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Bistability and bacterial infections.

Malka R, Shochat E, Rom-Kedar V - PLoS ONE (2010)

Bottom Line: However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity.We call such a behavior type II dynamics.We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel.

ABSTRACT
Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.

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The bifurcation diagram: the bacterial equilibrium points (EPs) as a function of neutrophil concentration.Solid line indicates a branch of stable EPs, and dashed line indicates a branch of unstable EPs. (a) Type I dynamic has a unique stable branch of EPs for all  values. The black arrows demonstrate that for any positive initial value of the bacteria, for any ,  converges to the corresponding unique stable EP (the intersection of the solid black curve with a vertical line). Bifurcation diagram is drawn for Eq. (1) with  (b) Type II dynamic has a region of bistability: when , the final state of  depends on whether the initial bacterial concentration is above or below the critical bacterial curve of unstable EPs (dashed line). The bifurcation diagram is drawn for Eq. (1) with . (c–d) Time plots of the two initial bacterial concentrations notated by red up and blue down arrows in (b) for a fixed  value (notice the different time scales).
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pone-0010010-g001: The bifurcation diagram: the bacterial equilibrium points (EPs) as a function of neutrophil concentration.Solid line indicates a branch of stable EPs, and dashed line indicates a branch of unstable EPs. (a) Type I dynamic has a unique stable branch of EPs for all values. The black arrows demonstrate that for any positive initial value of the bacteria, for any , converges to the corresponding unique stable EP (the intersection of the solid black curve with a vertical line). Bifurcation diagram is drawn for Eq. (1) with (b) Type II dynamic has a region of bistability: when , the final state of depends on whether the initial bacterial concentration is above or below the critical bacterial curve of unstable EPs (dashed line). The bifurcation diagram is drawn for Eq. (1) with . (c–d) Time plots of the two initial bacterial concentrations notated by red up and blue down arrows in (b) for a fixed value (notice the different time scales).

Mentions: In the type I parameter regime, the model has no critical dependence on neutrophil concentration . That is, for all levels of neutrophils there is a single stable equilibrium point (EP) which depends gradually on : for low values it corresponds to the high concentration point associated with the maximal capacity branch–the branch of stable equilibria that emanates from the point , where is the maximal capacity state of the natural bacterial dynamics. As the neutrophil level increases, this EP gradually lowers till, for sufficiently high neutrophil level (), it reaches the origin (see Fig. 1a and Models section; for simplicity of presentation, we consider here the zero influx and show in the Bacterial Influx section that the results are only slightly modified for small bacterial influx, see also Fig. 2).


Bistability and bacterial infections.

Malka R, Shochat E, Rom-Kedar V - PLoS ONE (2010)

The bifurcation diagram: the bacterial equilibrium points (EPs) as a function of neutrophil concentration.Solid line indicates a branch of stable EPs, and dashed line indicates a branch of unstable EPs. (a) Type I dynamic has a unique stable branch of EPs for all  values. The black arrows demonstrate that for any positive initial value of the bacteria, for any ,  converges to the corresponding unique stable EP (the intersection of the solid black curve with a vertical line). Bifurcation diagram is drawn for Eq. (1) with  (b) Type II dynamic has a region of bistability: when , the final state of  depends on whether the initial bacterial concentration is above or below the critical bacterial curve of unstable EPs (dashed line). The bifurcation diagram is drawn for Eq. (1) with . (c–d) Time plots of the two initial bacterial concentrations notated by red up and blue down arrows in (b) for a fixed  value (notice the different time scales).
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Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC2864736&req=5

pone-0010010-g001: The bifurcation diagram: the bacterial equilibrium points (EPs) as a function of neutrophil concentration.Solid line indicates a branch of stable EPs, and dashed line indicates a branch of unstable EPs. (a) Type I dynamic has a unique stable branch of EPs for all values. The black arrows demonstrate that for any positive initial value of the bacteria, for any , converges to the corresponding unique stable EP (the intersection of the solid black curve with a vertical line). Bifurcation diagram is drawn for Eq. (1) with (b) Type II dynamic has a region of bistability: when , the final state of depends on whether the initial bacterial concentration is above or below the critical bacterial curve of unstable EPs (dashed line). The bifurcation diagram is drawn for Eq. (1) with . (c–d) Time plots of the two initial bacterial concentrations notated by red up and blue down arrows in (b) for a fixed value (notice the different time scales).
Mentions: In the type I parameter regime, the model has no critical dependence on neutrophil concentration . That is, for all levels of neutrophils there is a single stable equilibrium point (EP) which depends gradually on : for low values it corresponds to the high concentration point associated with the maximal capacity branch–the branch of stable equilibria that emanates from the point , where is the maximal capacity state of the natural bacterial dynamics. As the neutrophil level increases, this EP gradually lowers till, for sufficiently high neutrophil level (), it reaches the origin (see Fig. 1a and Models section; for simplicity of presentation, we consider here the zero influx and show in the Bacterial Influx section that the results are only slightly modified for small bacterial influx, see also Fig. 2).

Bottom Line: However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity.We call such a behavior type II dynamics.We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel.

ABSTRACT
Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.

Show MeSH
Related in: MedlinePlus