| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
NORDITA (Nordic Institute for Theoretical Physics), DK-2100 Copenhagen, Denmark
Correspondence: Address reprint requests to Audun Bakk, E-mail: audunba{at}nordita.dk.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
). In other words, genetic switches regulate the cell and become the key elements in the synthesis of certain proteins. To understand the performance of a genetic switch, it turns out to be important to obtain detailed knowledge about the physical and chemical properties of the system (Alberts, 1998). A well-studied regulatory system is the bacteriophage lambda (phage 
) in the bacterium Escherichia coli, which under physiological conditions exhibits an extremely high stability (Brooks and Clark, 1967; Little et al., 1999).
Upon phage infection of E. coli, either one phage genome (prophage) is introduced into the host genome and silently replicated for generations, which is called the lysogenic track; or it becomes massively replicated by use of the host cell chemistry and the E. coli cell bursts (lyses), called the lytic track. The latter is also the outcome when a lysogen (E. coli cell on the lysogenic track) becomes irradiated with ultraviolet light (DNA becomes damaged) (Ptashne, 1992).
The right operator (OR) is playing an important role in the fate of the bacterium after infection. As shown in Fig. 1, OR consists of three operator sites, each potential binding sites for dimers of the regulatory proteins CI (commonly called repressor) and Cro. RNA polymerase (RNAP) is able to bind either to a region including OR1 and parts of OR2 (promoter PR in Fig. 1), and thereby initiates cro transcription, or it can bind to a region including OR3 and parts of OR2 (promoter PRM in Fig. 1), initiating cI transcription. (Nomenclature: genes are denoted with italicized letters and their protein products with Roman letters where the first letter is capitalized.) In a lysogen, OR1 and OR2 are usually occupied by one CI dimer each, exhibiting a cooperative interaction, and PRM is occupied by RNAP such that CI is continuously expressed that maintains repression of cro.
|
genome (Johnson et al., 1979; Takeda et al., 1992; Darling et al., 2000b). We will use experimental data of the affinities (protein-operator binding energies) of CI, Cro, and RNAP within a statistical-mechanical approach similar to Ackers et al. (1982). This provides us with the probabilities for RNAP occupancy of the promoters that are proportional to the activities that in turn are proportional to the production rates of CI and Cro, respectively. We study the sensitivity of the activities of PRM and PR to variations of the operator affinities within their experimental uncertainty. Furthermore, from experiments it is known that a single point mutation of operator DNA typically corresponds to a shift of ±2 kcal/mol in the binding affinity (Burz and Ackers, 1996; Little et al., 1999). One mutation may in principle be compensated through another mutation corresponding to a shift of another affinity (suppression).
Below, we systematically perturb the affinities ±2 kcal/mol, one by one, to mimic mutations. Thus, such (large) perturbations may be regarded as hypothetical mutations and in the following we for simplicity term these ±2 kcal/mol perturbations as mutations. Note that these mutations are not directly linked to experimental data, but serve us to assess the stability of the -switch. In our analysis, we check for possible suppressors for the affinities where the mutations correspond to a significant change in activity (>25% in absolute value relative to wild-type activity). We study the system in three different regimes: 1), lysogen, where CI dominates; 2), during induction, where CI and Cro are at comparable concentrations; and 3), after induction, where Cro dominates.
We find that the activity is not very sensitive within the experimental error in the lysogenic regime at PRM; however, in regimes 2 and 3, the activities turn out to be more sensitive at both promoters. The strength of the RNAP affinities appears to be important to maintain the activity. Interestingly, we find a number of suppressors in all regimes.
In the following, we first introduce the thermodynamics and the models involved, whereupon the sensitivity in three different concentration regimes is investigated before drawing our conclusions.
|
MODELING THE SYSTEM |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
). This means that mRNAs (and proteins) are produced on a timescale of minutes. Furthermore, Aurell et al. (2002) estimate that protein association typically occurs in fractions of a second. Thus, with regard to the production of mRNAs in a cell, it is a reasonable assumption that the protein association with DNA is in equilibrium.
