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How relevant are in vivo and in vitro studies for clinical sepsis? A mathematical model of LPS signaling based on endotoxin tolerance
C. Vasilescu, M. Olteanu, P. Flondor (Chirurgia, 104 (2): 195-201)
Introduction
Endotoxin, i.e. lipopolysaccharide (LPS), is a molecule found in almost all Gram negative bacteria in the outer cell wall. Endotoxin is a main trigger of Gram-negative septic shock and its biological effects are mediated by cytokines. The uncontrolled release of inflammatory cytokines may cause extensive organ damage and death.
Endotoxin tolerance was initially described as an in vivo phenomenon whereby a sublethal injection of LPS abrogated the response to a subsequent endotoxin exposure. In the late 1940s Beeson showed that repeated daily injections of LPS to rabbits resulted in a progressive reduction in the febrile response (1,2). As the main role of TNF-a in LPS - mediated response become appreciated in the 1980s, the reduced expression of this mediator in endotoxin tolerance was noted (1). These findings were similar to those observed after in vitro experiments with monocytes/macrophages. TNF-a is the best marker of endotoxin tolerance as assessed by its dramatically reduced releasing after a second LPS challenge in animals in contrast to its peak response to a first LPS stimulation (3).
In septic patients the proinflammatory reaction is often followed by an anti-inflammatory response resulting in an immunoparalytic state (4). A return to normal cytokine production was noted in patients that recovered from sepsis (5). Monocytes from immunoparalyitic patients show an impaired production of pro-inflammatory cytokines when stimulated with LPS in vitro (the so called ex vivo stimulation). LPS tolerance could be considered to be an experimental model for immunoparalysis.
The LPS signaling field is evolving rapidly over the last years (6,7,8). Several molecules have been identified that negatively regulate the LPS signaling pathway (6) and is highly probable that these molecules play a role in the signal transduction alterations in endotoxin tolerance. LPS is known to initiate multiple signal transduction cascades in monocytes/macrophages and recently certain steps in key pathway have been found to be down regulated in endotoxin tolerance (5,6). Some aspects of the tolerance can be attributed to the down-modulation of TLR4. ST2, SHIP, IRAK-M, MyD88, IRAK2, SIGIRR, SOCS1, TIRAF, TRIAD3, A20, A1, Dok1 and Dok2 have all been implicated in negative feedback regulation of the LPS pathway (6). There are some discrepancies between the data obtained from in vitro, in vivo systems of evaluation of endotoxin tolerance and data from septic patients. The information provided by these studies remains controversial depending on the experimental approach being utilized. There are studies that suggest that in vitro models cannot fully explain the phenomenon of endotoxin tolerance noted in experimental animals and the immunoparalysis observed in septic patients (9).
The paper will explain the low-level TNF-a concentration in most of the septic patients and will discuss the relevance of the in vivo and in vitro models of endotoxin tolerance for the clinical sepsis. We hypothesized that the low level of TNF-a concentration in septic patients can be understood, at least in part, by two particularities of these patients: (1) the kinetics of LPS plasma levels, and (2) the existence of a “endotoxin tolerance-like phenomenon” (a particular type of down regula-tion of LPS signaling).
The main aim of the present approach was to obtain an approximate model of LPS signaling that agrees well with experimental data.
The machinery of LPS signaling is with certitude a complex network of cytokines (8). But even considering only the LPS kinetics we would be able to explain the particularities of the cytokines plasma levels determined in septic patients.
In order to gain insights on the LPS signaling machinery we tried to simulate these experimental conditions in silico (10-13).

