Feedback Circuits in Hepatitis B Virus Infection

FEEDBACK CIRCUITS IN HEPATITIS B VIRUS

INFECTION

Claire Martinet-Edelist

Virologie moléculaire et structurale, CNRS, 91198 Gif/Yvette-Cedex, France.

Email: [email protected]

Contribution from the 2000 Meeting of the French Society of Theoretical Biology

Guest Edited by Jean-Pierre Mazat and Jean-François Hervagault

ABSTRACT

A simplified model using kinetic logic is proposed to approach the problem after Hepatitis B

viral (HBV) infection. It accounts for several stable regimes or attractors corresponding to the

essential dynamic behaviour of the replication of the Hepatitis B virus. Infection with the virus

can result in viral clearance, fulminant hepatic failure and death, or chronic transmissible

infection, that is multistationarity corresponding to the existence of the positive feedback circuit

in our modelling. Another implication of this model is the existence of oscillations or

homeostatic mechanisms, sometimes observed in the viral cycle, consistent with the existence of

the negative feedback circuit.

Thus, this report shows how a simple model of kinetic logic may be used to account for the

variety of manifestations of HBV infection. This model implies the presence of the Hepatitis B e

antigen, whose conservation suggests that it plays an important role in the life cycle of

hepadnaviruses. Its function in the viral cycle is still unknown, but our model suggests that this

antigen could explain the passage from one state of the viral infection (acute or latent) to

another, as well as the oscillatory behavior which may account for the intermittent symptoms of

hepatitis observed in some patients. Furthermore, this model shows a virgin state. This state is

also reached after recovery. The model proposed demonstrates that starting from a viral acute

infection, the host’s immune response, depending on the immunological status of the patient,

can lead to viral clearance, or to periodic spontaneous reactivation.

1. INTRODUCTION

The Hepatitis B virus (HBV) genome is a partially double-stranded DNA. Since

this genome is very small, 3.2 kb (Sitterlin et al., 2000 for a review), most of the viral

proteins are multifunctional, implying several feedback loops or circuits. Feedback,

the circular process of influence where action has an effect on the actor (Bar-Yam,

2000), is needed to describe most biological regulations, especially viral cycles.

Furthermore, feedback circuits, or closed oriented pathways, are the operators which

adjust the production rate of the various elements of biological regulatory systems

(Thomas, 1993; Thomas and Thieffry, 1995).

In this paper, we will present a model with three circuits, two positive and one

negative, for the HBV cycle. This model supposes the action of the Hepatitis B e

antigen (HBeAg) in the viral cycle. This small non-structural protein was found in the

serum of patients infected by HBV by Magnius and Espmark (1972), but its role is still

246 MARTINET-EDELIST

unknown (Ganem, 1996; for a review, Messageot et al., 2001). Nevertheless, the

conservation of this antigen in all hepadnaviruses indicates that it probably plays a

crucial role in their life cycle, whereas its role in infectivity or viral multiplication has

been excluded (Chang et al., 1987; Tong et al., 1990; Chen et al., 1992). Interestingly

enough, HBeAg shares most of its amino acid sequence with the HBV capsid protein

(Messageot et al., 1998; Günther et al., 1999).

Kinetic logic (Thomas, 1973, 1978, 1983; Thomas and d’Ari, 1990), an

intermediate between purely verbal and differential representation, seems to be a

suitable description of such a system because it takes time into account through

various on and off delays, with one unique hypothesis peculiar to this method, the

existence of threshold(s) of activity for each variable. Furthermore, this representation

which takes into account only a few parameters is particularly adapted for the HBV

cycle, where the function of some non structural viral proteins is still unknown.

Using this type of modelling, various attractors or steady states, depending on

HBeAg gene efficiency in the infecting virus, were found in the viral cycle.

Furthermore, the model presented here leads to the prediction of oscillations in the

viral cycle equally observed in some clinical data (Sallie, 1997). The model is capable

of explaining all the main features known in viral infections with HBV: viral

clearance, acute and chronic infection, periodic reactivation of asymptomatic hepatitis

B. Therefore, HBeAg seems to be sufficient to account for the complexity of the HBV

life cycle.

