General Model of the Innate Immune Response

Transcription

1 General Model of the Innate Immune Response Katherine Reed, Kathryn Schalla, Souad Sosa, Jackie Tran, Thuy-My Truong, Alicia Prieto Langarica, Betty Scarbrough, Hristo Kojouharov, James Grover Technical Report

2 General Model of the Innate Immune Response Katherine Reed Kathryn Schalla Souad Sosa Jackie Tran Thuy-My Truong Alicia Prieto Langarica Betty Scarbrough Hristo Kojouharov James Grover May 19, 2011 Abstract Within the human body, pathogenic can invade the bloodstream or adapt to an intracellular or communal environment, such as a biofilm. To counteract l invasion, the body s defense system elicits an immune response, and phagocytic cells are sent to destroy the pathogens by eating them. Although this process is one of the most important defense mechanisms against infection, little is known about its kinetics. A mathematical model has been constructed in terms of and two types of pathogenic : susceptible to and with an intracellular or biofilm refuge that cannot be eaten. The analysis of the dynamics of -host cell interactions predicts that in the presence of a l refuge state, a constant input of to the infection site is required for an infection to clear. 1 Introduction The human microbial defense system can be categorized into 3 levels: anatomic and physiologic, innate immunity, and adaptive immunity. Anatomical and physiological barriers include the skin and mucous membranes. The skin consists of epithelial surfaces acting as a physical barrier that is impermeable to most infectious agents. In addition, skin cells are constantly being sloughed off so that microbes cannot colonize on the surface for too long. The skin is also dry and acidic, which are not favorable conditions for l growth [2]. However, the skin is not the only barrier and some can infiltrate the body and activate the innate immune system. Innate immunity is inherent in all animals and This research was supported by an NSF UBM-Institutional grant DUE# as part of the UTTER Program at UT Arlington ( Department of Biology, The University of Texas at Arlington, P.O. Box 19498, Arlington, TX Department of Mathematics, The University of Texas at Arlington, P.O. Box 19408, Arlington, TX

3 humans. At any time, there are about two billion circulating through the body, regardless of whether an infection is present or not. A phagocyte does not just randomly encounter a bacterium, but rather, the phagocyte has receptors to detect the foreign components of that are not normally present in the human body [4]. The innate immune system also acts as an activator and a controller of the adaptive immune system. Within the body, can exist in either a commensalistic habitat or in a pathogenic environment. Free-floating are able to reproduce logistically according to the availability of nutrients in the body. However, can be detected by the immune system and are vulnerable to phagocytosis. When reach a certain population density, a recruitment process delivers more into the infection site via phagocyte signaling. Signaling allows to detect infections and move toward them. A larger amount of in the infection site work to eliminate the infection more quickly [6]. In response to stress, the population of can enter several different states as a strategy for survival depending on the species of. The two most common stress-induced states include entering a biofilm-like formation or becoming intracellular. Free can aggregate into dense clusters by secreting extracellular polymeric substances, producing a biofilm. Established biofilms can result in infections that are resistant to antimicrobials, and other immune defenses such as phagocytosis, as opposed to of the same species. Furthermore, biofilms are known to only be susceptible to immune responses within the first twentyfour hours; past that, they have enough tolerance to confer resistance. Bacteria inside a biofilm have been known to detach from the cluster and re-enter the bloodstream. Whether the detachment is spontaneous or directed requires more research [3]. Unlike, pathogenic are found inside the cells or within a biofilm. Most pathogens reside in a vesicular compartment while others occupy nonvesicular compartments. The overall theme of pathogens will be discussed without details associated with intracellular life, including entry, survival, replication, and exits because they are host-microbe specific interactions. Vesicular compartments include phagososomes, vacuoles that form around the pathogen via phagocytosis. Phagosomes then fuse with lysosomes, which contain enzymes that are capable of digesting the pathogens. In the general infection process, enzymes would be able to digest the pathogen contained within the phagosomes but those that can survive in a vesicular compartment have specialized mechanisms that can facilitate phagosomal escape. Nonvesicular compartments include intracellular sites where the pathogen is not enclosed by a membrane [1]. A mathematical model is constructed and the results are simulated and analyzed. The analysis of the dynamics of -host cell interactions shows that the presence of a l refuge state, a constant input of to the infection site is required for an infection to clear. Simulations explore possible parameters and initial conditions for a feasible infection to exist. This work extends on a previous work by Reynolds, et. al. [6], among others. 2

