A Quantitative Study of λ-Phage SWITCH and Its Components

Abstract

We propose what we believe is a new model to quantitatively describe the λ-phage SWITCH system. The model incorporates facilitated transfer mechanism of transcription factor, which can be simplified into a two-step reaction. We first sequentially obtain two indispensable parameters by fitting our model to experimental data of two simple systems, and then apply them to study the natural λ-SWITCH system. By incorporating the facilitated transfer mechanism, we find that in RecA⁻ host Escherichia coli, the wild-type λ-lysogenic state is in a monostable regime rather than in a bistable regime. Furthermore, the model explains the weak role of Cro protein and probably sheds light on the evolution of λ-Cro protein, which is known to be structurally distinct from the other Cros in lambdoid family members.

INTRODUCTION

One of the paradigms for quantitative study of living organisms is λ-phage, which has two phenotypes: lysogeny and lysis. In the lysogenic state, its DNA is integrated into the genome of host cell; whereas in the lytic state it is duplicated inside the host until destroying the host and releasing its progeny (1). Upon ultraviolet induction, λ-phage will exit the lysogenic state and enter the lytic state (1). It is worthy to note that this transition is unidirectional, i.e., transition from lysis to lysogen does not exist. Thus lysogeny and lysis are not good indicators for the possible bistable system.

Among λ-phage genome, there is one element, called SWITCH, which is the most important regulation module for the life cycle of the infected Escherichia coli. As described in Fig. 1, the SWITCH consists of two genes (cI and cro), two promoters (P_R and P_RM), three operators (O_R1, O_R2, and O_R3) in the O_R region, and three other operators (O_L1, O_L2, and O_L3) in the O_L region. The molecular mechanism of the SWITCH has been elaborated for a long time, although the detail was modified recently (1). As shown in Fig. 1 a, when O_R3 is free, gene cI can be transcribed by P_RM promoter; its activity can increase 10-fold if O_R2 is further occupied by CI₂. When both O_R1 and O_R2 are free, gene cro can be transcribed from P_R promoter by RNA polymerase. The O_L region participates in the SWITCH's regulation via DNA looping as shown in Fig.1, b and c. The DNA loops between the O_R and O_L region is mediated by a CI octamer, which can repress the activity of the P_R promoter. When an additional CI tetramer is presented beside the octamer, the activity of the P_RM promoter will be repressed, too.

FIGURE 1.

λ-SWITCH system and the process of O_L participation in the SWITCH. (a) SWITCH is composed of O_R and O_L promoter region and cI, cro genes. O_R region consists of O_R1, O_R2, and O_R3. P_R completely overlaps O_R1 and partially overlaps O_R2, whereas P_RM completely overlaps O_R3 and partially overlaps O_R2. (b and c) A schematic picture indicating the transition between unlooping configuration and looping configuration.

In the past 50 years, extensive experimental data have been accumulated on the behavior of the SWITCH and its components (1–7). Correspondingly, many mathematical models have formulated (4,7–15). These theoretical studies help us to understand the λ-SWITCH. Meanwhile, quantitative inconsistencies between numerical simulations and experimental measurements exist. For example, Bakk's model states that the concentration of free CI₂ (effective part of CI protein) is <10 molecules per cell in the lysogenic condition. In other words, merely 10 dimers are available for controlling expressions of P_R, P_L, and P_RM (12). Considering the fluctuation of protein number in cells (16), such a small number of the effective protein certainly leads to an unstable lysogenic state. In contract, it is observed that the lysogenic state of λ-prophage can sustain more than 5000 years (17). There must be other mechanisms that are responsible for the stable lysogenic state (12).

One of the possible revisions of the models is the distal regulation by DNA looping (18). Another mechanism of the stable lysogenic steady state should be facilitated transfer mechanism (FTM) of transcription factors (TFs) to their operators. FTM had been proved to exist extensively (19–25) and recently received increasing theoretical studies (26–31). It includes several microscopic processes: sliding along DNA contour, hopping along the DNA cylinder, and intersegment transfer between different segments (when the DNA exists crossover) within one DNA polymer (19,32). These three processes play important roles in the process of TFs searching for their binding sites. The mechanism has been raised in light of two experimental results. First, LacI repressor can bind to its specific site at a rate of 10¹⁰ M⁻¹s⁻¹, which is much larger than the calculated diffusion-controlled limiting rate for a one-step protein-DNA association in three-dimensional space, 10⁷∼ 10⁸ M⁻¹s⁻¹ (19). Second, there are experimental evidences that more than 90% of RNA polymerase attach on the nonspecific DNA site instead of existing freely in cytoplasm (33). These evidences imply that nonspecific binding may make a qualitative contribution to the process of TFs finding their target sites.