Thermodynamics
As mentioned, we apply the statistical-mechanical (equilibrium) approach presented by Ackers et al. (1982). Binding of CI dimers (CI2), Cro dimers (Cro2), and RNAP to OR of phage can occur in 40 different combinations s as listed in Table 1. The associated probability fs for finding the system in one of these 40 states s is (Hill, 1960; Ackers et al., 1982)
 | (1) |
G(s) is the Gibbs free energy difference (binding energy) between state s and the unoccupied state (s = 1). [CI2], [Cro2], and [RNAP] are the free (unbound) concentrations of CI dimers, Cro dimers, and RNAP, respectively. is 
{0, 1, 2, 3}, js {0, 1, 2, 3}, and ks {0, 1, 2} are the number of CI dimers, Cro dimers, and RNAP bound to OR in the state s. For instance, from Table 1 the state s = 23 corresponds to i23 = 1, j23 = 0, k23 = 1, and G(23) = -22.0 kcal/mol.
|
|
, G1 = G(2) is the free energy associated with the binding of CI2 to OR1, etc., and G1' = G(5) is the free energy associated with the binding of Cro2 to OR1, etc. (Table 2). Furthermore, two CI dimers at neighboring operator sites are supposed to obtain an additional cooperative free energy (see Fig. 2 in Shea and Ackers (1985) for a more detailed explanation). Thus, G12 is the cooperative free energy associated with the binding of CI2 to both OR1 and OR2, and G23 is the cooperative free energy associated with the binding of CI2 to both OR2 and OR3, provided that no repressor is bound to OR1. The data in Table 1 are all obtained in vitro in 200 mM KCl, resembling "physiological" conditions (Kao-Huang et al., 1977; Ackers et al., 1982).
|
assume no cooperativity between Cro dimers bound to vicinal operator sites. The more recent results reported by Darling et al. (2000b) state that Cro exhibits cooperativity as well. However, as these experiments are performed at 20°C, we prefer in this work to apply the Cro data from Shea and Ackers (1985) measured at 37°C. To check for possible implications of such cooperative interactions between Cro dimers in our perturbations, we introduce the tetrameric interactions, i.e., dimer-dimer interactions, G12' and G23', and the hexameric Gibbs free energy term G123' (see Table 1). The latter term takes into account the additional Gibbs free energy when all three operators are occupied by Cro2 (Darling et al., 2000b). Let us stress again that such Cro cooperativity is not assumed in the wild-type data (unperturbed), as stated in Table 2.
The Gibbs free energy of RNAP association at PR and PRM is GR = -12.5 kcal/mol and GRM = -11.5 kcal/mol, respectively. The latter values have an accuracy of ±0.5 kcal/mol (Shea and Ackers, 1985). Even though more details on RNAP have recently been obtained from in vivo experiments (Ptashne and Gann, 2002), to our knowledge there do not exist more accurate data of RNAP affinities. The overall resulting Gibbs free energies associated with the 40 states of the DNA associations of CI, Cro, and RNAP are listed in Table 1. Throughout this work we have for simplicity assumed a constant free RNAP concentration of 30 nM, as applied by Shea and Ackers (1985).
The free concentrations of CI monomers and dimers ([CI1] and [CI2]) are supposed to be in equilibrium, with a dissociation constant Kd = [CI1]2/[CI2] = 18 nM (Koblan and Ackers, 1991). Furthermore, Cro is in this work only supposed to occur in the dimeric form; this was also the assumption of Shea and Ackers (1985) during their analysis of the Cro affinity data (which we apply in this work). However, introduction of a nonzero dissociation constant of Cro does not modify our main results.
The total concentration of CI molecules, in monomeric equivalents, yields
 | (2) |
 | (3) |
Activity
The main purpose of this article is to estimate the effects of perturbations of the protein-operator and RNAP-operator affinities on the production rates (activities) of the regulatory proteins CI and Cro. Ptashne et al. (1980) point out that transcription initiation is the rate-limiting step in protein synthesis. More specifically, it is apparently the step taking the RNAP-DNA complex from the closed to the open form (isomerization step) that is limiting the rate with respect to repressor and Cro synthesis in a lysogen, and during induction of lysis (McClure, 1980). Thus, activity may be defined as the product of isomerization rate times the probability of RNAP occupancy of the promoter. In what follows, we use the same rate constants as Shea and Ackers (1985) in enumerating the activities. Thus, the rate constant we apply for cro isomerization is kR = 0.014 s-1, whereas the rate constant for cI isomerization is split into two terms: one stimulated rate when OR2 is occupied by CI2, kRM1 = 0.011 s-1, and one unstimulated rate when OR2 is not occupied by CI2, kRM2 = 0.001 s-1. The ratio kRM1/kRM2 = 11 is according to Hawley and McClure (1982).
