J. Cell. Mol. Med. Vol 8, No 2, 2004 pp. 269-281

Kinetic logic: a tool for describing the dynamics of infectious disease behavior

Claire Martinet-Edelist *

Virologie moléculaire et structurale, CNRS, Gif sur Yvette, France

Received: March 30, 2004; Accepted: June 3, 2004

Abstract

Most of the infectious diseases imply biological regulations controlled by several feedback loops or circuits. Kineticlogic, which is a method easily accessible to biologists or physicians, and which takes time and thresholds of activ-ity into account, seems a convenient method for building simplified models related to this field. This implies usual-ly qualitative predictions concerning the dynamics of such biological systems, leading to a movement back and forthbetween experimentation or observation and logical description. Here, we illustrate this simple modelling method inbuilding elementary models concerning prion infection to demonstrate how to proceed. We also discuss and sum-marize how this method has been used for studying several viral diseases. As an example, we show how predictionsrelated to the rhabdovirus cycle, were experimentally verified.

Keywords: feedback loops • models and modelling • kinetic logic • prion and viral infections

* Correspondence to: Claire MARTINET-EDELISTVirologie moléculaire et structurale, CNRS, 91198 Gif/Yvette- Cedex, France.

Tel.: 33(1) 69 82 38 36, Fax: 33(1) 69 82 43 08E-mail: edelist@vms.cnrs-gif.fr

Special Article

• Introduction• Formal modelling of the prion infection

– Biological description of prion infection andfeedback loops

– Logical formalisations– Model analysis

• Experimental verification of model predic-tions, concerning rhabdovirus– Biological description of rhabdovirus infection– Logical formalisations– Predictions

• Discussion and conclusions

Introduction

Intricate multiple feedback loops, the circular pro-cess of influence where action has an effect on theactor [1] describe efficiently infectious diseases.Several formal models could be built to approachthe dynamics of their behavior.

Starting from a verbal description of the diseasestudied, a matrix of interaction and a visual syn-thetic figure which contains the signs and thethresholds of interactions between the elements,could be proposed. Kinetic logic takes time intoaccount through various on- and off- delays, withone unique hypothesis, the existence of threshold(s)of activity for each variable [2] and seems to be par-ticularly suitable when only qualitative data areknown and their variations have a sigmoid shape. Itallows to write semi-logical equations, correspond-ing to the feedback loops proposed, and to draw lit-eral state tables up. The logical parameters aredetermined according to the biological and mathe-matical constraints. The efficiency of each circuit ischecked thanks to the condition of steadiness ofeach feedback loop. Then, numeric state tables arededuced. They allow to predict the dynamics ofinfectious disease behavior.

On the one hand, we apply here the first steps ofthe method to prion infection, a system poorlyunderstood, where the exact function of the cellularisoform is still unknown [3, 4]. Various steadystates and oscillations are found. These results arecompared with the experimental observations andseveral models are rejected. Some predictions con-cerning the biological properties of the system andsome new experiments are proposed.

On the other hand, we will show that logicaldescription already done for infections caused bymicro-organisms or viruses, has been helpful toprovide several biological predictions [5-9]. Byway of illustration, we will enter more into thedetails of the rhabdovirus cycle and after a briefdescription of the macromolecules synthesesinduced by the viral genome, emphasize on somepredictions of the model, experimentally confirmedlater on.

Thus, at several steps of this process, there is avery fruitful movement to and fro between logicalmodels and observations. Therefore this methodallows the users to elaborate an experimentationstrategy-directed by models.

Formal modelling of the prioninfection

Biological description of prion infection andfeedback loops

The agent responsible for the infection and propa-gation of transmissible spongiform encephalo-pathies (or prion diseases) seems to be a pathogenicconformational isoform or scrapie prion protein(PrPSc) of a protein normally encoded by a hostgene (PrPC), the cellular prion protein [10]. In thispaper, we focus our attention on the two isoforms ofthe prion protein and their interactions. Since theinfectious prion protein PrPSc is generated from itscellular isoform PrPC [11], the presence of PrPC

seems to be necessary for the synthesis of PrPSc,whereas the presence of PrPSc seems to decrease theamount of PrPC(action 1, below). Therefore there isa negative feedback loop between the two isoformsof prion protein, with an additional autoregulationcircuit of PrPSc (action 2, below), since the presenceof PrPSc is generally considered necessary for itsown synthesis, implying a positive regulation.