Material and Method
Biological background
Transduction of the LPS signal is known to be mediated by TLR4. TLR4 interact with LPS and activate downstream signal transduction pathways and induces the production of inflammatory cytokines. Some aspects of the tolerance can be attributed to the down-modulation of TLR4. Several other molecules have been identified that down-regulate the LPS signaling. ST2, SHIP, IRAK-M, MyD88, IRAK2, SIGIRR, SOCS1, TIRAF, TRIAD3, A20, A1, Dok1 and Dok2 have all been implicated in negative feedback regulation of the LPS pathway (6,14). For example, one of the best characterized molecules is ST2, a member of the interleukin-1 receptor family that negatively regulates TLR4 (it acts at the cell surface) (15). Brint and colab. (16) showed that ST2 produced his inhibitory effect by sequestering MyD88 through its TIR domain and impairing signaling by receptors that use this adapter. On the other hand, ST2 suppresses the inflammatory cytokines production also by inhibition of IkB degradation induced by the LPS signal (15). Therefore, ST2 seems to have a key function in endotoxin tolerance (16). The complex of down-regulating factors can be interpreted as a brake effect on the inflammatory cytokines production. The complex of down regulation factors (“brake system”) includes two classes: the intracellular (NfkB and other cytoplasm’s protein as MyD88, IRAK, TRAF6, TIRAP/Mal or TRIP) and extra cellular (as IL-10 and TGF-b) factors. The extracellular components of the “brake system” are removed by washing the cell cultures. This is an important particularity of the in vitro experiments.
TNF-a is a cytokine that mediates a wide range of immunological, inflammatory and cytotoxic effects (17). The reduction in LPS-stimulated TNF-a is such a “universal” feature of endotoxin tolerance that some authors equate inhibition of TNF-a with the tolerant state.
IL-10 and TGF-b: negative regulators of LPS response
Cytokines induced by LPS can influence the production of inflammatory mediators by autocrine feed-back pathway (13,18). TNF-a has the potential for amplifying its own effects (19). On the other hand, it is well documented that the anti-inflammatory cytokines IL-10 and TGF-b are involved in mediating the process of endotoxin tolerance (17). Both IL-10 and TGF-b have similar inhibitory effects on TNF-a.
Endotoxin tolerance seems to be mediated partially by induction of IL-10 and TGF-b, extra cellular components of the brake system.
Neutralizing IL-10 and TGF-b during LPS priming led to a normal TNF-a response after LPS rechallenge (avoidance of the endotoxin tolerance) (20).
IL-10 induces several genes that could possibly interfere with LPS signaling and transcription of proinflammatory mediators.
IL-10 targets proinflammatory cytokine production probably by changing the composition of NF-kB from trans-activating p65/p50 heterodimers to inhibitory p65/p50 homodimers (20,21).
However, IL-10 seems to inhibit late events in cytokine regulation.
A mathematical model of LPS signaling based on endotoxin tolerance
In the present approach the mathematical model is based on the hypothesis that endotoxin-tolerance is a chain of LPS – stimulated enzymatic reactions following the law of mass action and so modeled by a Michaelis-Menten (Hill) system. Let us stress that, in the model, LPS is a time-dependent control parameter of the system. As a consequence, the system to consider is nonautonomous (as a matter of fact some other time-dependent coefficients are needed). Some other connected papers containing mathematical models of acute inflammation are Chan et al. (13), Kumar et al (10), Day et al. (11), and Vodovotz (12).
Let us describe the experiment of endotoxin tolerance in a more specific way.
First step:
i. At (the initial) time t0 the concentration of TNF-a is measured; let it be a0.
ii. Exposure to LPS challenge; say of concentration b0.
iii. At time t1>t0 the concentration of TNF-a is measured again; one finds a1>a0 (and the difference being quite significant).
iv. At time t2>t1 the concentrations of TNF-a and LPS are measured; for TNF-a one obtains a0 (or very close to it) and for LPS a concentration very close to 0.
Before describing the second step let us remark the following:
a) The time intervals t1-t0, t2-t1 are specific, well-known and reported. We are not interested in their exact values at this moment because of our qualitative description.
b) The term “challenge” (or, in some other references “attack”) is somewhat fuzzy but, this is not a concern for now.
c) In (iv) we note that the system returns to the “initial condition” (in TNF-a) eliminating LPS and stabilizing TNF-a; it is tacitly suppose that, at t0 no LPS was present before the challenge.
An interpretation of the first step could be that the point (a0, 0) is a stable equilibrium of the TNF-a-LPS system which was perturbed by a sudden change of the state (under an external influence). The system comes back to the equilibrium in due time. This seems compatible with the intuition of the role of the innate immune system as a protector against attacks.
Second step:
v. At time t2 (or very close after it) a new exposure to the LPS challenge is considered.
vi. After the same time interval of (iii) the concentration of TNF-a is measured. The (surprising) result is that, this time, the concentration is close to a0.
We say that the conclusion is surprising because the system is in the same (TNF-a) initial condition as in the first step and is perturbed in the same way. This comes in contradiction with a deterministic autonomous TNF-a -LPS system. The experiment stops here but one knows that after some time the ability of reacting as in the first step is regained (if nothing “abnormal” happened).
Details of the mathematical model
The mathematical model is based on Michaelis-Menten (Hill) equation, modified according to our problem and approach [for a related approach see (13)]. The model is a non auto-nomous first order differential system of two equations with two unknowns, i.e. the concentrations of TNF-a and of the “brake system”, denoted by and, respectively:

