Mathematical Modelling of Bacterial Quorum Sensing: A Review.

Bacterial quorum sensing (QS) refers to the process of cell-to-cell bacterial communication enabled through the production and sensing of the local concentration of small molecules called autoinducers to regulate the production of gene products (e.g. enzymes or virulence factors). Through autoinducers, bacteria interact with individuals of the same species, other bacterial species, and with their host. Among QS-regulated processes mediated through autoinducers are aggregation, biofilm formation, bioluminescence, and sporulation. Autoinducers are therefore "master" regulators of bacterial lifestyles. For over 10 years, mathematical modelling of QS has sought, in parallel to experimental discoveries, to elucidate the mechanisms regulating this process. In this review, we present the progress in mathematical modelling of QS, highlighting the various theoretical approaches that have been used and discussing some of the insights that have emerged. Modelling of QS has benefited almost from the onset of the involvement of experimentalists, with many of the papers which we review, published in non-mathematical journals. This review therefore attempts to give a broad overview of the topic to the mathematical biology community, as well as the current modelling efforts and future challenges.

Microenvironmental Niches and Sanctuaries: A Route to Acquired Resistance.

A tumor vasculature that is functionally abnormal results in irregular gradients of metabolites and drugs within the tumor tissue. Recently, significant efforts have been committed to experimentally examine how cellular response to anti-cancer treatments varies based on the environment in which the cells are grown. In vitro studies point to specific conditions in which tumor cells can remain dormant and survive the treatment. In vivo results suggest that cells can escape the effects of drug therapy in tissue regions that are poorly penetrated by the drugs. Better understanding how the tumor microenvironments influence the emergence of drug resistance in both primary and metastatic tumors may improve drug development and the design of more effective therapeutic protocols. This chapter presents a hybrid agent-based model of the growth of tumor micrometastases and explores how microenvironmental factors can contribute to the development of acquired resistance in response to a DNA damaging drug. The specific microenvironments of interest in this work are tumor hypoxic niches and tumor normoxic sanctuaries with poor drug penetration. We aim to quantify how spatial constraints of limited drug transport and quiescent cell survival contribute to the development of drug resistant tumors.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h'(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h'(u) is constant k, (2) h'(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

Eco-Evolutionary Effects of Bacterial Cooperation on Phage Therapy: An Unknown Risk?

If there is something we have learned from the antibiotic era, it is that indiscriminate use of a therapeutic agent without a clear understanding of its long-term evolutionary impact can have enormous health repercussions. This knowledge is particularly relevant when the therapeutic agents are remarkably adaptable and diverse biological entities capable of a plethora of interactions, most of which remain largely unexplored. Although phage therapy (PT) undoubtedly holds the potential to save lives, its current efficacy in case studies recalls the golden era of antibiotics, when these compounds were highly effective and the possibility of them becoming ineffective seemed remote. Safe PT schemes depend on our understanding of how phages interact with, and evolve in, highly complex environments. Here, we summarize and review emerging evidence in a commonly overlooked theme in PT: bacteria-phage interactions. In particular, we discuss the influence of quorum sensing (QS) on phage susceptibility, the consequent role of phages in modulating bacterial cooperation, and the potential implications of this relationship in PT, including how we can use this knowledge to inform PT strategies. We highlight that the influence of QS on phage susceptibility seems to be widespread but can have contrasting outcomes depending on the bacterial host, underscoring the need to thoroughly characterize this link in various bacterial models. Furthermore, we encourage researchers to exploit competition experiments, experimental evolution, and mathematical modeling to explore this relationship further in relevant infection models. Finally, we emphasize that long-term PT success requires research on phage ecology and evolution to inform the design of optimal therapeutic schemes.

Drug-Induced Resistance in Micrometastases: Analysis of Spatio-Temporal Cell Lineages.

Resistance to anti-cancer drugs is a major cause of treatment failure. While several intracellular mechanisms of resistance have been postulated, the role of extrinsic factors in the development of resistance in individual tumor cells is still not fully understood. Here we used a hybrid agent-based model to investigate how sensitive tumor cells develop drug resistance in the heterogeneous tumor microenvironment. We characterized the spatio-temporal evolution of lineages of the resistant cells and examined how resistance at the single-cell level contributes to the overall tumor resistance. We also developed new methods to track tumor cell adaptation, to trace cell viability trajectories and to examine the three-dimensional spatio-temporal lineage trees. Our findings indicate that drug-induced resistance can result from cells adaptation to the changes in drug distribution. Two modes of cell adaptation were identified that coincide with microenvironmental niches-areas sheltered by cell micro-communities (protectorates) or regions with limited drug penetration (refuga or sanctuaries). We also recognized that certain cells gave rise to lineages of resistant cells (precursors of resistance) and pinpointed three temporal periods and spatial locations at which such cells emerged. This supports the hypothesis that tumor micrometastases do not need to harbor cell populations with pre-existing resistance, but that individual tumor cells can adapt and develop resistance induced by the drug during the treatment.