One associates each concentration, (noted by alower case letter: c for cellular prion protein, s forscrapie prion protein) with its corresponding syn-thesis (noted by the capital letters: C and S respec-tively). We will report on a model with a one-ele-ment positive circuit (circled by a thick line, action2) and one negative loop (circled by a dotted line),with only one negative interaction (action 1) asindicated in the following matrix.

We consider that the two distinct actions ofPrPSc may not be at the same threshold concentra-tion. Therefore, since little is known about priondisease, two models are proposed, as indicated inFigures 1-a and 1-b, depending on the threshold ofaction of PrPSc.

At this step of the study, it is difficult to forecastthe evolution of the system since the positive cir-

270

cuits are involved in multistationarity, whereas thenegative circuits can produce oscillations [12-16, 2].

After contamination by scrapie of a normal indi-vidual, there is a transition from PrPC to PrPSc

either self acting, or consecutive to the action of anunknown factor X proteic [17], or nucleic [18].Therefore, without PrPC to start, the putative cellu-lar factor would be ineffective. This transition couldbe represented by an external variable, as shown inFig. 1.

Our models have to account for several otherbiological observations:1. normal prion protein is present in non contami-

nated normal individual at an intermediate level[10],

2. except for the prion protein-less individualswhich, in addition, are unable to develop thedisease after contamination by the scrapie prionprotein [19, 20].

3. Infectious prion protein was observed at a highlevel in contaminated normal individual [10].

4. Some clinical observations reported mutationsof the PrP gene leading to dominantly inheritedprion diseases in humans [10].

Logical formalisations

Thanks to generalised kinetic logic [21, 22] wherevariables and functions can have more than two lev-els and according to Fig. 1-a and 1-b, we candescribe prion infection by two systems of equa-tions of the semi-logical functions (C and S) asso-ciated with their corresponding pseudo-logical, ormemory variable representing the concentration (cand s); dC and dS account for the transformation ofthe k’s, real values, into integer values, the K’s. Forinstance:

dS(ksc) = Ksc or dS(ksc + kss) = Ks(c+s)We consider s as a three-level pseudo-logical

variable associated with two thresholds, whereasthere is no indication that normal prion protein (c)

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Fig. 1 Model of theprion infection implyingtwo feedback circuitsfrom opposite sign.

+ indicates an activation(green), - an inhibition(red). The numbers, oneach arrow, indicate theorder of the threshold con-centration above whicheach interaction is effec-tive. Figures 1-a and 1-bdiffer only by these val-ues. An external variableX, increasing the amountof PrPSc could be added,according to [17, 18].

a

b

has more than two levels. Thus only one thresholdis used for this memory variable. Therefore, in themodels proposed, variables and functions can take alimited number of integer values (0, 1, 2 for s and 0,1 for c) and the “threshold” values (1t, 2t) separatingthese integer values.

This leads to the following relations between thelogical parameters:

0 ≤ Kcs ≤ 10 ≤ Ksc ≤ Ks(c+s) ≤ 20 ≤ Kss ≤ Ks(c+s) ≤ 2In all systems of equations, the bar above s in the

equation of C indicates the negative action of thescrapie prion protein on the cellular prion proteinconcentration.

The following equations (model a) account forFig. 1a:

The following equations (model b) account forFig. 1b:

Model analysis

On the basis of the logical equations, we can drawup two literal state tables corresponding to eachmodel (Tables 1a and 1b). They give, after dis-cretization, the values of the functions in terms ofsemi-logical parameters, for each set of values ofthe memory variables. After contamination byscrapie of a normal individual, there is a transitionfrom PrPC to PrPSc either self acting, or consecutiveto the action of an unknown factor X ineffectivewithout PrPC to start. This transition could be rep-resented by an external variable, as shown in Fig. 1and Table 1, or in other words by a change of thevalue of the Ks’s.