The coeffients represent:
e1: baseline for the function representing the TNF-a positive feedback on rate of the TNF- production;
e2: baseline for the function representing the inhibitor positive feedback on rate of the inhibitor production;
a: threshold for the TNF-a positive feedback on rate of the TNF-a production;
b: threshold for the inhibitor negative feedback on rate of the TNF-a production;
g: threshold for the brake system positive feedback on rate of the inhibitor production;
d1: clearence rate for TNF-a;
d2: clearence rate for the brake sytem; in this paper we shall suppose that d2 is a function of time. The main reason for this choice is that in some experiences, the brake system is eliminated in different ways;
n, m: Hill coefficients (generally, m£n; in our experiments n = 3, m = 2 seem to fit well enough).
The function a1 measures the interaction between the concentration of endotoxin and the production of TNF-a; the function measures the interaction between the concentration of endotoxin and the production of the inhibitor. One can suppose that (in a first approximation) they are proportional to the concentration of the endotoxin, so if A is the function representing the concentration of endotoxin, then there are constants m and n such that and a1 (t) = mA(t) and a2 (t) = nA(t) Using the technique of nondimensionalization (by an appropriate change of variables), an equivalent non-autonomous differential system is obtained. The new dimensionless variables are:

By substituting into the equations, the equivalent system is:

The new parameters are:

and the function
.
The numerical experiments were performed with MatLab 6.5.

Results
In order to test the above model the following scenarios concerning the LPS signaling were considered.
In the cases 1-3 the LPS input A(t) is a smooth function obtained by glueing together exponential and constant zero functions.
Case 1.
TNF-a dynamics following a single LPS stimulus:
After endotoxin stimulus the proinflammatory cytokines (i.e. TNF-a) are rapidly induced. TNF-a quickly appears in the plasma, reaches a peak concentration at 1-2 hours and returns to baseline levels by 4 hours (fig. 1).
Case 2.
Modeling of the in vivo experiments:
The experiment has two steps:
Step one: same as case 1.
Step two: a new LPS challenge (few hours after the first LPS stimulus) induces a negligible TNF-a production (phenomenon of endotoxin tolerance) (fig. 2).
The coefficients in cases 1 and 2 are: E1=0.15, E2=0.2, F1=2.8, F2=1.4, D1=12, D2=0.1, x0=0.01, y0=0.01.
Case 3.
Modeling of the in vitro experiments
For the method of an in vitro experiment see Schröder (17). The duration of LPS treatment has ranged from 4 hours to 2 days. There is no change in the concentration of LPS during the experiment (1). The LPS concentration in the cell culture medium rise suddenly and reach a plateau (step 1). In the step 2 the LPS kinetics is the same, but the TNF-a answer is negligible (endotoxin tolerance) (fig. 3). The coefficients are: E1=0.5, E2=0.2, F1=2.8, F2=1.4, D1=14, x0=0.01, y0=0.01 D2 is an interpolation with minimum value 0.1 and maximum value 3.0.
Case 4.
Modeling of clinical sepsis
The data for the LPS kinetics in this simulation were those of (22). The variations of LPS concentration are not followed by TNF-a production (fig. 4). In this case the function A(t) is an interpolation of the values reported by (22).
The coefficients are: E1=0.15, E2=0.2, F1=2.2, F2=1.0, D1=12, D2=0.5, x0=0.01, y0=0.01.
Figure 1
Figure 2
Figure 3
Figure 4