One should note that the origin of the stimulated transcription is unresolved, e.g., it is argued that the increased cooperativity transcription is due to higher promoter affinity of RNAP because the CI dimer at OR2 touches the polymerase and thereby enhances cI transcription (pages 21–22 in Ptashne (1992)). We will in this work use the traditional approach of Hawley and McClure, as mentioned above.
 | (4) |
 | (5) |
Since we are using a different data set for the CI affinities in this work compared with Shea and Ackers (1985), it is interesting to compare the promoter activities emerging from the two data sets, as shown in Fig. 2. Although both sets have the same qualitative behavior, they differ quantitatively. Employing the CI data from Shea and Ackers (1985) instead of the CI data of Koblan and Ackers (1992), Cro activity (PR) at lysogenic level ([CIt] 
200 nM and [Crot] 0) is elevated to a nonzero value, whereas the CI activity (PRM) is reduced by a factor 0.8 at the same protein level. However, given this observation, it is not straightforwardly possible to conclude what effect the changes of the individual CI affinities have on the total activity. We therefore systematically perturb the wild-type affinities in the following.
Dynamics
The dynamics of the system is quantified by the promoter activity and through subsequent intracellular production and degradation of CI and Cro versus time. From this, we obtain a rough estimate of the protein levels around the induction point, where the total CI and Cro levels are comparable. We use the dynamical equations and parameters of Shea and Ackers (1985). CI and Cro production rates are proportional to the promoter activities in Eqs. 4 and 5 above. In the rate equation for CI production, we also introduce a degradation term that is introduced to model RecA-mediated cleavage of repressor monomers, which causes that the repressors are unable to dimerize and thereby bind to the operator (Ptashne, 1992). We ignore cell growth in our simulations, but as the determination of the induction point is not crucial, this approximation will not significantly influence on our main results.
The two dynamical rate equations we obtain for CI and Cro (both of the form d([CI] or [Cro])/dt 
probability for RNAP occupancy of PRM or PR), which are equivalent to Eqs. 2 and 3 of Shea and Ackers (1985) and therein described in detail, are solved simultaneously by means of the fourth order Runge-Kutta method (numerical time step algorithm) (Dahlquist and Björk, 1974). This simulation yields the curves in Fig. 3. The initial conditions (time = 0) are [CIt] = 200 nM and zero Cro concentration, which may be regarded as typical concentrations for a lysogen. It is the protease- (RecA) mediated cleavage of repressor monomers that reduces the repressor concentration and makes it possible for Cro concentration to increase and eventually dominate as shown in Fig. 3.
|
|
RESULTS AND DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
GRM and GR (associated with RNAP) are systematically perturbed ±1 kcal/mol in the three different concentration regimes: lysogeny (1), around induction (2), and after induction (3). We note that with the footprint titration technique Koblan and Ackers (1992) applied to determine the CI affinities, the deviations of the affinities range up to 0.7 kcal/mol with ±67% confidence intervals. Thus, a perturbation of ±1 kcal/mol is reasonable to take into account experimental uncertainties and to probe for their potential effects.
We also study the effect of large perturbations by systematically changing each individual affinity ±2 kcal/mol, representing a typical operator mutation (Burz and Ackers, 1996; Little et al., 1999). Finally, we check for possible suppressors counteracting these large perturbations (termed mutations for simplicity) in all three different concentration regimes. We stress that the ±2 kcal/mol mutations (and their suppressors) are not directly linked to experimental data, but may rather be regarded as a prediction or indication of the effect such perturbations (mutations) have upon the activity.