This model confirms that a negative circuit (with an odd number of negative

interactions) of three elements generates periodicity, and a positive circuit (with an

even number of negative interactions) leads to multistationarity (Demongeot et al.,2000; Thomas and Kaufman, 2001 a and b).

2. MODELLING HEPATITIS B VIRUS REPLICATION

Biological Stages at the Organism Level

As described in Figure 1, infection with HBV can result in asymptomatic infection,

acute infection or fulminant infection and death. The severity of the initial phase of

disease seems to be determined solely by the immune response of the patient. After

completion of the replication cycle in hepatocytes, the newly-made virions are

transported into the patient’s serum through the secretion pathway. Thus, in most

cases, infected liver cells were not destroyed and the infection remains unapparent in

spite of the production of HBV particles. However, in about 20% of the cases, an

exacerbate immunological response leads to the lysis of the infected hepatocytes

(acute hepatitis). Whether the infection remains unapparent or leads to acute hepatitis,

an apparent total viral clearance is usually observed within a few months, as a

consequence of the host’s immune response. Nevertheless, in some cases, a periodic

reactivation of HBV is observed (Hollinger, 1990; Sallie, 1997). When HBV clearance

occurs, high titre antibodies directed against the virus proteins (anti-HBeAg, anti-

hepatitis B capsid protein, anti-hepatitis B envelope proteins) are observed in the sera

of patients, in agreement with a reduced viral replication. However, the different

classes of antibodies do not appear at the same time. Detection of the antibodies

against the hepatitis B capsid protein and decrease of the synthesis of viral proteins

FEEDBACK CIRCUITS IN HEPATITIS B VIRUS INFECTION 247

occured two months after infection with HBV while anti-HBeAg antibodies and anti-

hepatitis B envelope proteins appeared later (respectively after around four to five

months, and six months, according to Hollinger, 1990).

HBV can sometimes (5-10% of the cases as indicated in Figure 1) persist in liver

cells for many years, leading to cirrhosis, and eventually to an hepatocarcinoma

(Buendia, 1992). Interestingly enough, during chronic HBV infection, the periodic

spontaneous reactivation of symptomatic hepatitis B could be observed (Hollinger,

1990; Raimondo et al., 1990a; Sallie, 1997).

Figure 1. Variety in outcomes of HBV Infection. The biological stages of HBV infection at the

organism level can be summarised as indicated in the figure above, according to Hollinger

(1990) and Sitterlin et al. (2000). A periodic reactivation of symptomatic hepatitis B starting

from a chronic HBV infection was mentioned by Hollinger (1990) and carefully observed by

Sallie (1997) in a patient followed over 20 years. The figures in brackets refer to some of the

steps of the model developed, encountered in the biological stages.


Biological Events at the Molecular Level

In this paper, we focused our attention on a regulatory protein of HBV, the HBeAg

that can be involved in two apparently opposite functions:

i) the cytotoxic T lymphocyte response (Gil-Torregrosa et al., 1998) which is

essential for the viral clearance;

ii) the establishment of a persistent infection (Ou, 1997, for a review).

In fact, HBeAg seems to be required only to establish viral persistence, since in

chronic hepatitis, variant viruses that are not able to produce HBeAg rapidly become

predominant (Raimondo et al., 1990 a and b). It has recently been suggested that the

function of the HBeAg in the establishment of a persistent infection, may result from

the HBeAg precursor’s inhibiting viral replication (Scaglioni et al., 1997). This

explains the overproduction of virions observed in some chronic hepatitis B, in the

absence of HBeAg. Furthermore, at the beginning of the viral cycle, HBeAg is

synthesized in the presence of virion production (Hollinger, 1990), whereas later on it

can appear in the serum of an infected patient without detectable virus production

(Sallie, 1997).

After a complex processing, HBeAg derives from a precursor, the precore protein,

a 25-kDa protein. This precursor is encoded by the entire HBV C open-reading frame

and translated from the pre-C AUG, on the pre-C RNA, whereas the core protein, the

subunit of the capsid, is translated from the C AUG on the pregenomic RNA, a

slightly shorter transcript that does not include the pre-C AUG (Messageot et al., 2001

for a review).