The other biological observations, above-men-tioned, imply:1. Kcs = 1, to account for the presence, at an inter-

mediate level, of normal prion protein in noncontaminated normal individual,

2. Kcs = 0 for the prion protein-less individuals,unable to develop the disease after contamina-tion by the scrapie prion protein

3. Ks(c+s) = 2, to account for the high level of theinfectious prion protein in contaminated normalindividual,

4. a modification of Ksc to account for the muta-tions of the PrP gene leading to dominantlyinherited prion diseases in human.)( 21 skckdSS sssc •+•=

)( 1skdCC cs •=

)( 11 skckdSS sssc •+•=

)( 2skdCC cs •=

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Tables 1. Literal state tables: Table 1-a. corresponds to model a, Table 1-b to model b.In these Tables, obtained according to models a and b, the K’s are the semi-logical parameters resulting from thediscretization of the real parameters (the k’s) present in the equations. It must be pointed out that the conjugatedaction of two substances is not necessarily the sum of each action. X = 1 accounts for the action of a putative fac-tor X leading to the transition from PrPC to PrPSc [17, 18].

X = 0 X = 1

c s C S C S

0 0 Kcs 0 Kcs KsX

0 1 Kcs Kss Kcs Ks(s+X)

0 2 0 Kss 0 Ks(s+X)

1 0 Kcs Ksc Kcs Ks(c+X)

1 1 Kcs Ks(c+s) Kcs Ks(c+s+X)

1 2 0 Ks(c+s) 0 Ks(c+s+X)

X = 0 X = 1

c s C S C S

0 0 Kcs 0 Kcs KsX

0 1 0 0 0 KsX

0 2 0 Kss 0 Ks(s+X)

1 0 Kcs Ksc Kcs Ks(c+X)

1 1 0 Ksc 0 Ks(c+X)

1 2 0 Ks(c+s) 0 Ks(c+s+X)

a b

Model aNormally, some more relations exist concerning theKi’s since s and c act on s respectively at the firstthreshold for s and the unique for c, leading to var-ious possible numeric tables (Tables 2, a-1 to a-4),without external contamination:

0 ≤ Ksc ≤ 1 and 0 ≤ Kss ≤ 1The corresponding results are plotted in Fig. 2

(a-1 to a-4), with all steady states, regular and sin-gular, and the prediction of evolution resulting fromthe logical transitions. In all these cases, the nega-tive feedback circuit is functional: the attractor is acycle surrounding the characteristic steady state1tc2ts. This could correspond to the presence ofscrapie prion protein, that is the pathogenic isoformof this protein, with some fluctuation.

In two models (a-1 and a-2) there is a singularsteady state 11ts, characteristic of the positive cir-cuit, unstable, according to [23] and a regularsteady state 10, corresponding to normal prion pro-tein alone i.e. non contaminated normal individual.On the contrary, models a-3 and a-4 have to be dis-carded since they do not describe normal prion pro-tein in a non contaminated normal individual.

When the appearance of the disease results fromthe contamination of a healthy individual by thescrapie prion protein, several representations arepossible:1) For both remaining models starting from normal

prion protein alone, that is the state 10, if somescrapie prion protein is added (contamination),as predicted by 11/12, the system enters into thecycle. This is in good agreement with the bio-logical observations on the scrapie propagation,since after this transient action, the systemwould come back to the preceding schemes (a-1or a-2). Whether the scrapie propagates or notwill depend on the time delays starting from thestate 01 in a-1, whereas it will always propagatein a-2, with a complete separatrix. Therefore, the