Regime 1
We first consider the lysogenic regime. This state is characterized by a negligible Cro concentration and [CIt] 200 nM in monomeric equivalents. Fig. 4 illustrates that a perturbation of G1 has hardly any effect on the activity in a lysogen. Table 3 presents the results of systematic perturbations.
|
|
The perturbation of G3 has the largest effect on the activity at PRM among the CI affinities. This is reasonable because a more negative G3 makes CI repress its own synthesis (Ptashne, 1992) (a decrease in Gibbs free energy is equivalent to a stronger binding). Similarly, an increase in GRM leads to a decreased activity because RNAP then visits, and thereby transcribes, PRM less frequently. Naturally, the activities at PRM and PR are expected to be sensitive upon perturbations of GRM and GR, respectively.
By perturbing the affinities systematically ±2 kcal/mol, which we term as (hypothetical) mutations, we find in regime 1 that it is only the +2 kcal/mol mutation of GRM and the -2 kcal/mol mutation of G3 and GRM that leads to >25% change of the activity at PRM. Perturbations of the Cro affinities do not change the activity due to zero Cro concentration in this regime.
Regarding a mutation of +2 kcal/mol, GRM has no suppressors, i.e., this mutation cannot be compensated by another mutation (of another affinity), such that wild-type activity is restored. Conversely, all CI affinities and GR are suppressors for a mutation of -2 kcal/mol of GRM. Consequently, the binding strength of RNAP at PRM seems to be crucial for maintenance of wild-type activity.
Even though the impact of the perturbations of G1 at PRM under lysogenic conditions is negligible, we see in Fig. 4 that the effect is more pronounced at lower CI concentrations. In other words, the impact of perturbations will strongly depend on the respective concentrations, and thus motivates us to consider perturbations at other protein concentrations.
Regime 2
To estimate typical protein concentrations at the transition when the switch turns over such that CI production is replaced by Cro production, we perform a simulation as described in Dynamics. In Fig. 3, we display the dynamics after the introduction at time = 0 of CI monomer degradation mediated by protease RecA. At 45 min the total protein concentrations of CI and Cro are comparable (24 nM). At this concentration the Cro level starts to rise substantially, indicating that the system is committed to the lytic pathway. We call this crossover the induction point, but note that this definition of the induction point is somewhat arbitrary.
The induction point in the simulations of Shea and Ackers (with other CI affinity data), in comparison, occurs at 22 min corresponding to total concentrations of 43 nM. Thus, the numerical values of the simulated protein levels during induction are sensitive with respect to the affinity data. Furthermore, in our simulations (Fig. 3), we see that the repressor level has to be substantially lower compared with the lysogenic level (20%) to induce derepression at PR and thereby enhance Cro production. This behavior is also reported on in vivo experiments by Bailone et al. (1979).
In Fig. 5, we plot the promoter activities versus repressor level. Around induction both activities associated with PRM and PR are significant, and both activities are influenced by the perturbation of G1. In Table 4, we present the results from a systematic perturbation scheme. We find that the changes in both activities are relatively large for perturbations of G1, G2, G12, and GR. We also note that the activity of PRM changes considerably due to the perturbations of G3' and GRM. This is interesting, because the perturbations of the affinities around induction then have different effects on the corresponding activities compared with the perturbations in the lysogenic regime, in particular at PRM. Except for the high sensitivity of G3' at PRM, sensitivity of Cro is low in this regime.
|
|
G1, G2, and GR, and the +2 kcal/mol mutation of G23' change the activities at both promoters more than 25%. Interestingly, in regime 2 we find common suppressors that restore both wild-type activities within 5% for all these latter mentioned affinities. The only exception is the +2 kcal/mol mutation of GR. This means, for instance, that a +2 kcal/mol mutation of G2 can be compensated by a -2 kcal/mol mutation of G12, whereupon the wild-type activity is restored (within 5%) at both promoters.
