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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 3, 1151-1156, 1 August 2008

doi:10.1529/biophysj.108.129825

Biophysical Theory and Modeling

Are DNA Transcription Factor Proteins Maxwellian Demons?

Longhua Hu^*, Alexander Y. Grosberg^*, ,   and Robijn Bruinsma^†

^* Department of Physics, University of Minnesota, Minneapolis, Minnesota
^† Department of Physics and Astronomy, University of California, Los Angeles, California

Address reprint requests to Alexander Y. Grosberg, Tel.: 612-624-7542.

The ability of bacteria to respond within minutes to changes in their environment relies on genetic switches that are controlled by transcription factors (TFs). TFs are proteins that—after activation by an environmental change—are able to locate a specific region (the operator sequence) along the bacterial genome and bind to it, thereby regulating the expression of a gene (or group of genes) adjacent to that region 1,2. The number of copies of a TF protein associated with a specific gene varies, but typically it is in the range of 10², corresponding to a concentration in the range of 0.1μM. Because bacterial genomes have a size in the range of 10⁷ sites, a TF must be able to scan the DNA for the target site at a rate of 10⁵ sites per second or faster to at least one of them to reach the target site within seconds. Note that following the search for the target site, the TF still has to bind to the target site to regulate the expression of the gene.

A series of classical articles on the search process 2,3,4 culminated in the work of Berg et al. 5,6 and von Hippel and Berg 7 who showed—for the canonical case of the lac repressor protein of the bacterium E. coli—that the search process takes place not by straightforward three-dimensional diffusion to the target binding site but rather by a slide-jump combination of one-dimensional diffusional sliding along the DNA chain alternating with three-dimensional diffusional jumps between different DNA segments. By restricting part of the search to the one-dimensional target space, the binding rate is effectively enhanced with respect to a pure three-dimensional search, while the three-dimensional jumps reduce the repetitive visits to the same sites that characterize purely one-dimensional diffusive searches. This scenario is made possible by a modest, nonspecific electrostatic affinity between the TF and duplex DNA. More recently, one-dimensional diffusional sliding on DNA was directly observed in single molecule experiments 8,9,10. Berg et al. 5,6 and von Hippel and Berg 7 also provided evidence that, under physiological conditions, the search time has a minimum with respect to the strength of this nonspecific affinity, which may be the result of evolutionary optimization under selective pressure. Subsequent structural studies 11 have shown that the DNA-binding domains of the lac repressor are subject to strong conformational fluctuations when the protein is in contact with nonoperator DNA. If the binding domain is in contact with operator sequence DNA, then the protein can undergo a large-scale conformational change to a stable structure with direct contacts between the amino-acid side chains and the DNA bases.

It would seem obvious that the delay time between activation and binding of a TF to the operator sequence (i.e., binding time) is minimized by maximizing the one-dimensional diffusion constant D₁. However, simply increasing the transport rate will impair the accuracy, or fidelity, with which the protein can distinguish a right from a wrong site. Specifically, if the binding of a TF to the target site is characterized by a certain rate Ω, then the protein is likely to overshoot the target site if the jump rate D₁/a² between sites, with a the spacing between protein binding sites, is large compared to Ω. Similar conflicts between process speed and process fidelity are familiar from DNA duplication and transcription where increased reaction rates increase the number of duplication and transcription errors.

The search mechanism is known to involve a combination of three-dimensional diffusion through the bulk of the cell and one-dimensional sliding diffusion along the DNA. It is believed that the surprisingly high target binding rates of TF proteins relies on conformational fluctuations of the protein between a mobile state that is insensitive to the DNA sequence and an immobile state that is sequence-sensitive. Since TFs are not able to consume free energy during their search to obtain DNA sequence information, the Second Law of Thermodynamics must impose a strict limit on the efficiency of passive search mechanisms.