In general, FTM can be described by a sequential two-step reaction as Eq. 1. In contrast, the classical TF-operator interaction model uses two independent reactions as Eq. 2. In this article, we will adopt Eq. 1 instead of Eq. 2:

(1)

(2)

where [TF] is the concentration of free transcription factor, [D] is the concentration of nonspecific binding DNA site, [O] is the concentration operator of the transcription factor, and [TF − D] and [TF − O] represent, respectively, the concentrations of nonspecifically and specifically bound TFs. Under equilibrium condition, is the equilibrium constant of TF binding to a nonspecific site on DNA, is the pseudoequilibrium constant for the second step reaction in Eq. 1, and is the equilibrium constant of free TF binding to its operator.

In fact, a complete reaction picture should integrate the two equations into a circular reaction loop (Eq. 3). The main difficulty of using the whole reaction loop is that more parameters are needed to fit from quantitative experimental data, which are rare. So we have to adopt a reduced one. Our model reduction (Eq. 1) is based on the following: on the energy profile of the reaction, for a TF the switching from the nonspecific to specific binding mode is quite smooth; no entropy costs at all (25), but the process of directly binding to the operator from the free mode needs much higher activation energy (34). As a consequence, in the reaction loop parameters k₃(k₋₃) is much smaller than k₂(k₋₂) and the reaction characterized by k₃(k₋₃) can be neglected in the steady state. Difference of the parameters implies that even the equilibrium isn't held for the reaction of Eq. 2; the thermodynamic model still approximately works in the whole reaction:

. (3)

Our working outline in this article is the following: first, we use experimental data from a simple system (3) to determine an unknown parameter, then apply it in a more complicated system (4) that contains more unknown parameters. These parameters are induced by FTM or CI octamerization. Finally, we use these newly determined parameters in the model to study the λ-SWITCH system and to investigate its stability. We also discuss the role of Cro protein and raise a hypothesis about its evolution.

MODEL AND PARAMETER FITTING

Experimental systems

To obtain the essential parameters that are related to FTM and CI octamerization, we sequentially take account of three related experimental systems on λ-SWITCH (see Fig. 2):

1. A system only includes OR promoter regions and CI repressor (3) (see Fig. 2 a). In this system, LacZ reporter is under control of the PRM promoter, and the CI repressor is expressed from a plasmid. With the change of CI repressor concentration, the activity of PRM can be quantitatively determined by measuring the activity of the reporter gene LacZ.

2. The system is almost the same as the previous system, except that OL promoter regions are added (4) (see Fig. 2 b). Thus the octamer of CI possibly exists in this system.

3. The system is the wild-type λ-SWITCH system as described in Fig. 1 (Fig. 2 c).

FIGURE 2.

Three quantitative experimental systems. (a) The system involves O_R promoter region, CI₂ protein, and a reporter gene LacZ under P_R promoter controlling. (b) The system adding an O_L promoter region to the system (a) to incorporate the effect of CI octamerization. (c) The wild-type λ-SWITCH control element, in which CI₂ and Cro₂ was controlled, respectively, by P_RM and P_R promoters.

Using the model discussed below, we can fit the one free parameter in system a. Then we use it in system b and fit the remaining free parameter . And last, we take the two fitted parameters into system c and investigate the steady state of lysogen of the λ-phage.