As mentioned above, the induction point is sensitive upon affinity data. To check possible implications we choose another induction point that is equivalent of Shea and Ackers (1985) at total protein concentrations 43 nM, which is about twice the value previously discussed (24 nM). With this shift of induction point, PRM activity is reduced by a factor two and PR activity is reduced by a factor four, because the increased CI and Cro concentrations repress PRM and PR, respectively. Regarding the ±1 kcal/mol perturbations, with the new induction point, most of the activities change in a similar manner as listed in Table 4 (within 20%); however, for a few activities, the change in the activities is increased by a factor two, presumably due to the reduced wild-type activity that leads to an increased sensitivity upon perturbations of the affinities. We also find the same pattern of suppressors in this new situation with the induction point moved to total protein concentration of 43 nM. The only difference in this respect is that two new suppressors occur in the latter case (43 nM) compared to the original case (24 nM).
Regime 3
Finally, we introduce perturbations in the lytic regime. Here, the protein levels after induction are not known in vivo. However, as seen in Fig. 4, the repressor level is approximately zero for [Crot] above 100 nM, and we choose [Crot] 200 nM as a typical protein concentration after induction. Fig. 6 shows for large Cro levels that the activity at PRM is negligible. The ratio between PR and PRM is of order 102. Table 5 shows that the sensitivity of Cro affinities is in general high in this regime at both promoters.
|
|
GR, have suppressors. The latter observation shows the uniqueness of GR in this regime.
The recent data of Darling et al. (2000a) show that Cro has a nonzero dissociation constant at 20°C. We are not aware of any corresponding Cro affinity data measured at 37°C. Nevertheless, it is important to probe for the impact of such a monomer-dimer equilibrium. We find that at a given free Cro concentration, a nonzero Cro dissociation constant has no impact on the activity at any concentration. However, for a given total Cro concentration, a nonzero dissociation constant leads to less free Cro dimers, implying an effectively weaker Cro affinity associated with OR.
As previously mentioned (in Thermodynamics section), Darling et al. (2000b) measured nonzero Cro cooperative affinity terms—G12', G23', and G123' in Table 2. However, these data were measured at 20°C. In this work we study the system at 37°C and apply Cro data from Shea and Ackers (1985) without Cro cooperative terms. Nevertheless, we want to check implications of such Cro cooperative terms. Thus, these are given nonzero values, one by one, in our perturbation analysis. The ±1 kcal/mol perturbations in regime 1 and 2 of Cro cooperative terms lead to negligible changes of the activity. This is interesting in light of the size of the Cro cooperative affinities that are in the range 0.5–1 kcal/mol (Darling et al., 2000b). However, in regime 3, as shown in Table 5, Cro cooperativity has a nonnegligible effect upon both activities.
DNA binding of CI and Cro outside OR (nonspecific binding) may have impact on the free intracellular protein concentrations (Reinitz and Vaisnys, 1990; Johnson et al., 1981). Nonspecific binding leads to a larger effective cellular volume (Aurell et al., 2002). We test our simulation with regards to the perturbations performed at a volume increased by a factor 2. For a lysogen, the effect is negligible compared to original data, and in regime 3 the sensitivity is slightly reduced. However, our main conclusions about sensitivity remain unchanged. In regime 2, the situation is more complex, because nonspecific binding leads to another induction point and comparison to the original data is not obvious.
Another source of error, with regards to the relevance of our results in vivo, is the possibility for DNA loops formed by a more or less stable repressor octamer between the left operator and OR of phage that effectively reduces PRM activity (Dodd et al., 2001). Due to the fact that such mechanism is a recent finding and sufficient experimental details remain to be established, we do not here discuss the influence of such DNA looping.
|
SUMMARY AND CONCLUSION |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
genome. Experimental values of the protein-operator interactions are applied in a statistical-mechanical approach, with a probability for the different binding states given in Eq. 1. This is used to predict the activity (proportional to the rate of protein production) of the two competing promoters PRM and PR associated with OR.
Systematic perturbations (±1 kcal/mol) of the affinities in three different concentration regimes (lysogenic, around induction, and lytic) show that a lysogen at PRM is not very sensitive with respect to the activity, in contrast to inductive and lytic regimes. Thus, within experimental error of the affinities (±(0.5–1) kcal/mol), which may also reflect typical fluctuations of the cellular protein concentrations, lysogenic activity at PRM remains stable. The fact that the sensitivity is significant in the late lytic regime may not be a "problem" for the system to make lysis efficient, because at this stage other genes are important (Ptashne, 1992). In this respect, the sensitivity of regime 2 may turn out to be the most notable one.