Slutsky and Mirny 12 proposed that conformational fluctuations of the protein could ease the conflict between speed and fidelity. If some conformations of the TF are sensitive to the DNA sequence while others are characterized by rapid transport, then the TFs might be able to scan the genome efficiently by appropriately flipping between the two types of conformations. The mechanism proposed by Slutsky and Mirny would be easy to envision for an active searcher, which spends free energy to gather information from the underlying DNA sequence and uses it to decide when it has to switch from the sliding mode to the recognition mode. However, TF proteins do not hydrolyze ATP or consume other forms of free energy during their search. It thus would seem that the Slutsky and Mirny mechanism requires TF proteins to act as Maxwellian Demons, able to gather information without expending free energy, but this is not permitted by the Second Law of Thermodynamics. The Second Law of Thermodynamics is, therefore, expected to impose a rigorous limit on the search efficiency of passive searchers. The aim of this article is to analyze how close this mechanism can approach limits of search efficiency imposed by fundamental principles of thermodynamics. We will address this question by examining a simple model for the conformational fluctuations, similar to that of Slutsky and Mirny 12, where the TF is allowed to adopt only two conformations (+ and −) when in contact with nonoperator DNA. Since the binding of TF to DNA involves a significant deformation of the double helix, the + and −states should be interpreted as states of a joint protein-DNA complex. For brevity, we will continue to refer to “+” and “−” as states of the protein. As illustrated in Fig. 1, in the +state, the protein is less ordered and only loosely associated with the DNA while it can slide along the DNA chain. In the −state, the protein is more ordered, closely associated with the DNA and immobile. A specific realization of a conformational fluctuation spectrum of this type was recently discussed for the Ets-DNA system 13. If the TF is in contact with the target operator sequence then, in addition to these two states, it also can undergo an irreversible conformational transition from the −state to the fully ordered final bound state. We will show that the shortest possible binding time in this model is controlled by a dimensionless binding rate , with D₃ the protein diffusion coefficient in bulk solution, D₁ the diffusion coefficient for one-dimensional transport along the DNA in the +state, K the equilibrium constant for the nonspecific protein-DNA interaction, and b the DNA-protein capture radius (more precisely, b is defined as the radius of a cylinder surrounding the DNA duplex such that a protein will be captured by the DNA, for example by electrostatic attraction, if the center of the protein is located inside the cylinder). If the dimensionless binding rate is comparable to one—or larger than one—then we can show that, for a particular value of the energy difference ΔE_± between the + and −conformations, the binding time can approach an absolute lower bound that corresponds to proteins having infinitely fast final binding rates. In other words, if the internal degrees of freedom of the protein in the sliding state are properly matched to the final binding rate, then the binding time of a TF can approach the shortest possible value allowed by thermodynamics provided the dimensionless binding rate is sufficiently large.

Display large version of this figure

Figure 1

Schematic representation of the model. A protein moving diffusively through the cell volume (a) is adsorbed on genomic DNA (b) where it adopts one of two conformations: + and −. In the +conformation it is loosely associated with the DNA and can move by one-dimensional diffusion along the DNA chain (b) while in the −conformation (c) it is tightly associated with the DNA and is immobile. After returning to the +state, it restarts the sliding motion. The protein also can desorb from the chain (d) and return to three-dimensional diffusive motion. After a number of such cycles, the protein lands in the antenna region within a distance λ of the target binding site (e). After reaching the target site by one-dimensional diffusion it can undergo a large-scale irreversible conformational transition to the final bound state if it is in the −state (f).

To demonstrate these claims, assume a cell of volume V containing a DNA genome of length L. The cell also contains a certain (low) concentration c of TF proteins that can bind reversibly and nonspecifically to the DNA. A protein whose center is located inside a cylindrical tube of radius b surrounding the duplex DNA will be assumed to be nonspecifically associated with the DNA. The fraction ϕ of the total cell volume occupied by the tube is of the order of Lb²/V. There is also a single target site on the strand where the TF can bind irreversibly. We start by applying a fundamental theorem 14, which—in terms of our model—states that the mean waiting time for irreversible occupation of the target site, the quantity of interest to us, is equal to the inverse of a steady-state diffusion current of a different problem, namely one where the target site is replaced by a protein sink that constantly absorbs −state TF located at the target site at a rate Ω, while the protein concentration far from the target site is maintained at a certain fixed value c₃(∞). The steady-state diffusion current, denoted by J_3D, into the target site for this second problem can be obtained from straightforward solution of the diffusion equation, which leads to the well-known Smoluchowski relation for the reaction rate of diffusion-limited chemical reactions:

(1)

Our assumption that the antenna is straight is justified by the fact that the antenna length λ will turn out to be shorter than the DNA persistence length, which is close to 50nm. If the antenna length would have exceeded the persistence length, then DNA conformation fluctuations would play an important role, and intersegmental transfer and Levi-flight transport would have to be considered. For a discussion of this more general case, see the literature 15,16,17.

Assuming the antenna is straight, its length has to be determined self-consistently but first we must establish a relation between c₃(∞) and the actual protein concentration c.