Definition of the parameter ΔG_{basal_quasi 2d}^CI₂

We take the FTM into account of our model. For two TFs (CI, Cro) bound to their operators in the λ-SWITCH system, a two-step reaction (Eqs. 4 a and 4 b) is formulated respectively instead of the two independent reactions (Eqs. 4 c and 4 d). The major difference between the two mechanisms lies in which part of CI₂/Cro₂ (called effective factor) directly responsible for the formation of [CI₂ − O]/[Cro₂ − O] complex. In the previous models, the effective factor is the free CI₂ dimer; whereas in our model it is the CI₂-DNA complex. For Eqs. 4 a and 4 b, the first step reaction takes place in cytoplasm, so that the equilibrium constants are the same both in vitro and in vivo. But their second-step reactions are mediated by redundant DNA, and the quasi-equilibrium constant cannot be measured in vitro. In the following, we will make an effort to introduce an indispensable parameter to describe this quasi-equilibrium constant:

(4 a)

(4 b)

(4 c)

(4 d)

Because FTM exists in the process of TFs binding to their specific sites in vivo, i.e., in the second step of Eqs. 4 a and 4 b, the association rates that take the TFs to their operators are limited by diffusion, whereas the dissociation rates depend on the affinities between them (35,36). As a result, when a TF binds to two different operators in the same cell, the difference in their equilibrium constants, which equal the association rate divided by the dissociation rate, just depends on the difference in their dissociation rates, which are determined by their affinities (35). We assume that the difference in the affinities of a TF binding to two different operators is the same in vitro and in vivo, so that if we get the equilibrium constant of a TF to one of operators in vivo, we can deduce the equilibrium constants of the TF to other operators based on the existing affinities measured in vitro. Here we select, respectively, the constant of CI₂ and Cro₂ to O_R1 as the unknown parameters and ; thus the equilibrium constants of CI₂ binding to other operators can be calculated using , where represents . The same formula holds for Cro₂. To be consistent with the measured data that are listed in Table 1, we translate the constants to free energy forms and . For CI, the unknown parameter is fitted from to experimental data in Dodd et al. (3). Then using the measured data in Dodd et al. (4), we can deduct all the parameters (shown in Table 1). Unfortunately, there is no quantitative experimental data for Cro₂. We have to use as a free parameter to discuss the behavior of the SWITCH system.

TABLE 1.

Parameter used in the model

Parameter	Value (kcal/mol)	Parameter	Value (kcal/mol)	Parameter	Value (kcal/mol)	Activity of promoter	Value (LacZ units)
	−10.4*		−6.3†		−0.6**		1056*
	−7.9*		−5.1†		−3*		2*
	−7.4*		−7.7†		−10.4**		45*
	−11*		−6.3†		−3 ∼ −8**		406**
	−9.3*		−5.1†		−11.1†		265*
	−9.6*		−7.7†		−8.7†		0.5*
	−3*		−1†		−3.6‡
	−3*		−0.6†		−6.5§		6.0 nM/min¶
	−3*		−0.9†				4.7n M/min¶
	−2.5*		−1†				0.01732/min¶
	−2.5*		−0.6†	a	**		0.15/min‖
	−2.5*		−0.9†	[DNA]	6.76 × 10−3(mol/L)§		0.0/min¶

^*Calculated from Dodd et al. (4).

^†Calculated from Darling et al. (7) with choosing a fixed parameter =-6.3 kcal/M.

^‡Values from Bakk and Metzler (12) and their citation.

^§Values from Aurell et al. (43).

^¶Values from Reinitz and Vaisnys (9).

^‖Value from Arkin et al. (45).

^**Value from this model.

Introduction of parameter ΔG_oct

Parameter represents the released energy when two CI tetramers form a CI octamer between O_L and O_R promoter regions by DNA looping. The parameter has not been measured yet. We will deduce it using another quantitative experiment of Dodd et al. (4). Furthermore, when two CI dimers exist beside the CI octamer, they can interact with each other, and another part of free energy, , will be released (4). However one single CI dimer binding at the O_R region and another single CI dimer binding at the O_L region cannot interact with each other or form the DNA looping (4).