Discussion
According to the mediators theory (23) LPS exerts its lethal effects by cytokines. Among them the most studied is probably TNF-a, a paradigm for the pro-inflammatory cytokines (24). The mediator theory was verified in numerous experimental studies. According to it we expected high levels of TNF-a in plasma of septic patients. Surprisingly, there are sepsis patients with low TNF-a plasma levels. However, the results communicated until now look contradictory. Bozza (25) found that TNF-a plasma levels were not able to discriminate between survivors and non-survivors. Pinsky (1993)(26) reported TNF-a serum levels higher in septic than in non-septic shock, but the persistence of TNF-a in the serum rather than the peak levels predicts a poor outcome in patients with shock.
Martins et al. (2003) (27) found no difference in TNF-a production in different phases of septic patients or between septic and polytrauma groups. They concluded that TNF-a determination did not serve as marker of infectious disease.
One objective of the present work was to find a possible explanation for the case of septic patients with low plasma concentration of proinflammatory cytokines, respectively of TNF-a.
The endotoxin tolerance has been well documented in vitro, in vivo and ex vivo (removal of cells from an LPS-treated subject and then stimulated in vitro)(1,18,28).
Some studies suggested that data generated from in vitro systems are useful surrogates to further understanding of LPS signaling in vivo. A main question remains to be answered: how relevant are the in vitro and experimental in vivo models of endotoxin administration for clinical sepsis?
There are some particularities of the in vivo and in vitro models that fail to reproduce the conditions of clinical sepsis as seen in human disease. In order to gain insights on the LPS signaling machinery we tried to simulate these experimental conditions in silico.
The rationale, the biological significance of our model, was based on two factors: the kinetics of LPS and the existence of an “endotoxin tolerance-like phenomenon” (a brake-like effect of the proinflammatory cytokines production).
The novel mathematical model we are proposing can provide an explanation for the TNF-a releasing after different dosages and kinetics of LPS and can be used for predictions. This approach could show the different behaviors of the cell in in vitro, in vivo (experimental animals and human volunteers) and remains to be applied in other experimental settings (ex vivo - whole blood stimulation in septic patients, etc.). We focused on the relevance of the in vitro and in vivo experiments for the immune status of the septic patients.
About the mathematics of the model the basic ideas are:
a) the deterministic non-linear control system with variable coefficients based on Michaelis-Menten (Hill) philosophy;
b) reduction of the state-space dimension in order to get a better understanding but trying not to lose the characteristic reported properties;
c) some relevant experimental and clinical scenarios chosen in order to test the model;
d) the data were collected from published results of our team and of others (22).
We can imagine two, more or less, independent processes in progress after the first challenge with LPS: one process is that of the TNF-a-LPS-down regulating system cycle with the return to the steady state; the other, of the brake system production slow enough not to disturb the first but arriving at a concentration able to brake the production of TNF-a after the second challenge.
If one desires to keep the dimension of the state space as small as possible (not introducing, for example, TLR4, MyD88, etc.) and taking into account that the way in which the complex of down regulating factors are produced is not yet known, the solution to the problem of modeling endotoxin tolerance could be to give up to the autonomous character of the proposed system.
As a final remark let us mention that the model could be conceived for one “typical” cell only.
Experiments performed with the mathematical model
Case 1
In this scenario we checked the concordance of the mathematical model of LPS signaling with the kinetics of the TNF-a after a single LPS stimulus. Our results are in line with the observations of others (10,11). The dynamics of TNF-a plasma levels is similar in mouse and human (29).
Case 2
Endotoxin tolerance was induced “in vivo” (animal experiments or human volunteers) by various regimens (1). The reason of low TNF-a level could be explained in our model by the increased level of the brake system (above a given threshold).
For in vitro studies the duration of LPS treatments have ranged from 4 hours to 2 days (1). Beeson (2) obtained a maximum pyrogenic tolerance in rabbits by repetitive, closely scheduled injections. It is difficult to find studies dealing with the kinetics of the plasma endotoxin concentration from septic patients (22).
Case 3
This experiment asks for a time dependent brake clearance. The model seems to be concordant with the experimental data.
It is appropriate to consider the limits of the in vitro model and the relevance of the information obtained. In the in vitro systems the released products from the LPS stimulated cells, if not broken down, accumulate with time. Such products may be cleared in in vivo systems. By washing the system after the step 1 the LPS is cleared (as well as the extracellular components of the brake system). After all we have to observe that in the case of in vitro experiments the system is open.
Case 4
Based on the results of the first 3 scenarios we tried to check the mathematical model in the clinical situation of sepsis patients. This scenario is of a somewhat different nature. We have to take into account the particularities of the LPS kinetics, respectively that in sepsis there are sustained LPS challenges. This is explained in our model by the high levels of the brake system (above the threshold) quite similar with a sustained state of endotoxin tolerance.
The immune status of the septic patient may be evaluated by ex vivo stimulation of the whole blood (30). A blood sample is stimulated ex vivo with a known LPS quantity; thereafter the cytokines level in the stimulated sample is determined. In healthy subjects, and important elevation of the TNF-a concentration is observed. On the contrary, in most sepsis patients there is no significant TNF-a after LPS stimulation of the whole blood sample. The monocytes are no longer able to answer with a TNF-a releasing. Similar to case 4, our model explains this by the existence of an endotoxin-tolerance like phenomenon.

Conclusion
Our hypothesis, that in LPS signaling the kinetics of LPS as well as the existence of an endotoxin-like phenomenon are crucial features, may explain the contradictory data obtained in in vitro, in vivo and ex vivo stimulation of whole blood from septic patients. However, endotoxin tolerance is a dynamic, not a static, state characterized by continuously changing levels of circulating mediators (1).
Our results show good agreement with experimental and clinical data. Therefore, the model proved to be a good approximation of the real situations.

Acknowledgements
This work was supported in part by the research grant CNCSIS CEEX 187/2006.

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