathbb{T}$, for suitable functions $p,q$, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space $L^2(mathbb{T}_kappa)$, in such a way that the resulting operator is self-adjoint, with compact resolvent (here, "self-adjoint" means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as self-adjoint, but have not demonstrated self-adjointness in the standard functional analytic sense

Translational control of gene expression via interacting feedback loops

Translation is a key step in the synthesis of proteins. Accordingly, cells have evolved an intricate array of control mechanisms to regulate this process. By constructing a multi-component mathematical framework for translation we uncover how translation may be controlled via interacting feedback loops. Our results reveal that this interplay gives rise to a remarkable range of protein synthesis dynamics, including oscillations, step-change and bistability. This suggests that cells may have recourse to a much richer set of control mechanisms than was previously understood.Comment: Supplementary Material Available on Reques

<i>Bacillus subtilis</i> extracellular protease production incurs a context-dependent cost

Rosazza T, Eigentler L, Earl C, Davidson FA, Stanley-Wall NR. Bacillus subtilis extracellular protease production incurs a context-dependent cost. Molecular Microbiology. 2023.Microbes encounter a wide range of polymeric nutrient sources in various environmental settings, which require processing to facilitate growth. Bacillus subtilis, a bacterium found in the rhizosphere and broader soil environment, is highly adaptable and resilient due to its ability to utilise diverse sources of carbon and nitrogen. Here, we explore the role of extracellular proteases in supporting growth and assess the cost associated with their production. We provide evidence of the essentiality of extracellular proteases when B. subtilis is provided with an abundant, but polymeric nutrient source and demonstrate the extracellular proteases as a shared public good that can operate over a distance. We show that B. subtilis is subjected to a public good dilemma, specifically in the context of growth sustained by the digestion of a polymeric food source. Furthermore, using mathematical simulations, we uncover that this selectively enforced dilemma is driven by the relative cost of producing the public good. Collectively, our findings reveal how bacteria can survive in environments that vary in terms of immediate nutrient accessibility and the consequent impact on the population composition. These findings enhance our fundamental understanding of how bacteria respond to diverse environments, which has importance to contexts ranging from survival in the soil to infection and pathogenesis scenarios

Founder cell configuration drives competitive outcome within colony biofilms

Bacteria can form dense communities called biofilms, where cells are embedded in a self-produced extracellular matrix. Exploiting competitive interactions between strains within the biofilm context can have potential applications in biological, medical, and industrial systems. By combining mathematical modelling with experimental assays, we reveal that spatial structure and competitive dynamics within biofilms are significantly affected by the location and density of the founder cells used to inoculate the biofilm. Using a species-independent theoretical framework describing colony biofilm formation, we show that the observed spatial structure and relative strain biomass in a mature biofilm comprising two isogenic strains can be mapped directly to the geographical distributions of founder cells. Moreover, we define a predictor of competitive outcome that accurately forecasts relative abundance of strains based solely on the founder cells’ potential for radial expansion. Consequently, we reveal that variability of competitive outcome in biofilms inoculated at low founder density is a natural consequence of the random positioning of founding cells in the inoculum. Extension of our study to non-isogenic strains that interact through local antagonisms, shows that even for strains with different competition strengths, a race for space remains the dominant mode of competition in low founder density biofilms. Our results, verified by experimental assays using Bacillus subtilis, highlight the importance of spatial dynamics on competitive interactions within biofilms and hence to related applications