4 Figure 1: Flow diagram of the innate immune response in the human body. 2 Mathematical Model The simplified model includes (b f ), (b i ), and (p). Free can be found in the blood and are able to reproduce at a constant rate r. Inaccessible refer to those that have entered the cytosol of cells or formed a biofilm and can potentially cause an infection. Free die off at a natural death rate ξ > 0, but are also dependent on p because they are susceptible to phagocytosis. Inaccessible have a natural death rate of δ > 0, independent of p because being within a cell or inside a biofilm shelters them from being engulfed by. Independently, will cycle throughout the body at a constant rate θ. Free becomes at a constant rate µ, and can also revert back to at a constant rate λ. When detect, they recruit to the infection site at a rate ρ > 0. Phagocytes also leave the infection site at rate ω, for reasons not accounted for by the model. The equations of the model were derived from the flow diagram (Figure 1). 3

5 db f dt db i dt ( = rb f 1 b ) f + λb i µb f ξb f p, k = µb f λb i δb i, (1) where dp dt r = birth rate (r > 0), = ρb f + θ ωp, ξ = death rate (ξ > 0), δ = death rate (δ > 0), µ = becoming (µ > 0), λ = becoming (λ > 0), θ = constant supply of (θ > 0), ω = dispersion of from infection site (ω > 0), ρ = recruitment rate due to signaling that allows to detect (ρ > 0). 3 Theoretical Analysis 3.1 Constant Phagocyte Immigration (ρ = 0, θ > 0) The equilibria are solved by setting the differential equations equal to zero in Mathematica. The trivial equilibrium is E 0 = (b f, b i, p ) where b f = 0, b i = 0, (2) p = θ ω. The non-trivial equilibrium is E 1 = (b f, b i, p ) where b f = k δ θ ξ k θ λ ξ + k r δ ω + k r λ ω k δ µ ω, r (δ + λ) ω b i = µ ( k δ θ ξ k θ λ ξ + k r δ ω + k r λ ω k δ µ ω) r (δ + λ) 2, ω (3) p = θ ω. 4

6 3.1.1 Feasibility Conditions The trivial equilibrium is always feasible because p = θ is positive. Since θ ω and ω are both positive values, p must also have a positive value. The feasibility condition of the non-trivial equilibrium is based on the value of b f since b i = µ δ + λ b f. In order for the non-trivial equilibrium to be feasible > 0, which is equivalent to: b f k δ θ ξ k θ λ ξ + k r δ ω + k r λ ω k δ µ ω r (δ + λ) ω > 0. Therefore, the feasibility condition for the non-trivial equilibrium is: Stability Conditions δ + λ. (4) The Jacobian matrix for System (1) with ρ = 0 is: ( r 1 b ) f b f r k k µ pξ λ b f ξ J = µ δ λ 0. (5) 0 0 ω Trivial Equilibrium The Jacobian matrix (5) for the trivial equilibrium has the following eigenvalues: x 1 = ω, x 2 = 1 2ω (A 0 A 2 0 B 0), x 3 = 1 2ω (A 0 + A 2 0 B 0), (6) where: A 0 = θ ξ + r ω δ ω λ ω µ ω, and B 0 = 4 ω (δ θ ξ + θ λ ξ r δ ω r λ ω + δ µ ω). There are two cases to determine the stability of the trivial equilibrium: 1. B 0 < 0 means A 2 0 B 0 > A 2 0 which would cause x 3 > 0. So, the trivial equilibrium is unstable under the following condition: δ + λ. 2. B 0 > 0 means A 2 0 B 0 < A 2 0. Also, A 0 < 0 must be true, otherwise x 3 > 0. B 0 > 0 gives the following stability condition: r < p ξ + δµ δ + λ, 5