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Tables 2. Numeric state tables corresponding to model a. The regular steady states are circled. The superscript -indicates that the concentration will tend to decrease, whereas the superscript + that it will tend to increase.a-1 corresponds to Kcs = 1, Kss = 0, Ksc = 0, Ks(c+s) = 2a-2 corresponds to Kcs = 1, Kss = 1, Ksc = 0, Ks(c+s) = 2a-3 corresponds to Kcs = 1, Kss = 0, Ksc = 1 or 2, Ks(c+s) = 2a-4 corresponds to Kcs =1, Kss = 1, Ksc = 1 or 2, Ks(c+s) = 2a-5 corresponds to Kcs = 1, KsX = 0, Ks(s+X) = 2, Ks(c+X) = 0, Ks(c+s+X) = 2a-6 corresponds to Kcs = 1, KsX = 0, Ks(s+X) = 2, Ks(c+X) =1 or 2, Ks(c+s+X) = 2

c s C S+0 0 1 0+ -0 1 1 0

-0 2 0 01 0 1 0

+1 1 1 2-1 2 0 2

c s C S+0 0 1 0+ -0 1 1 0

-0 2 0 0

+ 1 21 0 1 1

+1 1 1 2-1 2 0 2

c s C S+0 0 1 0+ +0 1 1 20 2 0 21 0 1 0

+1 1 1 2-1 2 0 2

c s C S+0 0 1 0+ +0 1 1 20 2 0 2

+ 1 21 0 1 1

+1 1 1 2-1 2 0 2

c s C S+0 0 1 0+0 1 1 1

-0 2 0 11 0 1 0

+1 1 1 2-1 2 0 2

c s C S+0 0 1 0+0 1 1 1

-0 2 0 1

+ 1 21 0 1 1

+1 1 1 2-1 2 0 2

{ {

{

a-1

a-3

a-5

a-2

a-4

a-6

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situation described above seems to be coherentwith the biological observations of the appear-ance of the disease in both models.

2) Another way is to consider that contaminationcorresponds to the action of an unknown cellularfactor. Thereby at least, the synthesis of scrapieprion protein is increased: leading to Ks(s+X) = 2.This model, represented in Table 2-a-5 and Fig.2-a-5, with its two regular steady states (10 and02, corresponding respectively to normal prionprotein alone and the disease) seems very attrac-tive. The appearance of the disease results fromthe contamination, a major perturbation of ahealthy individual by the scrapie prion protein,allowing to jump over the separatrix. Startingfrom normal prion protein alone, that is the state10, the addition of scrapie prion protein leads to11/12 → 12/02 → 02/02 and the system reachesthe steady state 02, according to the biologicalobservations on the scrapie propagation.We will see if these remaining models fulfill

the other biological observations, correspondingto mutants of the PrP protein gene; the variousmutations can be represented by change of onethe K’s. Mutations of the PrP gene leading todominantly inherited prion diseases in humanscorrespond to the appearance of some synthesis ofPrPSc (S=1 or 2), that is Ksc = 1 or 2, or in otherwords a passage from a-1, a-2 and a-5 to respec-tively a-3, a-4 and a-6. As shown in Fig. 2, thesystem evolves from one basin of attraction to theother, that is from 10, the regular steady state,corresponding to the normal prion protein alone,to the point 11/12. In Fig. 2 -a-3 and -a-4, itbelongs to a cycle which could correspond to thedisease, whereas in Fig. 2-a-6 this leads to thesteady state 02. Variations of the time delays will

explain the different times of appearance of thedisease. All these facts are in agreement with thediscovery of dominantly inherited prion diseasesin humans.

Mutations of the host PrP gene implying theabsence of disease after contamination by thescrapie prion protein of a prion protein-less indi-vidual correspond to Kcs = 0. In any case, derivingrespectively from a-1, a-2 and a-5, a new steadystate appears (00) in good agreement with the bio-logical observations concerning this type ofmutant. Fig. 2-a-7, corresponding to model a-1,shows an evolution conform to the biological real-ity: 02/00 → 01/00 → 00/00 and is completely sat-isfactory. On the contrary, models with Kss = 1 (a-2) or 2 (a-5) would be rejected since they presenta stable steady state 01 or 02 when Kcs = 0.Nevertheless, it has to be pointed out that, in factin a-5, factor X intervenes, and this factor necessi-tates the presence of the normal prion protein toexpress, implying for the mutant Ks(s+X) = 0 asrepresented in Fig. 2-a-8, predicting an individualwithout prion protein. Thus, up to now, only mod-els a-1 and a-5 are compatible with all the biolog-ical observations known to date.