Re-thinking the Etiological Framework of Neurodegeneration.

Neurodegenerative diseases are among the leading causes of disability and death worldwide. The disease-related socioeconomic burden is expected to increase with the steadily increasing life expectancy. In spite of decades of clinical and basic research, most strategies designed to manage degenerative brain diseases are palliative. This is not surprising as neurodegeneration progresses "silently" for decades before symptoms are noticed. Importantly, conceptual models with heuristic value used to study neurodegeneration have been constructed retrospectively, based on signs and symptoms already present in affected patients; a circumstance that may confound causes and consequences. Hence, innovative, paradigm-shifting views of the etiology of these diseases are necessary to enable their timely prevention and treatment. Here, we outline four alternative views, not mutually exclusive, on different etiological paths toward neurodegeneration. First, we propose neurodegeneration as being a secondary outcome of a primary cardiovascular cause with vascular pathology disrupting the vital homeostatic interactions between the vasculature and the brain, resulting in cognitive impairment, dementia, and cerebrovascular events such as stroke. Second, we suggest that the persistence of senescent cells in neuronal circuits may favor, together with systemic metabolic diseases, neurodegeneration to occur. Third, we argue that neurodegeneration may start in response to altered body and brain trophic interactions established via the hardwire that connects peripheral targets with central neuronal structures or by means of extracellular vesicle (EV)-mediated communication. Lastly, we elaborate on how lifespan body dysbiosis may be linked to the origin of neurodegeneration. We highlight the existence of bacterial products that modulate the gut-brain axis causing neuroinflammation and neuronal dysfunction. As a concluding section, we end by recommending research avenues to investigate these etiological paths in the future. We think that this requires an integrated, interdisciplinary conceptual research approach based on the investigation of the multimodal aspects of physiology and pathophysiology. It involves utilizing proper conceptual models, experimental animal units, and identifying currently unused opportunities derived from human data. Overall, the proposed etiological paths and experimental recommendations will be important guidelines for future cross-discipline research to overcome the translational roadblock and to develop causative treatments for neurodegenerative diseases.

Differential Equations Models to Study Quorum Sensing.

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Pyocyanin Restricts Social Cheating in Pseudomonas aeruginosa.

Quorum sensing (QS) in Pseudomonas aeruginosa coordinates the expression of virulence factors, such as exoproteases and siderophores, that are public goods utilized by the whole population of bacteria, regardless of whether they invested or not in their production. These public goods can be used by QS defective mutants for growth, and since these mutants do not contribute to public goods production, they are considered social cheaters. Pyocyanin is a phenazine that is a toxic, QS-controlled metabolite produced by P. aeruginosa. It is a redox-active compound and promotes the generation of reactive oxygen species; it also possesses antibacterial properties and increases fitness in competition with other bacterial species. Since QS-deficient individuals are less able to tolerate oxidative stress, we hypothesized that the pyocyanin produced by the wild-type population could promote selection of functional QS systems in this bacterium. Here, we demonstrate, using competition experiments and mathematical models, that, indeed, pyocyanin increases the fitness of the cooperative QS-proficient individuals and restricts the appearance of social cheaters. In addition, we also show that pyocyanin is able to select QS in other bacteria such as Acinetobacter baumannii.

Selection of Functional Quorum Sensing Systems by Lysogenic Bacteriophages in Pseudomonas aeruginosa.

Quorum sensing (QS) in Pseudomonas aeruginosa coordinates the expression of virulence factors, some of which are used as public goods. Since their production is a cooperative behavior, it is susceptible to social cheating in which non-cooperative QS deficient mutants use the resources without investing in their production. Nevertheless, functional QS systems are abundant; hence, mechanisms regulating the amount of cheating should exist. Evidence that demonstrates a tight relationship between QS and the susceptibility of bacteria against the attack of lytic phages is increasing; nevertheless, the relationship between temperate phages and QS has been much less explored. Therefore, in this work, we studied the effects of having a functional QS system on the susceptibility to temperate bacteriophages and how this affects the bacterial and phage dynamics. We find that both experimentally and using mathematical models, that the lysogenic bacteriophages D3112 and JBD30 select QS-proficient P. aeruginosa phenotypes as compared to the QS-deficient mutants during competition experiments with mixed strain populations in vitro and in vivo in Galleria mellonella, in spite of the fact that both phages replicate better in the wild-type background. We show that this phenomenon restricts social cheating, and we propose that temperate phages may constitute an important selective pressure toward the conservation of bacterial QS.