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How Proteins Search for Their Specific Sites on DNA: The Role of DNA Conformation

Tao Hu ^*, A. Yu. Grosberg ^* and B. I. Shklovskii ^* 

^* Department of Physics, and William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota

Correspondence: Address reprint requests to A. Yu. Grosberg, E-mail: grosberg{at}physics.umn.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

 
It is known since the early days of molecular biology that proteins locate their specific targets on DNA up to two orders-of-magnitude faster than the Smoluchowski three-dimensional diffusion rate. An accepted explanation of this fact is that proteins are nonspecifically adsorbed on DNA, and sliding along DNA provides for the faster one-dimensional search. Surprisingly, the role of DNA conformation was never considered in this context. In this article, we explicitly address the relative role of three-dimensional diffusion and one-dimensional sliding along coiled or globular DNA and the possibility of correlated readsorption of desorbed proteins. We have identified a wealth of new different scaling regimes. We also found the maximal possible acceleration of the reaction due to sliding. We found that the maximum on the rate-versus-ionic strength curve is asymmetric, and that sliding can lead not only to acceleration, but also in some regimes to dramatic deceleration of the reaction.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

 
The problem
Imagine that while you are reading these lines a -phage injects its DNA into Escherichia coli bacteria. For the infected cell, this sets a race against time: its hope to survive depends entirely on the ability of the proper restriction enzyme to find and recognize the specific site on viral DNA and then cut it, thus rendering viral DNA inoperable and harmless. If restriction enzyme takes too long to locate its target, then the cell is dead.

Restriction-modification system in bacteria, based on the endonuclease and methyltransferase enzymes, defends the prokaryotic cells against phage infection while also protecting the cell's own genome (1). This fact has already been recognized (2) as just one example of the molecular-biological system whose function hinges on the ability of certain proteins to locate their respective specific target sites on the DNA, and do that quickly. Another system on which fast protein-DNA recognition was studied quantitatively was lac-repressor, again in E. coli. This protein was found (3) to be able to locate its specific target on DNA up to two orders-of-magnitude faster than the so-called Smoluchowski limit, which corresponds to the search based solely on diffusion in three dimensions (4). This paradoxical result is sometimes referred to as faster-than-diffusion search.

As a matter of fact, the insufficiency of three-dimensional diffusion as a search mechanism in molecular biology was recognized even before any experiments, on purely theoretical grounds, by Delbrück (5), who also suggested resolving this problem by reducing the search dimension, i.e., by nonspecific adsorption on a one-dimensional macromolecule or two-dimensional membrane and then diffusing in this smaller space. Interestingly, as pointed out in the work (6), the idea that reduced dimension speeds up chemical reaction can be traced even further back to Langmuir, who noticed that adsorption of reagents on a two-dimensional surface can facilitate their diffusive finding each other (the ideas of Langmuir are nicely presented in (7)).

Capitalizing on the ideas of Delbrück, the authors Richter and Eigen (8) explained the lac-repressor experiment (3) by saying that three-dimensional diffusion has to bring protein into an elongated space region extended along DNA quite far from the specific target itself, where protein can nonspecifically adsorb on DNA and then slide along the DNA.

The field kept attracting intensive attention for many years. A nice recent review of various strategies employed to address the problem experimentally can be found in the work by Halford and Szczelkun (9). Based on the summary of experimental evidence, authors of this review conclude, that the process is not just a naive one-dimensional sliding, but rather a delicately weighted mixture of one-dimensional sliding over some distances and three-dimensional diffusion. This conclusion was further reiterated in the even more recent experimental work by Gowers et al. (10). A theorist also could have guessed the presence of a crossover between one-dimensional sliding and three-dimensional diffusion, because sliding along coiled DNA becomes very inefficient at large scale: having moved by t^1/2 along DNA after one-dimensional diffusion over some time t, protein moves in space by only t^1/4 if DNA is a Gaussian coil. This is very slow subdiffusion. This is the situation requiring theoretical attention to how three-dimensional and one-dimensional diffusion can be combined, and how their combination should be manifested in experiments.

On the theoretical front, a major contribution to the field is due to Berg et al. (11). As an outcome of their theory, these authors formulated the following nice prediction, partially confirmed by their later in vitro experiments (12): the rate at which proteins find their specific target site on DNA depends in a nonmonotonic fashion on the ionic strength of the solution. In this context, ionic strength is believed to tune the strength of nonspecific adsorption of proteins on DNA, presumably because a protein adsorbs to DNA via positively charged patch on its surface. Thus, in essence one should speak of the nonmonotonous dependence of the rate on the energy of nonspecific adsorption of proteins on the DNA.