7 while A 0 < 0 gives the following stability condition: r < p ξ + δµ λµ + (δ + λ + δ + λ δ + λ ). So, the trivial equilibrium is stable under the stronger condition: r < p ξ + δµ δ + λ. Non-trivial equilibrium For the non-trivial equilibrium, the Jacobian matrix (5) has the following eigenvalues: 1 x 1 = ω, x 2 = 2ω(δ + λ) (A 1 A 2 1 B), x 1 3 = 2ω(δ + λ) (A 1+ A 2 1 B 1), where: A 1 = δ θ ξ + θ λ ξ r δ ω δ 2 ω r λ ω 2 δ λ ω λ 2 ω + δ µ ω λ µ ω, and B 1 = 4 (δ ω + λ ω) ( δ 2 θ ξ 2 δ θ λ ξ θ λ 2 ξ + r δ 2 ω + 2 r δ λ ω + r λ 2 ω δ 2 µ ω δ λ µ ω ). There are two cases to determine the stability of the non-trivial equilibrium: 1. B 1 < 0 means A 2 1 B 1 > A 2 1 which would cause x 3 > 0. So, the non-trivial equilibrium is unstable under the following condition: r < p ξ + δµ δ + λ. 2. B 1 > 0 means A 2 1 B 1 < A 2 1. Also, A 1 < 0 must be true, otherwise x 3 > 0. B 1 > 0 gives the following stability condition: δ + λ. A 1 < 0 gives the following stability condition: λµ (δ + λ + δ + λ δ + λ ). So, the non-trivial equilibrium is stable under the stronger condition: δ + λ. Since is the feasibility condition for the non-trivial equilibrium to exist, the non-trivial equilibrium is stable if it is feasible, and the trivial δ + λ equilibrium is unstable. In summary, when the non-trivial equilibrium is not feasible, the trivial equilibrium is stable. Similarly, when the non-trivial equilibrium is feasible, it is stable and the trivial equilibrium is unstable. (7) 6

8 3.2 Responsive Phagocyte Immigration (ρ > 0) To find the equilibria, the differential equations are set equal to zero and solved in Mathematica. The trivial equilibrium is E 0 = (b f, b i, p ) where b f = 0, b i = 0, (8) p = θ ω. The trivial equilibrium (8) for ρ 0 is identical to the trivial equilibrium (2). The non-trivial equilibrium is E 1 = (b f, b i, p ) where b f = b i = p = k (δ θ ξ + θ λ ξ r δ ω r λ ω + δ µ ω), (δ + λ)(k ξ ρ + r ω) k µ(δ θ ξ + θ λ ξ r δ ω r λ ω + δ µ ω) (δ + λ) 2, (k ξ ρ + r ω) k δ µ ρ + r (δ + λ) (θ + k ρ). (δ + λ)(k ξ ρ + r ω) (9) Feasibility Conditions The trivial equilibrium is always feasible. The non-trivial equilibrium of the system with ρ 0 has the same b f and b i expressions as the non-trivial equilibrium case where ρ = 0 (3) but multiplied by the following factor: 1 + k ξ ρ r ω. (10) Therefore, the feasibility condition for the non-trivial equilibrium is the same as in the case ρ = 0, i.e., the original feasibility condition (4): Stability Conditions δ + λ. (11) The Jacobian matrix in the case ρ 0 is too complicated to analyze. Numerical simulations suggest that the non-trivial equilibrium is stable for at least some reasonable parameter values. 7