One very simplified description of the viral cycle is given in Figure 2 and will be

used in our model. The encapsidated HBV genome has a positive action on both the

synthesis of HBeAg and the virus production, while HBeAg inhibits the virus

production, which, in turn, increases the number of new HBV genomes. To test if such

a simplified description is sufficient to relate the main observations done on the HBV

life cycle, we have developed a model using kinetic logic to describe the feedback

circuits implied in this viral cycle.

Figure 2. Feedback circuits involved in

the HBV cycle. + indicates an activation, –

an inhibition. The encapsidated HBV

genome allows the synthesis of HBeAg

and the virus production which produces

new HBV genomes. HBeAg inhibits the

virus production. The numbers upon each

arrow account for the threshold of control.

Modelling the Hepatitis B Virus Cycle Using Generalized Kinetic

Logic

Since an overproduction of HBV was observed under certain conditions, we will

need three levels to describe this production: 0 (absence of virions), 1 (normal

production), 2 (overproduction). The encapsidated viral genome acts on virion

production and HBeAg synthesis, probably not at the same threshold. Furthermore, it


is not possible to distinguish between a real viral clearance and a pre-activation stage

in which the viral genome is undetectable but followed by a spontaneous reactivation.

Thus, we decided to use a four-level variable to represent the encapsidated viral

genome. We used the logical modelling method called generalized kinetic logic,

proposed by Thomas and D’Ari (1990), to describe the hepatitis virus cycle. When

necessary, variables and functions can have more than two levels in this method,

according to Snoussi (1989).

In our case, this concerns funtions P and G (respectively the production of virions,

for biological reasons, and the production of encapsidated viral genomes). The

corresponding memorization variables, p and g, are multivaluate variables with two or

three thresholds. 1p and 1g are Boolean variables related to the first threshold, 2p and 2g

with the second, 3g with the third. E, HBeAg synthesis and e its memory variable

(presence/absence of HBeAg) are respectively a Boolean function and a Boolean

variable, with only one threshold.

Considering that E, G and P are the functions and that e, g, p are their respective

memory variables, we can describe the viral cycle, according to the model shown in

Figure 2, with the following equations:

E = dE (keg . 2g)

G = dG (kgp . 1p)

P = dP (kpg3 . 3g + kpg1 .

1g + kpe . e).

The choice of the levels for the effects of g in these equations was determined

considering that an intermediate level of viral encapsidated genome (2g) is sufficient

for the synthesis of HBeAg, whereas virion production would suppose a higher level

(3g). 2g = 0 corresponds to the presence of undetectable up to now, encapsidated

genome or to not encapsidated viral DNA described by Guidotti et al. (1999) at the

beginning of the life cycle. This accounts for two cases: total absence of encapsidated

genome if 1g = 0; presence of undetectable encapsidated genome if 1g = 1. According

to these considerations, the levels of effect are indicated in Figure 2 on each arrow.

In the semi-logical equation, dX accounts for the discretisation of ki's, which have

real values, by analogy with the differential description, but are transformed into

integer values through dX. This method allows us to detect all the steady states found

in the differential description, on logical grounds. Snoussi (1989) introduced this

procedure, which gives rise to the discrete, easy-to-handle integer values of the

function.

On the basis of the logical equations, we can draw up the state table (Table 1),

which gives the values, after discretisation, of the functions in terms of semi-logical

parameters (integer values: 0 or 1 for E, 0, 1, 2, or 3 for G, 0, 1, or 2 for P), for each

set of the values of the memory variables.

Such a model is greatly limited by the possible values of each function and the

mathematical constraints on the values of the logical parameters. Some biological

constraints will be studied under “Study of an infection with a wild type virus”. We

have the following relations for mathematical and logical reasons:

0 Keg 1

0 Kgp 3


Kpg1 Kpg3.

Because p is a three-level (0, 1, 2) variable and the conjugated contribution is

always greater or equal to a simple action, but is not necessarily the sum of each

action, we have:

0 Kpe Kp(g1+e) Kp(g3+e) 2

0 Kpg1 Kp(g1+e) 2

0 Kpg3 Kp(g3+e) 2.

Table 1. State table for wild type

infection. In this table, obtained

according to the equations presented,

the K’s are the semi-logical parameters

resulting from the discretisation of the

real parameters (the k’s) present in the

equations. It must be pointed out that

for instance Kp(g1+e) can be different

from the sum Kpg1 + Kpe since the

conjugated action of two substances is

not necessarily the sum of each action.