In regime 1, at PRM, only perturbations of GRM (RNAP at PRM) significantly change the activity. In regime 3, at PR, the corresponding changes are linked to perturbations of PR-associated Cro affinities and GR (RNAP at PR). Around induction, where both promoters are active, the sensitivity of the activity is large upon perturbations of three CI affinities and one Cro affinity, and both RNAP affinities.
We also look at large perturbations of order ±2 kcal/mol that may resemble a typical shift in the binding energy upon a mutation (Burz and Ackers, 1996; Little et al., 1999). Thus, for simplicity, such large perturbations are here called mutations, but we stress that these are not linked to specific experimental data and should therefore be regarded as hypothetical ones. In particular we study mutations that alter the activity >25%.
Most affinities (in all three regimes) have one or more suppressors defined as a perturbation that compensates for a mutation (±2 kcal/mol) such that wild-type activity is restored. However, it is notable that a +2 kcal/mol mutation of GRM in regime 1, GRM and GR in regime 2, and GR in regime 3 have no suppressors. In other words the RNAP affinities cannot be weakened much without destroying the function of the -switch. Furthermore, in regime 2 there are several affinities that change the activity >25% at both promoters. Surprisingly, it is only the +2 kcal/mol perturbation of GR that cannot be suppressed, by the same compensating mutation, such that wild-type activity is restored at both promoters.
It is also interesting that our perturbations may to some extent incorporate intracellular (time) fluctuations and cell-to-cell (ensemble) variations, i.e., noise (Metzler, 2001; Aurell and Sneppen, 2002; Elowitz et al., 2002), because these variations may effectively be regarded as uncertainties of the affinities. Thus in regime 1, within this approximation, it is only noise that effectively influences RNAP affinity that has significant effect upon PRM activity. Following this argumentation, noise will in general influence the activities mostly around induction and in the lytic regime.
To our knowledge, we have for the first time presented a systematic study of the sensitivity of the regulatory system associated with OR in phage . The identification of a small number of affinities that have a high sensitivity is expected to shed new light on the operating principle of genetic switches, similarly the findings of suppressors.
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
Submitted on May 8, 2003; accepted for publication September 15, 2003.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONMODELING THE SYSTEMRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
phage repressor. Proc. Natl. Acad. Sci. USA. 79:1129–1133.Alberts, B. 1998. The cell as a collection of protein machines: preparing the next generation of molecular biologists. Cell. 92:291–294.[Medline]
Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson. 1994. Molecular Biology of the Cell, 3rd ed. Garland, New York. Chapter 9.
Aurell, E., S. Brown, J. Johanson, and K. Sneppen. 2002. Stability puzzles in phage . Phys. Rev. E. 65:051914.
Aurell, E., and K. Sneppen. 2002. Epigenics as a first exit problem. Phys. Rev. Lett. 88:048101.[Medline]
Bailone, A., A. Levine, and R. Devoret. 1979. Inactivation of prophage repressor in vivo. J. Mol. Biol. 131:553–572.[Medline]
Brooks, K., and A. J. Clark. 1967. Behavior of bacteriophage in a recombination deficient strain of Escherichia coli. J. Virol. 1:283–293.
Burz, D. S., and G. K. Ackers. 1996. Cooperativity mutants of bacteriophage cI repressor: temperature dependence of self-assembly. Biochemistry. 35:3341–3350.[Medline]
Dahlquist, G., and Å. Björk. Translated by N. Anderson. 1974. Numerical Methods. Prentice Hall, New Jersey.
Darling, P. J., J. M. Holt, and G. K. Ackers. 2000a. Coupled energetics of cro repressor self-assembly and site-specific DNA operator binding I: analysis of cro dimerization from nanomolar to micromolar concentrations. Biochemistry. 39:11500–11507.[Medline]
Darling, P. J., J. M. Holt, and G. K. Ackers. 2000b. Coupled energetics of cro repressor self-assembly and site-specific DNA operator binding II: cooperative interactions of cro dimers. J. Mol. Biol. 302:625–638.[Medline]
Dodd, I. B., A. J. Perkins, D. Tsemitsidis, and J. B. Egan. 2001. Octamerization of CI repressor is needed for effective repression of PRM and efficient switching from lysogeny. Genes Dev. 15:3013–3022.