Model bNormally, since c acts at its unique threshold: 0 ≤Ksc ≤ 1, it leads to four models of evolution (Table3 b-1 to b-4). Models b-1 and b-3, which are farfrom reality have to be discarded, whereas modelsb-2 and b-4, presenting the main features concern-ing the biological properties of the prion protein,and the appearance of the disease are shown inFig. 3 (b-1 and b-2):

- two basins of attraction corresponding to thebistability of the system,

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Fig. 2 Variable space and evolution of the system, corresponding to model a, for the different values of theK’s, according to Tables 2 Arrows show how the system evolves before reaching a stable steady state. The stablesteady states are surrounded with a rectangle, whereas all singular steady states are symbolized with a dark point.Separatrix are represented by dark lines.a-1 corresponds to Kcs = 1, Kss = 0, Ksc = 0, Ks(c+s) = 2a-2 corresponds to Kcs = 1, Kss = 1, Ksc = 0, Ks(c+s) = 2a-3 corresponds to Kcs = 1, Kss = 0, Ksc = 1 or 2, Ks(c+s) = 2a-4 corresponds to Kcs = 1, Kss = 1, Ksc = 1 or 2, Ks(c+s) = 2a-5 corresponds to Kcs = 1, KsX = 0, Ks(s+X) = 2, Ks(c+X) = 0, Ks(c+s+X) = 2a-6 corresponds to Kcs = 1, KsX = 0, Ks(s+X) = 2, Ks(c+X) =1 or 2, Ks(c+s+X = 2a-7 corresponds to Kcs = 0, and is related to Figures a-1,a-8 is related to Fig. 2-a-5, with Kcs = 0, KsX = 0, Ks(s+X) = 0, Ks(c+X) = 0, Ks(c+s+X) = 2

- a stable state 10, in b-1, or a cycle surround-ing the characteristic steady state 1tc1ts in b-2,accounting for the normal prion protein in a noncontaminated individual,

- a stable state 02 corresponding to the disease,with an unstable singular steady state 02ts, charac-teristic of s autoregulation.

The appearance of the disease results from thecontamination of a healthy individual by thescrapie prion protein, that is from the action of anexternal variable with two possible representa-tions, both corresponding to a good representationof the normal contamination:

1) Starting from normal prion protein alone,that is either from 10 (Fig. 3-b-1), or from thebasin of attraction (Fig. 3-b-2), if scrapie prionprotein is added, at a high level (2), the systemtends to the state 02; therefore synthesis of PrPSc

occurs at the expense of PrPC,2) If we consider the action of a cellular factor,

we have to change Ksc into Ks(c+X) = 2, leading toTable 3-b-5 and Fig. 3-b-3. This modificationallows to get over the separatrix and the systemtends to the state 02.

The development of familial prion diseaseobserved in humans corresponds to a mutation andstarting from PrPc, leading to spontaneous appear-ance of PrPSc (S = 2). Therefore we suppose Ksc =2 leading again to Table 3-b-5, which allows topredict the occurrence of the disease as indicatedby the stable steady state 02 in Fig. 3-b-3, in goodagreement with this type of mutation.

Nevertheless, models b-1 and b-2, owing to thestable state 02, without any way to reach state01/00, are not in good agreement with the obser-vations concerning the prion protein-less animalwhich is unable to develop the disease after addi-tion of PrpSc. Thus they are not completely satis-factory and have to be discarded. Therefore,among all models studied, only models a-1 and a-5 seem to be compatible with the biological obser-vations.

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Tables 3 Numeric state tables corresponding to model b. We use the same conventions as in Table-2.b-1 corresponds to Kcs = 1, Kss = 0 or 1, Ksc = 0, Ks(c+s) = 2b-2 corresponds to Kcs = 1, Kss = 2, Ksc = 0, Ks(c+s) = 2b-3 corresponds to Kcs = 1, Kss = 0 or 1, Ksc = 1, Ks(c+s) = 2b-4 corresponds to Kcs = 1, Kss = 2, Ksc = 1, Ks(c+s) = 2b-5 derives from b-2 and b-4 and corresponds to Kcs = 1, Kss = 2, Ksc = 2, or Ks(c+x) = 2, Ks(c+s) = 2, or Ks(c+s+X) = 2