Far outside the target sphere the DNA-protein system is, by assumption, nearly in local thermal equilibrium, so one can determine the concentrations of adsorbed and free proteins purely from equilibrium considerations. If one views the association of the TF with DNA as a simple chemical reaction, then the concentration of proteins adsorbed nonspecifically on the DNA and the concentration c₃(∞) of free proteins must be related to the reaction volume fraction ϕ by the Law of Mass Action for dilute chemical systems in thermodynamic equilibrium,

(2)

Deep inside the target sphere, the system is not in thermal equilibrium, with the adsorption rate of proteins from the bulk solution to the DNA exceeding the evaporation rate. The difference is matched by a one-dimensional diffusion current J_1D along the DNA chain toward the target site. To estimate this one-dimensional diffusional transport, note that if the interconversion rate between the + and −states is sufficiently rapid, then their respective occupancies can be approximated by the equilibrium Boltzmann distribution. The effective one-dimensional diffusion constant for transport along the chain—which we will denote by —is then proportional to the Boltzmann probability p(+) to find the protein in the +state. If , then p(+)=μ/(1+μ) and . Similarly, the effective target-site binding rate is, under these same conditions, proportional to the probability p(−)=1−p(+) to find the protein in the −state and .

Let c₁(0) be the one-dimensional concentration at the target site. If the final binding rate were infinitely fast, then c₁(0) would be zero but, because of the overshoot effect, this is no longer the case. If we view the surface of the target sphere as a matching region between the asymptotic regions far from the sink where the one-dimensional concentration approaches c₁(∞) and the region deep inside the target sphere near the sink where the one-dimensional concentration approaches c₁(0), then we can estimate the one-dimensional concentration gradient as . It follows that the one-dimensional diffusion current toward the sink equals

(3)

(4)

(5)

We have been assuming that the energy difference between + and −states, or μ, does not depend on the DNA sequence. The effects of sequence-dependent, one-dimensional transport are discussed in the literature 18,19.

Equating the one-dimensional and three-dimensional currents provides us with a self-consistency condition that determines both the size of the antenna length λ and the reaction rate. Solving for λ using Eqs. (1) and using gives the antenna length:

(6)

It will be helpful to express the binding rate in dimensionless units as A≡J_3D/(4πD₃ca), with 4πD₃ca the Smoluchowski limiting rate of a conventional three-dimensional diffusive search for an absorber target of radius a (the spacing between binding sites), so A can be viewed as a reaction amplification or enhancement factor. This enhancement factor can be expressed as a simple function of the dimensionless binding rate and the Boltzmann factor

(7)

Display large version of this figure

Figure 2

Contour plots of the transport enhancement factor A as a function of the equilibrium constant K and the occupation probability ratio μ of the + over −states for D₁=10⁻⁹cm²/s, D₃=3×10⁻⁷cm²/s, a=0.34nm, b=5nm, ϕ=0.01, and Ω=3×10³Hz. There is a shallow maximum at ∼μ=0.1 and K=10⁻³. The ratio of the transport enhancement factor at this maximum, A_opt, and the thermodynamic limiting enhancement factor, A_∞, equals 0.193. (Upper inset) Dependence of the ratio A_opt/A_∞ on the dimensionless binding rate ω. (Lower inset) Enhancement factor A against μ at the value of K corresponding to the maximum of the main figure.

If μ adopts the optimal value then the ratio A_opt/A_∞ of the optimal rate amplification factor and its maximum value is a function only of the dimensionless rate ω:

(8)

The dependence of A_opt/A_∞ on ω is shown in the upper inset of Fig. 2: A_opt is of the same order of magnitude as the theoretical limit A_∞ already for modest values of ω. This demonstrates our central claim: it is possible for the overall binding rate of a TF to approach the theoretical limiting value but only by a suitable choice of μ, and only if the dimensionless binding rate ω is of ∼1, or >1.

