Comparative analysis of bacterial essential and nonessential genes with Hurst exponent based on chaos game representation

Essential genes are indispensable for the survival of an organism. Investigating features associated with gene essentiality is fundamental to the prediction and identification of essential genes with computational techniques. We use fractal theory approach to make comparative analysis of essential and nonessential genes in bacteria. The Hurst exponents of essential genes and nonessential genes available in the DEG database for 27 bacteria are calculated based on their gene chaos game representations. It is found that for most analyzed bacteria, weak negative correlation exists between Hurst exponent and gene length. Moreover, essential genes generally differ from nonessential genes in their Hurst exponent. For genes of similar length, the average Hurst exponent of essential genes is smaller than that of nonessential genes. The results of our work reveal that gene Hurst exponent is very probably useful gene feature for the algorithm predicting essential genes.     

Stability and periodic solutions for a model of bacterial resistance to antibiotics caused by mutations and plasmids

Bacterial resistance is one of the most prominent public health problems affecting the entire world population. Although some infectious diseases are no longer a problem as they were in the past, the acquisition of bacterial resistance continues to increase. In particular, antibiotics have been losing their effectiveness after decades of misuse and overuse, which has generated an emergency situation. In this work, we formulate and analyse a deterministic model for the population dynamics of susceptible and resistant bacteria to antibiotics, assuming that drug resistance is acquired through mutations and plasmid transmission. Qualitative analysis reveals the existence of a bacteria-free equilibrium, a resistant bacteria equilibrium, an a coexistence equilibrium and a limit cycle arising from Hopf bifurcation. The stability of the equilibria are given in terms of the growth rate of bacteria, the acquisition of resistance, as well as the elimination of bacteria due to the immune system and the action of antibiotics. Numerical simulations corroborate our analytical results, and illustrate the temporal dynamics of the susceptible and resistant bacteria.     

Mathematical study of a bacteria–fish model with level of infection structure

In this paper, we propose a mathematical model to study a bacteria–fish system, based upon the interactions between Clostridium botulinum and tilapia, Oerochromis mossambicus. The fish population is divided into susceptible and infected, and the infected fish population is considered structured by the level of infection. The model is thus a system with the infected fish equation being an evolution equation, while those corresponding to the susceptible fish and bacteria in water are ordinary differential equations. The model is firstly transformed into a system with distributed delay for susceptible fish and bacteria and, further, under some assumptions, into a system with discrete delay. The study of this system gives us some results concerning the existence, uniqueness, positivity and boundedness of solutions; we also discuss the existence and stability of its equilibrium points, including conditions for the appearance of Hopf bifurcation. The theoretical results are illustrated by some numerical simulations.     

Modeling the spread of an infectious disease with bacteria and carriers in the environment

An SIS model with immigration for the spread of an infectious disease with bacteria and carriers in the environment is proposed and analyzed. It is assumed that susceptibles get infected directly by infectives as well as by their contacts with bacteria discharged by infectives in the environment. The growth rate of density of bacteria is assumed to be proportional to the density of infectives and decreases naturally as well as by bacterial interactions with susceptibles and carriers. The carrier population density is considered to follow the logistic model and grows due to conducive human population density related factors. It is assumed further that the number of bacteria transported by carriers to susceptibles is proportional to densities of both bacteria and carriers. The model study shows that the spread of the infectious disease increases due to growth of bacteria and carriers in the environment and disease becomes more endemic due to immigration.     

Simple test cases for validating a finite element unstructured grid fecal bacteria transport model

This paper presents analytical test cases for tracer advection–diffusion-decay problems. The test cases are used to validate a finite element, unstructured grid fecal bacteria transport model. The test cases include the following domains: one-dimensional infinitely long river, two-dimensional half plane and two-dimensional infinitely long channel. In this work the bacteria are considered to enter the domain only through point sources. Analytical solutions are derived using either a Dirac delta function or a variable-width Gaussian function as a point source. Both analytical derivations and numerical simulations suggest that the error is maximised at the source. We present formulae for estimating the error caused by replacing a Dirac source with a Gaussian function in the numerical model. Furthermore, numerical simulations suggest that the best approximation for a Dirac source is a Gaussian whose width parameter is one third of the local mesh size.     

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.     

Modeling and stability analysis of a microalgal pond with nitrification

Microalgae culture fed with ammonium may face the presence of nitrifying bacteria. The aim of this paper is to propose and analyze a nonlinear system which represents the dynamics of these two species (microalgae and nitrifying bacteria) in competition for nitrogen (present as ammonium and nitrate produced by nitrification) in a continuous process. The existence and local stability of system equilibria is studied. Reduction by conservation principle, perturbed systems and Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the system equilibria. Finally, we illustrate our analysis with a case study, showing which operating conditions (dilution rate and pond depth) can promote the presence of nitrifiers with microalgae.     

Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.     

一类具有时滞的病菌-免疫力模型的Hopf分歧

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性及方向.运用计算机数值模拟验证所得理论结果,为传染病的控制和预防提供了理论基础和数值依据.     

