Conversion of graded to binary response in an activator-repressor system
aa r X i v : . [ q - b i o . M N ] J a n Conversion of graded to binary response in anactivator-repressor system
Rajesh Karmakar ∗ November 14, 2018
Department of PhysicsA. K. P. C. MahavidyalayaSubhasnagar, Bengai, Hooghly-712 611, India.
Abstract
Appropriate regulation of gene expression is essential to ensure that protein synthesis occursin a selective manner. The control of transcription is the most dominant type of regulationmediated by a complex of molecules such as transcription factors. In general, regulatorymolecules are of two types: activator and repressor. Activators promote the initiation oftranscription whereas repressors inhibit transcription. In many cases, they regulate the genetranscription on binding the promoter mutually exclusively and the observed gene expressionresponse is either graded or binary. In experiments, the gene expression response is quantifiedby the amount of proteins produced on varying the concentration of an external inducermolecules in the cell. In this paper, we study a gene regulatory network where activatorsand repressors both bind the same promoter mutually exclusively. The network is modeledby assuming that the gene can be in three possible states: repressed, unregulated and active.An exact analytical expression for the steady-state probability distribution of protein levelsis then derived. The exact result helps to explain the experimental observations that in thepresence of activator molecules the response is graded at all inducer levels whereas in thepresence of both activator and repressor molecules, the response is graded at low and highinducer levels and binary at an intermediate inducer level.
PACS number(s): 87.10.Mn
I. INTRODUCTION
Gene expression, a fundamental cellular process whereby mRNAs and proteins are synthesized,is inherently stochastic in nature. There is a large number of theoretical and experimental studieswhich confirm the stochastic nature of gene expression [1]. The stochasticity or noise in geneexpression is due to the small number of molecules involved in the associated cellular processes.For example, the DNA molecule which gives an organism its unique genetic identity is present inone or two copies per cell. The small number of molecules taking part in the biochemical events of ∗ Electronic address: [email protected]
II. STOCHASTIC MODEL AND EXACT SOLUTION
Transcriptional regulation by activator and repressor molecules on binding the same promoter isan important regulatory mechanism of gene expression in living organisms. The activator (repres-sor) molecules activate (inhibit) the transcription by binding the appropriate site on the promoter.Here we consider a gene regulatory network where activators and repressors both regulate the genetranscription mutually exclusively [12]. This can happen in different ways and one such way maybe the overlapping binding sites on the promoter (Fig. 1). Therefore, the activator and repressormolecules cannot bind the promoter simultaneously, rather they compete for their binding sites toregulate gene transcription. This mechanism of transcriptional regulation is represented by a sim-ple reaction scheme (Fig. 2) where a gene can be in three possible states: G , G and G . G is theunregulated state and G ( G ) is the repressed (activated) state of the gene. The unregulated stateof the gene which is achieved when both the sites are empty. Activator (repressor) molecules, onbinding its specific site, help in transition from the unregulated state G to the active (repressed)state G ( G ) of the gene. There are random transitions taking place between the three statesof the gene. Activator and repressor molecules compete for the state G to take control of thenetwork. If activator molecule wins, the gene turns into active state and protein synthesis occurswith rate constant J p . Protein production does not take place from the unregulated ( G ) andrepressed ( G ) states of the gene. Degradation of proteins occur with rate constant k p and thisevent is independent of the states of the gene. Here transcription and translation are combinedtogether into a single step as done in earlier studies [17, 19]. The stochastic transition from G to G occurs with rate constant k a and that from G to G with k (Fig. 2). The rate constants k a and k are the functions of activator and repressor molecules respectively. Thus, in