They are three regulation, or feedback circuits: the three-element circuit E/P/G is

negative, two G/P positive. Therefore, there are only three characteristic states to

study. This allows us to establish the conditions of steadiness for the characteristic

states corresponding to each circuit, that is, to see if the circuit is efficient, since it was


demonstrated that each circuit is efficient only if its characteristic state is steady

(Snoussi and Thomas, 1993; Thomas, 1993). If we let the various thresholds be,

respectively, 1se,1sg,

2sg , 3sg,

1sp and 2sp for e, g, p, then to be efficient, the negative

feedback circuit must have a singular steady state at 1se2sg

1sp , that is, at the unique

threshold of e, at the first threshold of p and the second of g. This implies, according to

Table 1, Kp(g1+e) = 1 or 2, Kpg1 = 0, Kgp = 2 or 3 and Keg = 1.

To be efficient, one of the positive feedback circuits must have a singular steady

state at e3sg1sp , that is, at the third threshold of g and at the first threshold of p,

whatever the value of e. According to Table 1, and to be compatible with the previous

results, this implies for continuity reasons, Keg = 1, Kgp = 3 and Kpg3 = 1 or 2. The

other positive feedback circuit must have a singular steady state at e1sg1sp , that is, at

the first threshold of g and at the first threshold of p, whatever the value of e.

According to Table 1, and to be compatible with the previous results, this implies for

continuity reasons, Kpe = 0, Kgp = 3 and Kp(g1+e) = 1 or 2. This finding is in agreement

with a biological observation: in the absence of a viral genome (that is to say 1g = 0),

the production of virions is impossible, therefore P = 0, implying Kpe = 0.

Study of an infection with a wild type virus

This first study will be done without the effect of the host’s immune response, that

is, it is based on what occurs at the beginning of an HBV infection, since the immune

response appears only two months after exposure to HBV (Hollinger, 1990). In the

case of an infection with a wild type virus, the infected cells produce the enigmatic

HBeAg.

To account for biological knowledge, we have to choose the values for Ki's with

some new constraints.

Since three levels of virion production were observed experimentally during HBV

infection, we have Kp(g3+e) = 2. But production of the virus was observed at an

intermediate level as well, leading to the relations:

0 Kp(g1+e)< Kp(g3+e) implying the choice Kp(g1+e) = 1

0 Kpg3 < Kp(g3+e) implying the choice Kpg3 = 1.

Taking into account all these data, we obtain:

Kpe = 0 = Kpg1

Kpg3 = Kp(g1+e) = Keg = 1

Kp(g3+e) = 2

Kgp = 3

implying that the singular states 1se2sg

1sp, 13sg

1sp and 01sg1sp are steady. This leads to

the state table (Table 2). We note two regular stable steady states: 000/000 (recovery

that is clearance of infected cells, or virgin state if virus was never encountered by the

system) and 131/131 (acute hepatitis), the three singular steady states mentioned

above. The multistationarity observed shows the functionality of the positive feedback

circuit. Furthermore it has to be noted that, although the production of virion (P

function and p variable) has two levels of activation (1 for low and 2 for high) there

are no different effects between these two levels in the Table 2. This choice


corresponds to biologic observations and will lead to a different steady state in

Table 3, with a different level of P and p.

Table 2. Numeric state table for wild

type infection, without host immune

response. Regular stable states are

encircled.

As shown in Figure 3, oscillations were observed corresponding to the

functionality of the negative feedback circuit. After infection, the HBV life cycle

begins with a small encapsidated partially double-stranded DNA genome, with no

viral surface protein. As indicated in Figure 3, starting from state 020, three paths are

possible, one of which is very unlikely, since it was experimentally observed, that, at

the beginning of the viral cycle, HBeAg is synthesised only when complete virions are

produced (Hollinger, 1990). Therefore, the passage by 120 is very unlikely. It has to

be noted that time delays are still completely unknown in HBV life cycle. An early

step with a naked viral genome is known in the viral cycle, (Ganem, 1996; Guidotti et

al., 1999), which can correspond to 010, followed by 011 and 021. Depending on time


delays, the second step could also be state 021: Two choices are then possible,

depending on time delays again:

Figure 3. Evolution of the logical variables and functions after a wild type HBV infection,

without host immune response. Memorization variables are given in the following order: e, g, p.