The steady-state equation of λ-SWITCH phage

To formulate the thermodynamic model, we first analyze the possible microscopic configurations (also called states) for CI₂/Cro₂ binding to their operators in the three systems shown in Fig. 2. We calculate that system a has 8 states (see Table 2); system b has 73 = 64 + 9 states, including 9 looping states; and system c has 762 = 629 + 33 states, including 33 looping states. Note that the looping states represent the octamerized CI state existing between the O_R and O_L promoter regions; we do not exclude any possible looping state and corresponding unlooping state. For any sth state in anyone of the three systems, we employ Eq. 5 to represent its weight in the partition function:

(5)

where is the total binding affinity of the sth state, which sum over all protein-operator, protein-protein binding affinities that exist in the sth state; R is the universal gas constant; and T is the absolute temperature. Typically, and are the numbers of CI₂ and Cro₂ that bind to the regulation region in the sth state, respectively; [CI₂ − D] and [Cro₂ − D] are concentrations of the complex for CI₂ and Cro₂ binding to nonspecific DNA sites, respectively. These concentrations can be calculated using Eq. 6:

(6)

where [D] is the total E. coli chromosomal DNA concentration by basepair; and are the dimerizing affinities of Cro and CI, respectively; and and represent the nonspecific binding affinities of CI₂ and Cro₂ to DNA, respectively. All of the parameters are listed in Table 1.

TABLE 2.

States of system a in Fig. 2 and the free energy for each state

The corresponding partition function can be written as below, in which summation is over all possible states in the system:

(7)

The probability of the sth state is

(8)

Meanwhile, following Dodd et al. (4), we set and , respectively, to indicate the transcriptional activities of P_R and P_RM promoters in the sth state. There are four categories for P_RM (basal, stimulated no looping, stimulated with looping, and repressed) and two categories for P_R (basal and repressed) (Table 1). We adopt Dodd et al.'s empirical values, except that we reanalyze their data and properly change it in some cases. Thus we can obtain the activities (,) of P_R and P_RM promoters for a given system:

(8a)

In the previous models, the bistability of the λ-SWITCH (Fig. 2 c) is usually considered as equivalent to the coexisting λ-lysogenic and lytic states. In fact, the λ-SWITCH is just a part of the complex λ-regulation cascade, which is essentially responsible for the λ-lysogeny/lysis decision (17). We notice that when λ-phage exists in lysogeny, P_RM promoter is the only high active promoter in the whole λ-genome. Correspondingly, CI protein is continually expressed (1). Under this situation, the λ-SWITCH can be decoupled from the whole λ-phage network and completely take charge of the λ-phenotype (lysogeny). Thus the stability of lysogeny of host E. coli is determined by the stability of λ-SWITCH. We can use a set of ordinary differential equations (see Eq. 9) to describe its dynamical property as previous models (11,37):

(9)

The stability property of lysogeny is decided by the steady state of Eq. 9, which gives Eq. 10. The function and is added and equaled to zero to study the steady-state's properties. Furthermore, the kinetic process of the system is investigated by a stochastic simulation using Gillespie's algorithm (38) (the detail of simulation is described in the Appendix):

(10)

where a is the constant, which relates the activities of P_R and P_RM in Dodd et al.'s experiments (4) to the transcription rate in the wild-type λ-SWITCH. Its value is determined by the fact that, in the physiological lysogenic state, the CI's total concentration is and Cro's is close to zero. and represent the synthesis rate of CI and Cro, respectively; and represent the degraded rate of CI and Cro monomer, respectively. Here, we neglect the degradation of dimers because we take into account the effect of nonlinear degraded rate of proteins (39). is the dilution rate of and due to growth of E. coli; and represent, respectively, the total CI or Cro protein concentration; and and represent, respectively, the concentration of free CI or Cro monomer. All the parameters are listed in Table 1.

RESULTS AND DISCUSSION

We first fit the two parameters and using the quantitative experimental data of systems a and b in Fig. 2; the results are presented in Fig. 3. Using the quantitative data in experimental system a, we fit the parameter for CI₂ to be . Using this data, we obtain another parameter, , in experimental system b. The second parameter is slightly different with Dodd value −0.5k cal/mol (4). Note that in experimental system a, we adjust the empirical parameter of the P_RM activity from 360 to 406 LacZ units. Because the states that characterize the P_RM activity by never become absolutely dominant among all the possible states, the maximum value of their weight in the partition function is always <90%, thus we cannot directly take the highest experimental activity of P_RM as . Besides reconciling with the experimental data, these results resolve the puzzle about the fluctuation of the available CI dimer: the available CI dimer's number increases around ninefold by incorporating FTM, so that the amplitude of internal fluctuation is reduced.

FIGURE 3.

P_RM activity (LacZ units) versus the total CI concentration for system a (solid line) and system b (dashed line). The experimental data are kindly offered by Dodd et al. (3,4).