Hopf bifurcation of a bacteria-immunity system with delayed quorum sensing

On the basis of Zhang’s model (see [P. Fergola, J. Zhang, M. Cerasuolo, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, in: Collective Dynamic: Topics on Competition and Cooperation in the Biosciences: A Selection of Papers in the Proceedings of the BIOCOMP2007 International Conference, AIP Conference Proceedings, vol. 1028, 2008, pp. 215-232] for more details), we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells with bacterial quorum sensing mechanism. A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. Subsequently, the length of delay which preserves the stability of the positive equilibrium is estimated and Hopf bifurcation occurs when time delay crosses through a critical value are researched. Further, by using the normal form theory and center manifold theory, the explicit formulaes are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Mathematical analysis of a model describing the number of antibiotic resistant bacteria in a polluted river

In France, 90% of surface water suffer from antibiotic pollution that increases the number of antibiotic resistant bacteria. The first indicator of water quality is revealed by the fish quality. According to the Le Conseil Supérieur de la Pêche, only 15% of rivers in France are considered in good condition, whereas 22% are in very bad condition. The bacterial resistance to antibiotics is a public health problem as it affects humans through drinking water; the treatment of water is costly. Mathematical modeling may estimate and predict the quantity of bacteria in rivers. In this paper, we investigate properties of the mathematical model estimating the number of bacteria in a river presented by Lawrence, Mummert and Somerville. Global analysis of equilibria is presented, using a Lyapunov function. Moreover, the existence of positive periodic solutions is proven. Copyright © 2013 John Wiley & Sons, Ltd.     

一类具有时滞的病菌-免疫力模型的定性研究

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性.特别的,研究了无病平衡点E0 在奇异条件(R0=1)下的稳定性.数值模拟验证了所得理论结果.     

Gliding motion of bacterium in a non-Newtonian slime

Gliding bacteria are a taxonomically heterogeneous group of rod-shaped prokaryotes that adhere to and translocate on certain substrata by mechanism(s) that are unknown. Organelles of motility, such as flagella, have not been observed on these bacteria. Furthermore, their rigidified cell walls preclude amoeboid type locomotion. The present investigation analyzes a motility model that involves an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum. The non-Newtonian nature of the slime is taken into account by considering a specific fluid of the differential type, namely an incompressible homogenous fluid of grade three. The non-linear differential equation resulting from the balance of momentum and mass is solved analytically and numerically.     

One-dimensional free boundary problem for actin-based propulsion of Listeria

Some bacteria move inside cells by recruiting the actin filaments of the host cells. The filaments are polymerized at the back surface of the bacteria, and they move away, forming a “comet” tail behind the bacterium, which consists of gel network. We develop a one-dimensional mathematical model of the gel based on partial differential equations which involve the number of filaments, the density and velocity of the gel, and the pressure. The two end-points of the gel form two free boundaries. The resulting free boundary problem is rather non-standard. We prove local existence and uniqueness.     

Modelling the spread of bacterial disease: effect of service providers from an environmentally degraded region

In this paper, an SIS model for bacterial infectious diseases, like tuberculosis, typhoid, etc., caused by direct contact of susceptibles with infectives as well as by bacteria is proposed and analyzed. Here the demography of the human population is constant immigration and the cumulative rate of the environmental discharges is a function of total human population. Further this model is extended to the model for socially structured population (rich and poor) where poor people work as service provider in the houses of rich people but do not settle in the habitat of rich people. It is assumed that bacteria population does not survive in the clean environment of rich people and only affects the population in the degraded environment of the poor class. The stability of the equilibria is studied by using the theory of differential equation and computer simulation. It is concluded that the spread of the infectious disease increases when the growth of bacteria caused by conducive environmental discharge due to human sources increases. Also the spread of the infectious disease in rich class increases due to the interaction with service providers, who are living in relatively poor environmental condition, suggesting the need to keep our environment clean all around.     

Solving the controllability problem for a nonlinear three-dimensional system

The controllability problem is solved for a nonlinear system of three differential equations that simulates aerated biological wastewater treatment using the thermal metabolism of thermophilic aerobic bacteria. Properties of the phase variables of such a system, bounded and continuable to a given segment, are established. The corresponding attainability set is parameterized by the switching instants of piecewise-constant controls. Parameterization helps reduce the controllability problem for the relevant system to finite dimensional conditional minimization of the auxiliary function. The results from numerical calculations for the solution to the initial problem are given.     

Analysis of the reproduction operator in an artificial bacterial foraging system

In his seminal paper published in 2002, Passino pointed out how individual and groups of bacteria forage for nutrients and how to model it as a distributed optimization process, which he called the Bacterial Foraging Optimization Algorithm (BFOA). One of the major driving forces of BFOA is the reproduction phenomenon of virtual bacteria each of which models a trial solution of the optimization problem. During reproduction, the least healthier bacteria (with a lower accumulated value of the objective function in one chemotactic lifetime) die and the other healthier bacteria each split into two, which then starts exploring the search place from the same location. This keeps the population size constant in BFOA. The phenomenon has a direct analogy with the selection mechanism of classical evolutionary algorithms. In this letter we provide a simple mathematical analysis of the effect of reproduction on bacterial dynamics. Our analysis reveals that the reproduction event contributes to the quick convergence of the bacterial population near optima.     

Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism

Considering the mechanism of quorum sensing, we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see [9] for more details). A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In the sequel, the length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated. Further, by using the normal form theory and center manifold theory, the explicit formulae are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

On a strain-structured epidemic model

We introduce and investigate an SIS-type model for the spread of an infectious disease, where the infected population is structured with respect to the different strain of the virus/bacteria they are carrying. Our aim is to capture the interesting scenario when individuals infected with different strains cause secondary (new) infections at different rates. Therefore, we consider a nonlinear infection process, which generalises the bilinear process arising from the classic mass-action assumption. Our main motivation is to study competition between different strains of a virus/bacteria. From the mathematical point of view, we are interested whether the nonlinear infection process leads to a well-posed model. We use a semilinear formulation to show global existence and positivity of solutions up to a critical value of the exponent in the nonlinearity. Furthermore, we establish the existence of the endemic steady state for particular classes of nonlinearities.