The values of the corresponding functions allow us to predict the evolution of the system. A

virgin stable state which also accounts for recovery is found (000). When a memorization

variable differs from its function, we use a superscript over the memorization variable,

indicating that a command will increase (+), or decrease (–) its value. An infection starts at state

020, where only the viral encapsidated genome is present, at an intermediate level. This is

followed by other states depending on time delays. Three possibilities appear according to

Table 2, but one is very improbable (120), indicated by a dotted line, since HBeAg appears only

in the presence of virions at the beginning of the viral cycle. This allows us to predict

oscillations around the singular steady state 1se2sg

1sp, described by a cycle (dark lines), or a

stable steady state corresponding to acute infection (131).

- After several steps, this leads to 131 corresponding to acute infection where

HBeAg is present, the encapsidated viral genome is at its highest level and virion is at

an intermediate level. This seems to correspond to a greater level of replication than

occurs under basal conditions.

- The state (121) once again offers two choices, acute infection or the description of

a 6-element cycle, that is, periodic reactivation of the different components implied in

this model.

It is noteworthy that even if the homeostatic mechanism, corresponding to a

fluctuating viraemia, has begun, there are three states of the cycle where the system

can bifurcate to recovery or to an acute infection. This seems to correspond indeed to

chronic hepatitis B with viral clearance and periodic spontaneous reactivation, as

described by Sallie (1997), since he has found a periodic behavior in certain biological


parameters; the states not observed experimentally stand in bracket in the following

sequence, corresponding to the steps of the 6-element cycle observed in our model:

110 010 (011 021) 121 (120) 110 and so on.

The state 000 (recovery) is found in the model but not immediately reached after

the beginning of infection. It is frequently observed in the presence of the anti-HBeAg

antibody in the patient’s serum. This is in good agreement with the pathway (110

100 000) found in Figure 3, where HBeAg is the last viral component to disappear.

The appearance of recovery is found in this model, but will be observed more

frequently when the host’s immune response is taken into account (see under “Study

of an infection with a wild type virus followed by host immune response”).

Table 3. Numeric state table for an

infection with a mutant of the HBeAg

gene, without host immune response.

Regular stable states are encircled.


Study of an infection with a variant containing a mutation abolishing the synthesis of

functional HBeAg

Since HBeAg and the core protein genes share most of their coding sequences (Ou,

1997), two cases must be studied.

If the mutation affects both proteins, the two proteins are no longer functional.

Therefore, this mutation will lead to the absence of functional encapsidated HBV

genome and HBeAg and, finally, the appearance of recovery, even with no host

immune response.

Figure 4. Evolution of the logical variables and functions for a mutant of the HbeAg gene

infection, without host immune response. As shown in Figure 3, memorization variables are

given in the following order: e, g, p. The superscript over the memorization variables has the

same meaning as in Figure 3 and the values of the corresponding functions allow us to predict

the evolution of the system. An infection starts at state 020, (upper panel), where only the viral

encapsidated genome is present. This is followed by different steps, but whatever way it goes, it

finally leads either to 000 (recovery), or to 032, a state corresponding to a latent infection,

without HBeAg, and a high level of viruses, that is higher than in the acute infection. Starting

from the acute infection, 131, (lower panel) and assuming a mutation in the HBeAg gene has

appeared, the system evolves into 032, the state corresponding to a latent infection.


In the second case, the mutation peculiar to HBeAg (mutation in the preC region)

will be represented by the same equations as the wild type virus, but with Keg = 0. For

each set of values for the memory variables, we can draw up the state table (Table 3)

which provides the values of the corresponding functions.

Table 3 gives two regular steady states which are stable: 000/000 (recovery or

virgin state) and 032/032 (latent infection), which was not encountered in Table 2.

Here, only the positive feedback circuits G/P are still present, since the negative

feedback circuit aborted with the use of a mutant of the gene coding for HBeAg.