absence3IG. 1. Schematic diagram of transcriptional regulation by activator and repressor moleculeswhere both the molecules compete for their respective binding site.FIG. 2. Reaction scheme with the three states of the gene: repressed ( G ), unregulated ( G ) andactivated ( G ). From the activated state G proteins are synthesized with rate constant J p .of repressor (activator) molecules the transition from G to G ( G ) is not possible at all. Theassumption that there can be three possible states of the gene provides the basis for the minimalmodel of the activator-repressor system.Let p i ( n, t ) ( i = 1 , , ) be the probability that at time t, the gene is in the G i state with n number of protein molecules in the system. The Master equations for the biochemical reactionscorresponding to the Fig. 2 are given by ∂p ( n, t ) ∂t = k p ( n, t ) − k p ( n, t ) + k p [( n + 1) p ( n + 1 , t ) − n p ( n, t )] (1) ∂p ( n, t ) ∂t = k p ( n, t ) + k d p ( n, t ) − k p ( n, t ) − k a p ( n, t ) + J [ p ( n − , t ) − p ( n, t )]+ k p [( n + 1) p ( n + 1 , t ) − n p ( n, t )] (2) ∂p ( n, t ) ∂t = k a p ( n, t ) − k d p ( n, t ) + J p [ p ( n − , t ) − p ( n, t )] + k p [( n + 1) p ( n + 1 , t ) − n p ( n, t )] (3)Now the standard approach of the theory of stochastic processes will be used to determinethe steady-state probability density function for protein levels [22]. The generating functions aredefined as F ( z, t ) = X n z n p ( n, t ) , F ( z, t ) = X n z n p ( n, t ) , F ( z, t ) = P n z n p ( n, t ) and F ( z, t ) = X n z n p ( n, t ) (4)where F ( z, t ) = F ( z, t ) + F ( z, t ) + F ( z, t ) p ( n, t ) = p ( n, t ) + p ( n, t ) + p ( n, t ) (5)where F ( z, t ) and p ( n, t ) are the total generating function and total probability density functionrespectively. 4
10 15 20 25 30 n p H n L FIG. 3. Plot of p ( n ) versus n for the activator-repressor system for b = 16 and four different setsof parameter values: long dashed curve: s = 1 , s = 6 , s a = 10 , s d = 1 , for solid curve: s = 1 ,s = 4 , s a = 13 , s d = 2 , for short dashed curve: s = 2 , s = 6 , s a = 5 , s d = 1 and for dottedcurve: s = 1 . , s = 6 , s a = 10 , s d = 1 . . In terms of the generating functions (4), Eqs. (1), (2) and (3) can be written as ∂F ( z, t ) ∂t = k F ( z, t ) − k F ( z, t ) + k p (1 − z ) ∂F ( z, t ) ∂z (6) ∂F ( z, t ) ∂t = k F ( z, t ) + k d F ( z, t ) − k F ( z, t ) − k a F ( z, t ) + k p (1 − z ) ∂F ( z, t ) ∂z (7) ∂F ( z, t ) ∂t = k a F ( z, t ) − k d F ( z, t ) + J p ( z − F ( z, t ) + k p (1 − z ) ∂F ( z, t ) ∂z (8)In the steady state ( ∂F i ∂t = 0 , i = 1 , , ), addition of Eqs. (6), (7) and (8) results J p F ( z ) = k p ∂F ( z ) ∂z (9)With the help of the Eqs. (5), (6), (8) and (9), F ( z ) and F ( z ) can be expressed in terms of F ( z ) . Then, in terms of the generating function F ( z ) , the Eqs. (6), (7) and (8) can be written as ( z − F ′′′ ( z ) + { a ( z − − b ( z − } F ′′ ( z ) + { a − b b ( z − } F ′ ( z ) − b a F ( z ) = 0 (10)where a = (1 + s + s + s a + s d ) , b = J p /k p , a = s s a + s s d + s s d , b = a − s d , a = s s a , s = k /k p , s = k /k p , s a = k a /k p and s d = k d /k p .The solution of the Eq. (10) is a generalized hypergeometric function and is given by F ( z ) = C p F q [ g − g ; g + g ; h − h ; h + h ; b ( z − (11)where g = − + b , g = p ( b − − a , h = − + a , h = p ( a − − a , C is thenormalization constant and p F q ( a, b, c, d ) is the generalized hypergeometric function (GHF). Thenormalization constant can be determined easily from the condition F (1) = 1 . Differentiating Eq. (11) n times w.r.t. z at z = 0 , one can easily obtain the expression for thesteady-state probability density function p ( n ) as p ( n ) = C b n Γ( g + n ) Γ( g + n ) Γ( h ) Γ( h ) n ! Γ( h + n ) Γ( h + n ) Γ( g ) Γ( g ) p F q ( g + n ; g + n ; h + n ; h + n ; − b ) (12)5 (cid:13) 5(cid:13) 10(cid:13) 15(cid:13) 20(cid:13) 25(cid:13) 30(cid:13)0(cid:13)200(cid:13)400(cid:13)600(cid:13)800(cid:13)1000(cid:13)1200(cid:13) p ( n ) (cid:13) n(cid:13) FIG. 4. Plot of p ( n ) versus n obtained from stochastic simulation using Gillespie algorithm withthe rate constants k = 1 , k = 6 , k a = 10 , k d = 1 , J p = 16 and k p = 1 (same as the long dashedcurve of Fig. 3). For k p = 1 , s i = k i ( i = 1 , , a, d ) and b = J p .The plot of p ( n ) versus n for different values of s i ( i = 1 , , a, d ) with b = 16 is shown in Fig.3. Different curves in Fig. 3 show that