Elowitz, M. B., A. J. Levine, E. D. Siggia, and P. S. Swain. 2002. Stochastic gene expression in a single cell. Science. 297:1183–1186.
Hawley, D. K., and W. R. McClure. 1982. Mechanism of activation of transcription initiation from the PRM promoter. J. Mol. Biol. 157:493–525.[Medline]
Hill, T. L. 1960. Introduction to Statistical Thermodynamics. Addison-Wesley, London, England.
Johnson, A. D., B. J. Meyer, and M. Ptashne. 1979. Interactions between DNA-bound repressors govern regulation by the phage repressor. Proc. Natl. Acad. Sci. USA. 76:5061–5065.
Johnson, A. D., A. R. Poteete, G. Lauer, R. T. Sauer, G. K. Ackers, and M. Ptashne. 1981. repressor and cro—components of an efficient molecular switch. Nature. 294:217–223.[Medline]
Kao-Huang, Y., A. Revzin, A. P. Butler, P. O'Conner, D. W. Noble, and P. H. von Hippel. 1977. Nonspecific DNA binding of genome-regulating proteins as a biological control mechanism: measurement of DNA-bound Escherichia coli lac repressor in vivo. Proc. Natl. Acad. Sci. USA. 74:4228–4232.
Koblan, K. S., and G. K. Ackers. 1991. Energetics of subunit dimerization in bacteriophage cI repressor: linkage to protons, temperature, and KCl. Biochemistry. 30:7817–7821.[Medline]
Koblan, K. S., and G. K. Ackers. 1992. Site-specific enthalpic regulation of DNA transcription at bacteriophage OR. Biochemistry. 31:57–65.[Medline]
Little, J. W., D. P. Shepley, and D. W. Wert. 1999. Robustness of a gene regulatory circuit. EMBO J. 18:4299–4307.[Medline]
Metzler, R. 2001. The future is noisy: the role of spatial fluctuations in genetic switching. Phys. Rev. Lett. 87:068103.[Medline]
McClure, W. R. 1980. Rate-limiting steps in RNA chain initiation. Proc. Natl. Acad. Sci. USA. 77:5634–5638.
Ptashne, M. 1992. A Genetic Switch: Phage and Higher Organisms, 2nd ed. Cell Press & Blackwell Scientific Publications, Cambridge, MA.
Ptashne, M., and A. Gann. 2002. Genes & Signals. Cold Spring Harbor Laboratory Press, New York.
Ptashne, M., A. Jeffrey, A. D. Johnson, R. Maurer, B. J. Meyer, C. O. Pabo, T. M. Roberts, and R. T. Sauer. 1980. How the repressor and cro work. Cell. 19:1–11.[Medline]
Reinitz, J., and J. R. Vaisnys. 1990. Theoretical and experimental analysis of the phage lambda genetic switch implies missing levels of co-operativity. J. Theor. Biol. 145:295–318.[Medline]
Shea, M. A., and G. K. Ackers. 1985. The OR control system of bacteriophage lambda: a physical-chemical model for gene regulation. J. Mol. Biol. 181:211–230.[Medline]
Takeda, Y., P. D. Ross, and C. P. Mudd. 1992. Thermodynamics of Cro protein-DNA interactions. Proc. Natl. Acad. Sci. USA. 89:8180–8184.
This article has been cited by other articles:
 |
|
 |
|
  V. B. Teif General transfer matrix formalism to calculate DNA-protein-drug binding in gene regulation: application to OR operator of phage {lambda} Nucleic Acids Res., June 28, 2007; 35(11): e80 - e80. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 C. Lou, X. Yang, X. Liu, B. He, and Q. Ouyang A Quantitative Study of {lambda}-Phage SWITCH and Its Components Biophys. J., April 15, 2007; 92(8): 2685 - 2693. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams Target Search of N Sliding Proteins on a DNA Biophys. J., August 1, 2005; 89(2): 895 - 902. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