Only one characteristic state of the two positive feedback circuits is still steady

01sg1sp, whereas 03sg

1sp is not. The evolution of the system and its various states are

shown in Figure 4. This suggests that for a mutant of HBV which does not produce the

HBeAg, the system can evolve to 032 (latent infection) where the production of

virions is higher than in an acute infection as observed by Raimondo et al. (1990a and

b). It is worth noting that this conclusion will be identical whatever the initial

conditions (a new infection or an acute infection followed by a mutation in the

corresponding gene. According to Figure 4, recovery can also be observed after a new

infection by such a mutant, depending on the time delays.

Study of an infection with a wild type virus followed by host immune response

The effect of the host’s immune response appears only around two months after

exposure to HBV starting with the appearance of the antibody anti-HBV core protein,

then antibody anti-HBeAg and, finally, in some patients, the antibody anti-HBV

surface protein (Hollinger, 1990). This will modify Kis. Taking into account the

appearance of the antibody anti-HBV core protein alone, several possibilities can be

considered depending on the value of Kgp (0, 1, 2) and we obtain Table 4-a, if Kgp = 0

(high level of antibodies), Table 4-b if Kgp = 1 (intermediate level), Table 4-c if

Kgp = 2 (low level).

Starting from an acute disease (131), the system will evolve, as shown in Figure 5.

Several cases are observed, depending on the value of Kgp. The action of a high level

of antibody anti-HBV core protein (Figure 5-a) leads to the equilibrium state 000,

which corresponds to a total recovery as described in some clinical patterns

(Hollinger, 1990). Nevertheless, if only an intermediary level of antibody anti-HBV

core protein is synthesised (Figure 5b), two steady states are observed: 000 (recovery)

and 011. This last state can be considered as a persistent infection developed after

acute hepatitis B in accordance with some clinical data (Hollinger, 1990). If the level

of anti-HBV antibodies fluctuates, this will lead to fluctuating viraemia, according to

Figure 3, since this step 011 is found in the 6-element cycle. This seems to account for

the periodic reactivation observed clinically. When the cycle, corresponding to

fluctuating viraemia is established, the host’s immune response is not obvious, since it

was observed that the presence of anti-HBV antibodies also fluctuates (Sallie, 1997).

The fluctuating viraemia can also be explained by Figure 5-c, where the same

6-element cycle as in Figure 3 is observed. Nevertheless, in these conditions, a path

leads to a total recovery too (state 000). These three cases can account for the host’s

immune response, depending on the immunological status of the patient Likewise,

intrinsic viral mechanisms may down-regulate basal viral replication, whereas the

host’s immune response would have no important effect.


Table 4. Numeric state table for wild type infection, describing a maintained host’s immune

response two months after the initial infection. Regular stable states are encircled. Whatever the

level of antibody, a recovery (000) is observed except if the immune response was not

maintained. Table 4-a corresponds to a high level of antibody (that is Kgp = 0). Table 4-b

corresponds to an intermediate level of antibody (that is Kgp = 1). Table 4-c corresponds to a

low level of antibody (that is Kgp = 2).


Figure 5a. Antibody at a high level.

Figure 5b. Antibody at an intermediate level.


Figure 5c. Antibody at a low level.

Figure 5. Evolution of the logical variables and functions of an acute infection by a wild type

HBV, after a maintained host immune response. Memorization variables are given in the

following order: e, g, p. The values of the corresponding functions allow us to predict the

evolution of the system. The superscript over the memorization variables has the same meaning

as in Figure 3. Several levels of anti-HBV core protein antibody were investigated here. The

improbable paths, according to clinical data (Hollinger 1990), are in dotted lines: 111 is a more

likely state than 120, since antibody anti-HBV core protein appears before antibody anti-HBV

surface protein, implying the decrease of g before p. 011 is a more likely state than 110, since

antibody anti-HBeAg appears before antibody anti-HBV surface proteins, implying the

disappearance of e before p. Figure 5-a corresponds to Table 4-a (high level of antibody, that is

if Kgp = 0 ). Whatever the path, this leads to a final recovery (000) except if the immune

response was not maintained. Figure 5-b corresponds to Table 4-b (intermediate level of

antibody, that is if Kgp = 1). This seems to lead to a latent infection (011). The recovery (000) is

found also. Figure 5-c corresponds to Table 4-c (low level of antibody, that is if Kgp = 2). The

recovery (000) and the 6-element cycle described in Figure 3 are found.