the distributions of protein levels are bimodal in differentparameter regions with s i ( i = 1 , , a, d ) > . The binary response can also be observed in a regionof parameter values with s i ( i = 1 , , a, d ) < (not shown). The binary responses in the activator-repressor system cannot be observed for s , s d > (simultaneously). Figure 4 shows the binaryresponse in protein levels obtained from stochastic simulation using Gillespie algorithm [23] forthe biochemical reactions shown in Fig. 2 for the rate constants k = 1 , k = 6 , k a = 10 , k d = 1 , J p = 16 and k p = 1 (For k p = 1 , s i = k i ( i = 1 , , a, d ) and b = J p ).To understand the origin of bimodal distribution in protein levels in the present scenario wecalculate the components of probability density function p i ( n )( i = 1 , , in the steady state.Using Eqs. (9) and (11) one can easily obtain p ( n ) and is given by p ( n ) = C b n Γ( g + n + 1) Γ( g + n + 1) Γ( h ) Γ( h ) n ! Γ( h + n + 1) Γ( h + n + 1) Γ( g ) Γ( g ) p F q ( g + n +1; g + n +1; h + n +1; h + n +1; − b ) (13)In the steady state, differentiating Eq. (8) n times w. r. t. z at z = 0 we have p ( n ) = ( s d + b ) s a p ( n ) − b n +11 Γ( g + n + 2) Γ( g + n + 2) Γ( h ) Γ( h ) k p s a n ! Γ( h + n + 2) Γ( h + n + 2) Γ( g ) Γ( g ) × p F q ( g + n + 2; g + n + 2; h + n + 2; h + n + 2; − b ) (14)From Eq. (5) we have p ( n ) = p ( n ) − p ( n ) − p ( n ) (15)where p ( n ) and p ( n ) are obtained from Eqs. (13) and (14) respectively.Figure 5 shows the plot of total and component probability density functions p ( n ) and p i ( n )( i =1 , , respectively versus n , the number of proteins, for the rate constants s = 1 , s = 6 , s a = 10 ,s d = 1 and b = 16 (same as the dotted curve in Fig. 3). The bimodal distribution in proteinlevels is clearly the resultant of three unimodal functions p i ( n ) ( i = 1 , , (Fig. 5). This is alsotrue for other bimodal curves in Fig. 3. From the rate constants used to obtain the bimodal6
10 15 20 25 30 n p H n L ,p H n L p H n L p H n L p H n L p H n L FIG. 5. Plot of p ( n ) and p i ( n ) ( i = 1 , , ) versus n for the rate constants s = 1 , s = 6 , s a = 10 ,s d = 1 and b = 16 . The bimodal nature of the function p ( n ) is the resultant effect of threeunimodal functions p i ( n )( i = 1 , , .distributions in Fig. 3, it is clear that the probability of occurrence of the gene in the G and G states are higher. Once it is in the G state, the probability of transition to G state is lowerbecause of the lower value of s . On the other hand, at any instant of time, if the gene is in the G state, there can be two possibilities: gene can switch either to the active state ( G ) or to therepressed state ( G ) depending on the amount of activators or repressors present in the systemsince activator molecules modulate the transition from G to G state and repressor moleculesmodulate the transition from G to G state. Higher values of s a and s make the G and G states more probable than G and due to the lower values of s and s d , the gene spends most ofthe time either in the G or G state. Once the gene is in the G state, the transition to G stateis rare because of the lower value of s d . From the G state of the gene, proteins are synthesizedwith rate constant J p and there is enough time for the protein level to reach the steady value.This gives rise to high protein level in single cell and the peak in the distribution of protein levelat higher value. This is clearly observed in the curve for p ( n ) in Fig. 5. Now if the gene switchessuddenly to G state then protein level starts to decrease. The protein level keeps on decreasingas long as the gene is in the G state or switches to the G state. If the gene switches to the G state the protein level decreases and reaches zero value. This gives rise to low/zero protein levelin a single cell and the peak in the distribution of protein level occurs at low/zero value. This isobserved in the curve for p ( n ) in Fig. 5. From the G state, the gene can also switch back to the G state and this causes a rise of protein level again from an intermediate value. Therefore, thereis a finite probability to observe the protein level at the intermediate value. The curve for p ( n ) shows a finite value at the intermediate region of protein level (Fig. 5).Rossi et al. examined whether an interplay of transcription factors can convert a gradedto binary response in gene expression [12]. They