c s C S+0 0 1 0

-0 1 0 0

-

0 2 0 00 1

1 0 1 0- -1 1 0 0-1 2 0 2

c s C S+0 0 1 0

-0 1 0 0

-

0 2 0 00 1

+1 0 1 1-1 1 0 1-1 2 0 2

c s C S+0 0 1 0

-0 1 0 00 2 0 2

+1 0 1 2- +1 1 0 2-1 2 0 2

c s C S+0 0 1 0

-0 1 0 0

0 2 0 21 0 1 0- -1 1 0 0-1 2 0 2

c s C S+0 0 1 0

-0 1 0 0

0 2 0 2+

1 0 1 1-1 1 0 1-1 2 0 2

{

{

b-1

b-3

b-5

b-2

b-4

Experimental verification of modelpredictions, concerning rhabdovirus

Biological description of rhabdovirusinfection

By way of illustration, we will show predictionsconcerning the rhabdovirus cycle and thereforeneed to briefly describe the macromolecules syn-theses induced by the viral genome [24]. Theseviruses possess a single-stranded negative sense(-) RNA genome, that is complementary to viralmessenger RNA’s (mRNAs: +), associated with 5viral structural proteins (L, N, P, M and G). Afterinfection and entry into the host cell, through theplasmatic membrane, the rhabdovirion loses itsenvelope (M and G proteins). The remainingnucleocapsid, genome associated with three viralproteins (L, N and P), constitutes the active tem-plate both for transcription and replication of theviral genome. As shown in Fig. 4A, primary tran-

scription, the first step of the viral macro-molecules syntheses occurs. It gives rise to viralmRNAs (+), complementary to the viral genomesequence (-) principally M-G-Ψ, G-Ψ, N, M andP, as shown in Fig. 4B2-2 and 4B2-4 and a smallamount of L mRNA, which encode the five viralproteins. The second step, the translation of thesemRNAs into viral proteins using the cellularmachinery, can be inhibited by a drug, cyclohex-imide. The switch from transcription to replica-tion necessitates a new protein synthesis.Therefore, the third step, the replication of theviral genome (-) takes place only after translation,leading to production of antigenomic RNA (+)using the viral genome as a template followed bythe synthesis of new viral genomes using antige-nomic RNA as a template. Then, secondary tran-scription, using the new genomes as a templateoccurs, followed by new rounds of translation andreplication. Finally, maturation and virus buddingappear.

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Fig. 3 Variable space and evolution of the system,corresponding to model b, for the different valuesof the K’s, according to Tables 3. We use the same conventions as in Fig. 2.b-1 corresponds to Kcs = 1, Kss = 2, Ksc= 0, Ks(c+s) =

2, as in Table 3-b-2b-2 corresponds to Kcs = 1, Kss = 2, Ksc = 1, Ks(c+s) =

2, as in Table 3-b-4b-3 corresponds to Kcs = 1, Kss = 2, Ksc = 2 or Ks(c+X)