9 (a) Default parameters with b f = 1, b i = 1, p = (c) Default parameters with b f = 10 8, b i = 10 8, p = (b) Default parameters with b f = 10 8, b i = 10 8, p = x (d) Default parameters with r = 1.2, b f = 10 8, b i = 10 8, p = 10 8 Figure 2: Simulated dynamics of pathogenic and using original parameters with ρ = 0 4 Numerical Simulations To verify the theoretical results, a set of simulations has been generated (Figure 2). Unless otherwise noted, the following parameters have been used: r = 1.5 hr 1, k = cells/ml, ξ = ml/(phagocyte cell)/hr, δ = 0.5 hr 1, µ = 0.5 hr 1, λ = 0.5 hr 1, θ = cells/ml/hr, ω = 0.1 hr 1. Some of the parameter values have been based on comparable parameters in previous models [6], others have been set to biologically plausible values. The initial conditions and the reproduction rate r are varied in the simulations. In the first set of simulations (Figure 2), ρ is set at ρ = 0 to observe how the and the are affected by a constant inflow of. In the second set of simulations (Figure 3), ρ is fixed at ρ = 100 to see how the and change with the signaling and recruitment of. 8

10 (a) Default parameters with ρ = 100, b f = 1, b i = 1, p = (b) Default parameters with ρ = 100, b f = 10 8, b i = 10 8, p = (c) Default parameters with ρ = 100, b f = 10 8, b i = 10 8, p = (d) Default parameters with r = 1.2, ρ = 100, b f = 10 8, b i = 10 8, p = 10 8 Figure 3: Simulated dynamics of pathogenic and using the original parameters with ρ = Constant Phagocyte Immigration (ρ = 0) The system with constant phagocyte immigration (ρ = 0) has been simulated with default parameter values and several different initial conditions for free,, and (Figure 2). 4.2 Responsive Phagocyte Immigration (ρ 0) The system with responsive phagocyte immigration representing signaling and recruitment (ρ > 0) has been simulated with default parameter values and ρ = 100. Several different initial conditions for, and are used in the simulations (Figure 3). 5 Discussion and Conclusion When ρ = 0 there is a general production of which remain at a constant background value, while when ρ > 0 become responsive to the 9

11 presence of. The general expectation is that an increase in would clear an infection. However, the feasibility condition (11) for persistence of the infection implies that ultimate clearing of the infection is related only to the constant, background supply rate of. When this constant rate is high enough, the non-trivial equilibrium representing a persistent infection becomes infeasible, and only the trivial equilibrium is feasible and stable. It is observed in the numerical simulations that as the phagocyte immigration changes from constant ρ = 0 to responsive ρ > 0, the l infection continues to persist, but under lower levels than if there was no signaling. The simulations showed that the rate of clearing of the infection was dependent upon the l birth rate r, a constant inflow of pagocytes, and the signaling of ρ > 0. A low reproduction rate of as well as both the constant and signaled inflow of are necessary for the infection to be cleared quickly. Since the l infection cannot be eliminated with only responsive phagocyte migration, the immune system must also have an adequate rate of background phagocyte immigration θ coming into the infection. Future studies of the general model of the innate immune system would require additional parameters to more accurately explore how the immune system will react to pathogens. A scenario to explore would start with the trivial state free of pathogens, to which an amount of is introduced. It would be interesting to determine the various lengths of time it takes to clear an infection, or similarly, how long persist in the body before an infection peaks. Further investigation could determine the time it takes for to be called by the immune system to fight the infection, and would yield promising results in order to further analyze the effectiveness of the immune response. References [1] Casadevall, A., Evolution of Intracellular Pathogens. Annu. Rev. Microbiol. 62, [2] Cvitkovitch, D., Li, Y., Ellen, R., Quorum sensing and biofilm formation in Streptococcal infections. J. Clin. Invest. 112: [3] Donlan, R., Costerton, J., Biofilms: Surival Mechanisms of Clinically Relevant Microorganisms. Clin. Microbiol. Rev. 15(2): [4] Kievit, T., Iglewski, B., Vodovotz, Y., Chow, C., Bacterial quorum sensing in pathogenic relationships. Infect. Immun. 68(9), [5] Kumar, R., Clermont, G., Vodovotz, Y., Chow, C., The dynamics of acute inflammation. J. Theor. Biol. 230, [6] Reynolds, A., Rubin, J., Clermont, G., Daya, J., Vodovotz, Y., Ermentrouta, G.B., A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation. J. Theor. Biol. 242,