designed an experiment in which the ratio ofactivator and repressor molecules that bind to the same promoter can be modulated by a singleinducer molecule dox. Furthermore, the activator and repressor molecules bind the overlappingbinding sites on the same promoter mutually exclusively. They analyze the graded and binaryresponses to the inducer molecule by flow cytometery in large population of individual cells. Threedifferent cell populations viz. a dox regulated repressor (“repressor only”), a dox regulated activator(“activator only”) and both (“activator+repressor”) were generated in the experiment to study therole of positive and negative transcription factors. The flow cytometric analysis of the activatoronly and repressor only cell populations revealed a graded response (unimodal distribution) ofGFP expression at all dox concentrations. The binary response (two distinct sub-populations)was observed in cells containing both activator and repressor molecules for a range of intermediatedox concentrations. With increasing dox level, the increase in the number of cells with maximal7evel of GFP and decrease in the number of cells with low GFP level is observed. Therefore, anall-or-none (binary) response to the inducer level is observed in the experiment of Rossi et al.[12] when a combination of activator and repressor molecules act on the same promoter mutuallyexclusively. Moreover, since either factor independently produces a graded response, the binaryresponse observed in cells with both the regulatory molecules is not due to a dominant effect ofone factor over the other but rather to their combined effect.Our theoretical analysis of the activator-repressor system does not explicitly include the acti-vator and repressor numbers in the equations but are included in the rate constants s a and s . Therate constant s a increases with the increase in activator amount and s increases with the increasein repressor amount. Now, let us assume that the numbers of both molecules can be controlled bya single inducer molecule like dox, as in the experiment of Rossi et al. [12], so that s a increasesand s decreases with the increase of dox. Depending on the presence of regulatory molecules, thegene regulatory network can be divided into three categories: activator-only system (i.e., only ac-tivator molecules regulate the network), activator-repressor system (i.e., activators and repressorsboth regulate the network) and repressor-only system (i.e., only repressor molecules regulate thenetwork). In presence of only activator molecules the three-state gene activation process reducesto the two-state one. Random switching then takes place only between G and G states. Withthe two-state gene activation process, the graded and binary responses are observed for s a , s d > and s a , s d < respectively [19]. In one hand, with s d > if s a is varied from low to high value thenunimodal responses are observed. On the other hand, with s d < , if s a is varied from low to highvalue then first unimodal (for s a < s d ), then bimodal (for s a ≃ s d ) and then again unimodal (for s a > s d ) responses are observed [19]. Rossi et al. observed graded responses in activator-only sys-tem at all levels of inducer. To reproduce the experimental observations of Rossi et al. we choosethe parameter region s a , s d > . Let us assume that initially there are only activator molecules(with low copy number) activating the gene transcription and s d is fixed at . . Now, with thegradual increase of inducer molecules dox in the system, s a increases and the mean protein levelalso increases gradually. The probability distributions always remain graded (curves in the left col-umn of Fig. 6) because the values of s a and s d satisfy the condition of unimodal/graded response( s a , s d > ) for all values of dox [19] . Let us now consider the same regulatory network (same s d ) but with repressor molecules also present in the system. The gene can now switch betweenall three possible states and both the molecules compete for their binding site to take control ofthe gene transcription. Let us assume that initially there are large number of repressors i.e., s islarge and small number of activators i.e., s a is low. With the gradual increase of dox moleculesin the system, s decreases and s a increases gradually and simultaneously i.e., inhibition effectdecreases and activation effect increases simultaneously. This causes