= 2, Ks(c+s) = 2, or Ks(c+s+X) = 2, as in Table 3-b-5

278

Fig. 4 Viral RNA synthesis in rhabdoviruscycle. A - Main steps of viral macro-molecules synthesis in rhabdovirus infectedcell (for a review, see [24]).Primary transcription, the first step of the viralcycle, gives rise to viral mRNAs (+), comple-mentary to the viral genome sequence (–). Thesecond step, the translation of these mRNAsinto viral proteins using the cellular machinery,can be inhibited by a drug, cycloheximide. Thethird step, the replication of the viral genome(–) takes place leading to production of antige-nomic RNA (+) using the viral genome as atemplate followed by the synthesis of new viralgenomes using antigenomic RNA as a tem-plate. Then, secondary transcription, using thenew genomes as a template occurs, followed bynew rounds of translation and replication.B - Analysis of viral RNAs present in infect-ed cells by Northern hybridizationsTotal cytoplasmic RNA were subjected to elec-trophoresis on 8% agarose in MOPS buffer, asdescribed in [32]. The cells were harvested at20hr post infection B-1- Cellular RNAs, most-ly ribosomal, were estimated by staining gelwith ethidium bromide, and visualised by expo-sure to ultraviolet light. It allows to estimate theamount of cells for each sample. B-2- Northernblots of total RNA from BHK-21 cells infectedwith rabies virus, CVS strain, were hybridizedwith a mixture of viral gene-specific probes.The viral RNAs were determined by a specifichybridization of each probe (not shown) andare in agreement with previous observation[24]. The infection was performed with TsG1, atemperature sensitive mutant of rabies virus,altered in the glycoprotein, as already described[27] and one of its revertants. For the mutant,33°C and 39.5°C are respectively a permissiveand a non permissive temperature. Nohybridization could be detected with mocked-infected cells (5). (1) and (2) revertant infectedcells, (3) and (4) TsG1 infected cells, (2) and (4)experiment performed in presence of 100µg/ml of cycloheximide (inhibitor of proteinsynthesis) added in the culture medium, imme-diately after adsorption. In the permissive con-ditions, without cycloheximide, Viral genomicRNA and Viral mRNAs are present at a highconcentration, either in cells infected by rever-tant at both temperature (1), or by TsG1 at 33°C(3). On the contrary, with cycloheximide (2,4),or in cells infected by the mutant at restictivetemperature (3), there is no viral genomic RNAand viral mRNAs are present at a lower con-centration. It has to be pointed out that B-1shows a higher cell concentration at 39.5°C in4 than in 3, explaining the higher level ofmRNAs observed in B-2, for the correspondinglanes.

A

BB-1

B-2

Logical formalisations

Naïve kinetic logic, that is with a unique thresholdfor each variable and function, was first used todescribe the rhabdovirus cycle [7]. Starting from 15models of 7 equations, some were discarded bystudying the steady states: the comparison of thevarious models with biological data allowed us toreduce them to 6 systems of 7 equations. Then, aftera return to an experimental phase to determine thevalues of the different memory variables and thetime delays, simulations of the viral cycle, in vari-ous conditions, were performed on a computer andcompared with experimental data. This leads toonly two remaining models, corresponding to 2 sys-tems of 7 equations, which are indicated here under:

Transcription

OR

Translation

Replication

AND

where T is the transcription, t the correspondingmemory variable, that is the presence of mRNAs,G, M, L are the synthesis of viral G, M, L proteins,g, m, l the corresponding memory variables, that isthe presence of these proteins, N is the synthesis ofviral N and P proteins, n the corresponding memo-ry variable, that is the presence of these two pro-

teins which share nearly the same biological prop-erties, intervening as cofactors of the viral RNApolymerase RNA-dependant L, R1 is the synthesisof viral antigenome r1, R2 the synthesis of viralgenome r2, p the cellular machinery for translation,gg, gm, gl, gn correspond to the viral genes.

But it has to be pointed out that some hypertran-scription was observed with mutants of the M pro-tein [25], therefore two thresholds were introducedto describe transcription [8] and therefore general-ized kinetic logic was used for this equation.Finally, it is possible to replace the two preceedingequations by the following unique equation:

with, for biological reasons, the following valuesfor the Ki:K1 = K2 = K3 = K1+3 = K2+3 = 0, K1+2 = 1 (normaltranscription), K1+2+3= 2 (hypertranscription).

Predictions

Modelling by kinetic logic of the evolution of rhab-dovirus infection leads to several predictions, asalready described [7, 8]:

i) the viral cycle could occur only with the fastdisappearing of the viral matrix protein (M), as itwas experimentally verified [26],

ii) occurence of curing of the rhabdovirus infect-ed cells was predicted by the study of the steadystates and we effectively observed such a predeter-mination of the recovery of a persistent-infectedcells line of rabies virus [7],

iii) our models were based on the fact that amutation in the gene coding for the M protein wasnecessary to obtain persistent-infected cells, but, inaddition, they forecast that a mutation in the genecoding for the glycoprotein (G) could lead to thesame result, and we effectively observed this phe-nomenon with tsG1, a rabies virus mutant of theglycoprotein [27],

iv) M and G viral proteins are involved in thereplication, or synthesis of genomic and antigenom-ic RNA’s. This seems to be experimentally con-firmed since tsG1, a temperature-sensitive mutantof the rabies virus glycoprotein, looks to be unableto replicate properly its genome in restrictive con-ditions. In these conditions, Fig. 4 shows that themutant synthesizes viral mRNAs at the primary