3. DISCUSSION AND CONCLUSIONS

The generalised logical method used here for the replication of HBV leads to an

almost immediate analysis of the system described, in terms of feedback circuits, as

Thieffry and Thomas (1995, 1998) already demonstrated with models concerning

other systems. It must be pointed out that the model proposed here is a slightly

modified example of the “logical regulon” described by Demongeot et al. (2000),


since the junction between g and p is common to the two autocatalytic circuits. Once

again, our results show how such a simple scheme accounts for different situations: in

this system not only multistationarity (especially recovery, acute and two types of

latent infection) corresponding to the positive circuit is observed, but also a stable

periodicity related to a negative circuit length three.

When several pathways are possible, depending on the time delays, two hypotheses

can be proposed: either the delays fluctuate randomly around a mean value, or there is

an influence by the patient’s genetic background. It has to be pointed out that acute

infection is encountered only in the case of a wild type infection, before host’s

immune response (Figure 3). The stable steady states predicted by our model and

experimentally observed are indicated in Figure 1.

The homeostatic mechanism, probably ensuring virus propagation, seems a good

description of the chronicity observed during some HBV infections. It is observed in

several cases (Figures 3 and 5c). Such periodic reactivation of symptomatic hepatititis

B was observed by Sallie (1997) in a patient observed over 20 years. This leads him to

propose a model implying a negative feedback loop as well. Nevertheless, although his

model is more complex than the model developed here, it does not show the

appearance of any stable steady states.

Interestingly enough, if a mutation of the gene coding for HBeAg (pre-core region)

appears during an acute infection, the model indicates the evolution of the system from

this state to a latent infection (Figure 4), as was observed earlier (Carman et al., 1989;

Okamoto et al., 1990; Tong et al., 1990; Brunetto et al., 1991). The theoretical result

is compatible with the observation of the development of severe liver disease and

continuing viraemia in patients with anti-HBeAg detected after mutations in the pre-

core region (Benjelloun et al., 1993).

Total viral clearance, leading to patient’s recovery, seems to appear, in different

ways in our model, either directly, or after the host’s immune response to wild type

virus, or after a mutation affecting the overlapping open-reading frames of the core

protein and HBeAg genes. Furthermore, the recovery is found without the intervention

of anti-HBV surface protein antibody. This is in good agreement with the late

appearance of this antibody in the sera, usually six months after infection (Hollinger,

1990). Although the HBV life cycle was oversimplified in this qualitative model, since

the role of naked viral genome, for instance, or the HBV X protein was not described,

it must be noted that all the main features of HBV life cycle were found. Therefore, it

shows that the inhibition of virus production by HBeAg is sufficient to observe acute

infection, latent infection, and chronicity, providing a possible explanation of its

conservation in all Hepadnaviruses. Patient’s recovery resulting from a mutation in the

pre-core region or from the consequence of the host’s immune response was also

observed. Therefore, kinetic logic may prove a useful tool to simplify the study of

complex systems implying feedback circuits.

Comparing some more experimental data and the theoretical responses of the HBV

life cycle will help us to determine whether or not the theory correctly describes the

system’s behaviour.

Likewise, some of the attractors of viral infection at the organism level (latent

infection and acute infection) were encountered in the herpes virus life cycle (Pastoret

et al., 1986). Such steady states of viral infection were also noted when the system was

studied at the cellular level (Martinet-Edelist, 1994, 1999).


A complex quantitative model concerning HBV infection was previously

developed (Marchuk et al., 1991 a and b). But, contrary to our model, it does not take

into account the role of HBeAg. Furthermore, for simplification reasons, it emphasizes

the role of HBV surface protein even though the corresponding antibody appears six

months after infection only. Therefore, it is difficult to compare this model with ours.

ACKNOWLEDGEMENTS

I am very grateful to Professor J.M. Rossignol for his helpful insights and A. Tardy

for her careful reading of the manuscript. I thank the referees for having significantly

improved this paper through their valuable comments on previous versions. This work

was supported by the CNRS (UMR2472) and Université Paris-Sud, Centre d’Orsay

(UFR940).

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Feedback Circuits in Hepatitis B Virus Infection