the conversion of unimodal(for low dox i.e., low s a and high s ) to bimodal (for intermediate dox i.e., intermediate s a and s )and then again unimodal (high dox i.e., high s a and low s ) distribution of protein levels (rightcolumn of Fig. 6). The gradual increase in the inducer level causes a discontinuous change in themean protein level. Therefore, the response is bimodal/binary as the mean protein level is not acontinuous function of inducer but has only low and high values. These results (Fig. 6) are inqualitative agreement with the experimental observations of Rossi et al. [12] for activator-only andactivator-repressor systems.Rossi et al. [12] also observed the graded response when only repressor molecules regulatethe gene transcription. To reproduce the experimental observation for repressor-only case in thepresent scenario, one has to consider the basal rate of protein synthesis from the unregulatedstate of the gene ( G ). With the basal rate of protein synthesis, say J ( J < J p ), from the G state, the generation of graded response for repressor-only case is quite similar to that of theactivator-only case discussed above. In the presence of only repressor molecules in the system, the8
10 15 20 25 30 n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Unimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Unimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Unimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Bimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Bimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Bimodal n p H n L s a = .0 Unimodal n p H n L s a = .0, s = .0 Unimodal
FIG. 6. Distribution of protein levels p ( n ) versus n in activator-only system (left column) andactivator-repressor system (right column) for different level of inducer molecules i.e., for differentvalues of s a and s . For the activator-only system s is kept fixed at . and s a is varied asmentioned on the top of the figures (figures in the left column). For the activator-repressor system s and s d are kept fixed at . and . ( s d is same as in the activator-only system) respectivelyand the different curves are drawn for different values of s a and s mentioned on the top of thefigures (figures in the right column). For all curves the relative transcription rate constant b = 16 .9hree-state gene activation process reduces to the two-state one i.e., the gene can switch randomlyonly between G and G states. Repressor molecules help in transition from G to G state. Withthe finite basal rate of protein synthesis from the G state the response will be graded for s > and for all values of s [19] . The initial value of the rate constant s is large due to the presenceof large number of repressor molecules in the repressor-only system, the response in this case willbe unimodal since s = 1 . and s is large. Now with the gradual increase of dox concentration,the rate constant s decreases from a high to a low value but the response still remains gradeddue to s being greater than one ( s = 1 . ). Therefore, with the basal rate of protein synthesisfrom the unregulated state the graded response can also be observed for repressor-only system.But with the basal rate of protein synthesis from the state G the derivation of exact analyticalexpression for the probability density function of protein levels for the activator-repressor systemis very difficult. Though, with the help of stochastic simulation using Gillespie algorithm, it can beshown that the finite basal rate of protein synthesis from the state G does not change our resultsqualitatively. The qualitative nature of the curves drawn in Figs. 3 and 6 will remain unchangedwith the basal rate of gene expression from G taken into account. III. DISCUSSION
In this paper, we have studied an gene regulatory network where the positive and negativetranscription factors regulate the gene transcription mutually exclusively. Both the moleculescompete for their respective binding sites on the DNA to take control of the network (Fig. 1). Theactivator-repressor system is represented by a simple stochastic model where gene can be in threepossible states viz. inactive/repressed, unregulated and active. An exact analytical expressionfor the probability density function of the protein levels in the steady state is derived and is ageneralized hypergeometric function (GHF) (Eq. (12)). From the GHF, the bimodal distributionin protein levels