)( 3212 mknklkdTrT •+•+•••=

)(12 mgnlrR +•••=

)(21 mgnlrR +•••=

tpgN n ••=

tpgL l ••=

tpgM m ••=

tpgG g ••=

)(2 mnlrT +••=

)(2 mnlrT +••=

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transcription level, as it is obtained by addition ofcycloheximide a drug blocking protein synthesisand therefore the replication of the viral genome.Furthermore, no genomic viral RNA is detected incells infected in these both conditions.

Discussion and conclusions

The logical method used here for modelling prioninfection corresponds to examples of the “logicalregulon”, the minimal regulatory module with twoautocatalytic circuits, described in [2]. Our resultsshow how such a simple scheme accounts for differ-ent situations: not only multistationarity, correspond-ing to the positive circuit, is observed, but also stableperiodicity related to a negative circuit (not observedto date). Several of these models show bistable prop-erties corresponding to the process of prion propaga-tion described in the model developed in [28]. Mostof the findings encountered in this last paper arerecovered here although our models are very simple.Both types of models exhibit similar comportment ofthe system, probably as a consequence of the obser-vation by the authors of the crucial role of distinctthresholds. Furthermore, among the different modelswe proposed, a-1 and a-5, described in Fig. 2, appearto be better than the others since they are completelyin agreement with the facts known to date, concern-ing the prion infection. This implies an autoactiva-tion of PrPSc from its first threshold upward, findingperhaps sufficient to account for the autocatalyticaction of this isoform. The predictions of the modelcould be tested experimentally ex vivo by a study ofthe prion propagation following an artificial contam-ination performed either with the yeast prion proteindescribed in [29], or with the cells expressing ovineprion protein found by Vilette et al. [30]. Whether acellular factor is implied or not in prion replication,our model infers that the addition of PrPSc proteinupon a determined threshold to a cell line expressingonly PrPC isoform, will increase rapidly the transi-tion into PrPSc isoform, leading to a state, corre-sponding to the disease, with a much greater amountof PrPSc than it was added. On the contrary, themodel b described in [31] from a theoretical point ofview, without any biological connection, does not fitcompletely with the biological data, concerningprion infection.

The qualitative method, developed here to studyprion infection, demonstrates how such a descrip-tion, in terms of feedback loops, account for theunderstanding of complex regulatory networks andthe finding of all steady states stable and unstable aspreviously shown [12, 13].

This method is also very convenient for studyingand understanding regulation of the immuneresponse [5], virus infections either at the organismlevel in the herpes [6] and hepatitis B virus cycle [9]or at the cellular level with rhabdovirus [7, 8].Although the viral life cycle was oversimplified inall these qualitative models, they allow to find mul-tistationarity: recovery, acute and latent infection.The homeostatic mechanism, probably ensuringvirus propagation, seems as well a good descriptionof the chronicity observed during some viral infec-tions and was particularly displayed in the modelconcerning hepatitis B virus cycle [9]. Since thehypotheses are very clear cut in kinetic logic, thebiological implications appear without any ambigu-ity leading often to further experimentation to veri-fy the predictions of the model(s).

Although only descriptive and relatively simple,kinetic logic seems a powerful tool to predict allsteady states of complex regulatory networks andtheir dynamic behavior. Some predictions whichflow from the models will require further experi-mentation. Therefore, it seems to open new per-spectives in the study of complex systems implyingfeedback circuits and specially infectious diseases,at organism level as well at the cellular one.

Acknowledgements

I am very grateful to Y. Gaudin, N. Kellershohn and M.Laurent for their comments and suggestions concerningthis manuscript. This work was supported by the CNRS(UMR2472) and Université Paris-Sud, Centre d’Orsay.

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