For the wild-type λ-phage, our model predicts that its lysogenic state is the only steady state when its host cell is RecA⁻. We adopt all the parameters determined in the two experimental systems (a, b) plus some new parameters (see Table 1). Since there are not quantitative data that can be used to fit the parameter, we vary it from −8 kcal/mol to −3 kcal/mol and investigate the steady state of the system using Eq. 10. The range is proper if we consider that its in vitro value should be −5.5 kcal/mol. The calculation results show that, no matter how we change the free parameter in this range, wild-type λ-SWITCH system only has a single stable steady state. The state is characterized by high CI concentration and very low Cro concentration see (Fig. 4, a–c). At the same time, because the SWITCH can be decoupled form the whole complex λ-regulation network and completely take charge of the physiological lysogenic phenotype of λ-phage, the single stable steady-state is lysogenic state of the prophage, i.e., the lysogenic phenotype should be absolutely monostable in RecA⁻ condition. The similar result has been deduced by Santillan and Mackey (15), but their model does not consider the FTM or nonspecific binding protein. Notice that here we interpret the RecA⁻ condition as in the model (see Table 1), because the degraded rate of CI can be neglected compared with its dilution rate in the RecA⁻ lysogenic host E. coli (15).

FIGURE 4.

With the variation of parameter , a–c, plot in the versus plane of curve (thick line) and curve (thin line), the cross point of the two curves gives the steady state of the system. (d–f) The activity of and promoter change as a function of CI or Cro total concentration. The thick solid line represents , the thick shaded line represents , and the thin solid line represents . In these subfigures, the value of is −6.3 kcal/mol in a and d; −3 kcal/mol in b and e, and −8 kcal/mol in c and f.

So far the experimental results about induction of lysogen are not contrary to the results. It is reported that the lysogen is extremely stable. The spontaneous induced rate from lysogen to lysis is even smaller than the mutation rate of λ-genome (5). Under this condition, it is believed that the majority of spontaneously induced lysogenic cells are not wild-type ones, but mutants that change in the cI gene or other regulating elements (6). Even without taking genetic mutations into account, such a tiny rate cannot be considered as a transition between two stable steady states of the λ-SWITCH element, since the kinetic fluctuations in λ-phage are enough to cause the lytic phenotype induction. Once the lytic phenotype is induced, the system cannot revert to its lysogenic phenotype any more, because the lysis of the E. coli cell will destroy the primary system (1). Furthermore, the mutant of can simultaneously exist in immunity and anti-immunity states. Immunity state is characterized by high CI857 concentration and low Cro concentration; whereas anti-immunity state is characterized by low CI857 concentration and high Cro concentration (40). The reason for the bistability is the higher degraded rate of CI. In our model, the bistability will emerge with the increase of the degraded rate of CI (Fig. 5). To demonstrate the results, we first analyze the stability properties of the steady state and then implement the stochastic simulation. The results are compatible with each other (Fig. 5). With the change of control parameter, forms 0.0/min to 0.35/min, the SWITCH acquires and then loses the bistable property via twice saddle-node bifurcations. It is worth noting that the critical value of the control parameter in which the bistable state emerges or disappears cannot be used to give any prediction about the degradation rate of the CI monomer. As when the simulations are implemented, the free parameter is fixed to −7.5 kcal/mol.

FIGURE 5.

With the change of the control parameter , the stability of λ-SWITCH is changed. In a, d, and g, ; in b, e, and h, ; and in c, f, and i, . Panels a–c represent the solution line of Eq. 10 in the [CI_T] and [Cro_T] phase space. Panels d–f demonstrate the corresponding projections. Panels g–i indicate the corresponding stochastic simulations of the CI and Cro protein number per cell, in which the solid and shaded lines, respectively, represent the trajectories of CI and Cro protein numbers evolving. Each simulation implements 2 × 10⁶ steps.

The model also indicates that the Cro protein is a weak repressor in the λ-SWITCH compared to the CI repressor. To investigate the role of Cro protein, we use Eq. 8 to investigate the activity of the P_R and P_RM promoter as a function of Cro concentration, and the activity of the P_R promoter as a function of CI concentration. From Fig. 4, d–f, it is obvious that the decrease of these promoters' activity by CI is much sharper than by Cro. In this study, the parameter is changed from −8 kcal/mol to −3 kcal/mol and this variation doesn't qualitatively affect the difference (see Fig. 4, d–f).