Are these two conditions realistic for typical TF? Typical values for the diffusion constants of bacterial TF are, according to the literature 8,9,10, D₁≈10⁻⁹cm²/s and D₃≈3×10⁻⁷cm²/s. We can estimate the protein-DNA reaction volume fraction ϕ for E. coli by assuming it to be comparable to the DNA volume fraction (∼1% within cell volume of the order of 1 cubic micrometer). The equilibrium constant can then be determined from the relation and the fact that it is known that ∼10% of the lac repressor proteins of E. coli are in solution 20, which means that K must be ∼10⁻³. If we assume a to be equal to the basepair spacing 0.34nm, and estimate b as 5nm, then the dimensionless binding rate ω is ∼10⁻⁴Ω with the binding rate Ω expressed in Hz. A large-scale protein conformational change typically involves millisecond-to-microsecond timescales, from which it follows that ω must lie in the range of 0.1–100. Note, from Fig. 2 that the optimal value for K is close to 10⁻³ for Ω in the kHz range. We conclude that the second condition can be satisfied under typical conditions. Next, the optimal occupation ratio is in the range of 0.1–10 for ω in the range of 0.1–100. The corresponding optimal energy difference ΔE_± between the + and −states is then in the range of a few k_BT, with ΔE_± positive for ω<4 but negative for ω>4. In either case, the structure of optimized TF bound to nonoperator DNA should be subject to strong thermal fluctuations. As we saw, this is indeed the case of the lac repressor 11, while a recent modeling study of the Ets-DNA system arrives at the same conclusion 13. The first condition can thus be satisfied as well under reasonable conditions. Finally, the measured lac repressor binding rates 5,6,7 are comparable to the limiting rate imposed by thermodynamics. We conclude that, under reasonable conditions, the binding rate of TF proteins can be of the same order of magnitude as the thermodynamically imposed limiting rate if the energy spectrum of conformational fluctuations is determined, under selective pressure, by minimization of the overall binding time.

APPENDIX:

Optimal searches on coiled genomes

The above results were derived assuming that the antenna was straight. The role of coiled DNA conformation in the slide-jump search process was addressed systematically, for a variety of DNA conformations, in Berg et al. 6. The goal of this Appendix is to see how the optimization with respect to μ is implemented for coiled DNA conformations and, simultaneously, to demonstrate that the assumption of a straight antenna is justified in most physiological conditions.

The starting point is to consider the relation between antenna size, ξ, and antenna length along DNA, λ. For a straight antenna, ξ=λ, while for a Gaussian coiled antenna, where p is effective Kuhn segment of DNA, usually p≈100nm. In the general case, we can write with arbitrary power ν, 0<ν<1, and with suitably defined characteristic length . We note in passing that is not necessarily equal to the Kuhn segment p for instance, for the wormlike chain with excluded volume ν≈3/5 and where d is the chain thickness.

In the arbitrary ν-case, it turns out that there is still a single dimensionless criteria, which generalizes ω, and which can be written in the form Using as before the equations J_3D=J_1D=J_s4, but with a more general relation between ξ and λ, one can obtain the following equation for the normalized enhancement factor α=A/A_∞:

(9)

In the old case of a straight antenna, ν=1, this equation is quadratic and its solution is Eq. (7). In the general case this equation does not allow explicit solution, but we can still address optimization with respect to μ. It can be shown in a few lines of algebra that the optimized value of α=α_opt≡A_opt/A_∞ satisfies the equation

(10)

which reproduces Eq. (8) if ν=1, and which in general yields the following asymptotics:

(11)

Qualitatively, this is essentially the same behavior as illustrated in the upper inset of Fig. 2.

In the light of these results, we can now provide a more specific justification for the assumption of straight antenna which we used in the main text. For instance, for the in vitro situation of DNA in solution, DNA is straight (ν=1) at length scales up to about Kuhn segment p≈100nm and becomes a Gaussian coil (ν=1/2) at larger length scales:

(12)

Therefore, once we analyzed the straight antenna (ν=1) case and optimized the enhancement factor with respect to μ, we should check that the optimal antenna length is shorter than p. Omitting algebra, this yields the condition Similarly, once we have optimized the enhancement for ν=1/2, we should check that the optimal antenna length is longer than p. This condition upon some algebra leads to exactly the opposite inequality Thus, there is a smooth crossover between these two regimes. Given that the left-hand side of these two inequalities is a monotonically growing function of ω which approaches unity at very large ω, we arrive at the conclusion that the optimal antenna might become a Gaussian coil if and only if . It is not difficult to see the physical meaning of this inequality: the average number of desorption events while diffusing one Kuhn length should be less than unity. Taking the typical numbers (p=100nm, b=5nm, D₃=3×10⁻⁷cm²/s, D₁=10⁻⁹cm²/s), we see that which means the optimal antenna can become Gaussian only when . Since values of K below this threshold are unlikely, the optimal antenna—no matter how big is ω—is typically shorter than the Kuhn segment, thus justifying our assumption in the main text that the antenna is more or less straight.