is observed in a wide region of the parameter values. From the theoretical analysis,the experimental observation of Rossi et al. (i.e., the regulation only by activator moleculesproduce the graded response in the protein levels whereas binary responses are observed whenboth the activator and repressor molecules regulate the gene transcription by binding the promotermutually exclusively) can be reproduced very easily. Here we have considered only the parameterregion s i ( i = 1 , , a, d ) ≥ (Fig. 3). The binary response in protein level is more prominentfor s i ( i = 1 , , a, d ) < (not shown). This region is excluded from the present analysis becausewith s i ( i = 1 , , a, d ) < , the binary response can also be observed for two-state activator-onlysystem [19]. But the experiment of Rossi et al. [12] observed only graded response when onlyactivator molecules regulate the gene transcription. This experimental observation along with thetheoretical prediction of graded response in gene expression [19] helps us to choose the parameterregion for theoretical analysis. Rossi et al. observed the graded response also for repressor-onlysystem. Here we have not considered the repressor-only case because this requires a basal rate ofprotein synthesis from the unregulated ( G ) state. The basal rate of protein synthesis from theunregulated state brings difficulties in the analytical tractability of the model. In the presenceof the basal rate of protein synthesis from the unregulated state, it is very difficult to expressthe components of the generating functions F i ( z ) ( i = 1 , , ) in terms of the total generatingfunction F ( z ) and therefore the Chemical Master Equations (CME) cannot be expressed by asingle differential equation like Eq. (10). Again, the reduction of the CME into a single differentialequation does not lead to the exact solution of CME because of the unavailability of the analyticalsolution of the higher order differential equation. This shows the limited scope and applicability ofthe generating function technique used here to solve the CME. The difficulty increases when the10egulatory networks consist of nonlinear feedback loops.In the present analysis of activator-repressor system, we have combined the transcription andtranslation into a single step process. In the process of transcription mRNAs are produced from theactive gene and then mRNAs are translated into proteins. Therefore, the steady state probabilitydistribution (Eq. 12) derived here gives the correct description for mRNAs. The distributionin protein levels does not follow the bimodal mRNA distribution when the protein lifetime islonger than that of mRNA [24]. Despite the above limitations of the stochastic model, it containsimportant features necessary for an explanation of the binary response in an activator-repressorsystem and is graded in activator only system as observed in experiment [12]. The exact analyticalresult with three gene states is important and useful specially in the eukaryotic system. The geneactivation of the complex eukaryotic system consists of many unknown number of rate limitingsteps (chromatin remodeling, assembly of preinitiation complex etc.). The simplification of thecomplex gene activation process by the two-state one is the first approximation of the complicatedbiological process. The ’three-state’ assumption may be considered as the second approximationof the stochastic gene activation-deactivation process.The present analysis of the origin of binary responses in three-state model may be helpfulto explain the bimodal distribution in transcriptional silencing [25]. In transcriptional silencing,Sir proteins (Sir 2-4) are the key structural components of silenced chromatin and under theirregulation the silencer can be in two possible states: repressed and derepressed. The silencerhelps to assemble the Sir protein complex. This process of assembling is not a single step processbut rather consists of several reversible biochemical steps. The intermediate steps between therepressed and derepressed states of the chromatin make the effective rates of transitions very slowand these slow rate of transitions ultimately may lead to the bimodal distribution of protein levelsfrom reporter gene. Acknowledgement
This work is supported by the Minor Research Project Grant, UGC, India, under Sanction No.F. PSW-001/07-08 (ERO).
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