This result is consistent with the experiments. Several experiments indicate that Cro₂ is a weaker repressor for the P_R, P_L, and P_RM promoters compared to CI₂ (41,42). If we give up the two-step reaction constraint and just consider the binding energy of free CI₂/Cro₂ to their operators, we cannot obtain this result, because binding energy for CI₂ to its best operator is 12.5 kcal/mol, whereas it is 13.4 kcal/mol for Cro₂. As a consequence, Cro₂ should be a more effective repressor than CI₂ if the concentration of free Cro₂ and CI₂ is same. Even though two CI₂ dimers show slightly stronger cooperation, according to the previous theories (10–15,43) the repression efficiency of Cro₂ cannot be negligible compared to CI₂. One may argue that the dimerization ability of Cro is weaker than CI, causing a weaker role of Cro₂. But, in fact, λ-Cro is the only protein that has strong dimerization affinity in the Cro family of lambdoid phage. Its dimerizing affinity is 1000-fold of other Cros (44). So we cannot simply attribute the weak role of λ-Cro to the weaker dimerization.

In light of this model, we can raise a hypothesis about the physiological drive of the λ-Cro's secondary structure switching in the evolving process. Cordes et al. said that λ-Cro separated from other lambdoid CI/Cro protein family via an - to -secondary structure switching event during evolution history and obtained a stronger dimerization ability (37). But one puzzle remains: if the role of Cro is just a weak repressor, and the weak dimerizing affinity is enough, why does λ-Cro evolve to obtain strong dimerization ability and high nonspecific binding affinity? The answer may be that it provides an additional level of gene regulation, which increases the λ-phage's adaptation (44). It is possible that such auxiliary regulation is achieved by FTM. According to Eqs. 5 and 6, the local concentration of DNA around the operators of Cro₂ participate in the regulation, and are responsible for the repression ability of Cro₂. A difference in the local DNA concentration will result in a difference in repression ability of Cro. In nature, at least two situations can make the difference in the local DNA concentration: when λ-DNA freshly injects into E. coli cell or when the λ-DNA has been integrated into E. coli chromosome. This difference causes Cro playing a different role in the infection process and in the induction process. If the local concentration of DNA is higher in the integrated condition, Cro will play a more important role in the induction process than in the infection process, and vice versa.

In summary, we have presented what we believe is a new quantitative model of the λ-SWITCH, which has incorporated the facilitated transfer mechanism via a two-step reaction. Besides reconciling with experimental data, it can easily explain the stability of lysogen and the weaker role of Cro. Nonetheless the model is a rough one, which uses some empirical results and some indispensable parameters. We believe it is helpful to understand the λ-SWITCH system and other regulation systems.

APPENDIX: STOCHASTIC SIMULATION OF λ-SWITCH

To incorporate transcription and translation noise, we separate Eq. 9 into transcription step and translation step. The corresponding reactions that happen in a cell are shown in Eqs. A1 and A2. The reactions in Eq. A1 account for, respectively, transcription of cI/cro mRNA, translation of CI/Cro protein, degradation of cI/cro mRNA, degradation of CI/Cro monomer, and dilution of total CI/Cro protein due to the host E. coli cell growth. Equation A2 is the same as Eq. 3 in the main text. They are considered as very fast compared with Eq. A1 and easily reach equilibrium. Our simulation is performed with these two sets of coupled stochastic reactions using the Monte Carlo algorithm described by Gillespie (38). In here, O_PRM and O_PR, respectively, represent the PRM and PR promoters. mRNA_cI and mRNA_cro, respectively, represent the mRNA transcript of cI and cro. The parentheses represent degradation. All the parameters are converted from Table 1 and shown in Table 3.

(A1)

(A2)

TABLE 3.

Parameters for stochastic simulation

*

*

†

†

OPRM(OPR) = 2.5 molecules/cell‡

^*L_PRM and L_PR are defined in Eq. 8.

^†Converted from S_CI and S_Cro, respectively.

^‡Average E. coli chromosome number per cell and from Santillan and Mackey (15).