Although qualitatively consistent with experiment, the theory of Berg et al. (11) leaves several questions open. First and foremost, how does the search time of proteins finding their target, or the corresponding rate, depend upon the DNA conformation? In particular, is it important that the DNA is coiled at a length scale larger than the persistence length? Is it important that the DNA coil may not fit in the volume available, and then DNA must be a globule, like the nucleoid in a prokaryotic cell in vivo or under experimental conditions in vitro (13)? Second, a closely related aspect is that the theory of Berg et al. (11) does not answer the experimentally most relevant question (9) of the interplay between one-dimensional sliding and three-dimensional diffusion. In particular, one of the questions raised by experiments and not answered by the theory of Berg et al. (11) regards the correlations between the place where a protein departs from the DNA and the place where it readsorbs. The third aspect of the theory of Berg et al., although of a lesser importance and more dependent upon taste, is that it does not yield a simple intuitive explanation for nonmonotonic dependence of the rate on the strength of nonspecific adsorption. One may want to know whether a simple qualitative description of the rate exists, at least within some limits.

More recent refinement of the theory is given in Coppey et al. (14). The authors of this work follow Berg et al. in that they treat DNA in terms of domains—a concept having no unambiguous definition in the physics of DNA. Also, Coppey et al. (14) makes it very explicit that Berg et al. (11) and subsequent theories neglect correlations between the place where protein desorbs from DNA and the place where it adsorbs again—the approximation that clearly defies the polymeric nature and fractal properties of DNA. At the same time, this approximation leaves unanswered the experimentally motivated question of the interplay between one-dimensional and three-dimensional components of the search process.

In recent years, the problem was revisited by physicists several times (15–17), but the disturbing fact was that all of them attributed quite different results and statements to the findings of Berg et al. Bruinsma (15) says that according to Berg et al. (11), the search timescales as DNA lengths L rather than L², as in one-dimensional diffusion along DNA; Halford and Marko (16) state that proteins slide along DNA some distance—a distance that is independent of DNA conformation, regardless even of the DNA fractal properties. Slutsky and Mirny (17), however, concentrate on the role of the nonuniform DNA sequence, claiming that the time for three-dimensional diffusion must be approximately the same as the time for one-dimensional diffusion along DNA. A further, possibly even more disturbing fact is that none of these articles (11,14,15,17) make any clearly articulated explicit assumption about DNA conformation. Is it straight, or Gaussian coil with proper persistence length, or something else? Does the result depend on the DNA conformation? Interestingly, experimenters do discuss in their works (see (9) and references therein) the issue of correlated versus uncorrelated readsorption; these discussions call for theoretical attention and theoretical description in terms of correlations in fractal DNA, but so far, no proper theory has been suggested.

Motivated by these considerations, in this work we set out to reexamine the problem from the very beginning.

Model, approach, and limitations
We assume that within some volume , one (double-helical) DNA polymer is confined, with contour length L, persistence length p, and a target site of the size b.

Although in our theory we use a model of a single DNA molecule confined in the volume , all our results apply, without any modification, to a macroscopic solution of DNA, with concentration 1/ (in units of DNA chains per unit volume). For this solution, we assume that every DNA has to have contour length L and each DNA has to have one target site.

On a length scale smaller than that of the persistence length p, DNA is practically straight. In particular, if DNA contour length is shorter than p, i.e., L < p, DNA as a whole is rodlike. Long DNA, with L > p, coils up on the length scales exceeding p. We assume that the DNA coil is Gaussian, with overall size proportional to R (Lp)^1/2 (as opposed to the swollen coil described by the Flory index 3/5 (18)). That means that we neglect the excluded volume effect. For DNA, this is a reasonable approximation for most realistic cases, such as, e.g., -DNA, because of the large persistence length/diameter ratio of the double helix: excluded volume in the coil remains unimportant (19) up to DNA length L < p³/b² (up to 100,000 basepairs under normal nonexotic ionic conditions).

We do allow for the possibility that the length of DNA L is so large that DNA Gaussian coil does not fit into the confinement volume , R³ > . In this case, DNA has to reflect many times back into the volume after touching the wall. We refer to this situation as DNA being a globule. We assume, however, that the volume fraction of DNA inside volume , which is Lb²/, is sufficiently small even when DNA is a globule. In particular, we assume Lb²/ < b/p, because in a denser system liquid crystalline nematic ordering of DNA segments becomes likely (19).

In terms of DNA solution, rodlike DNA (with L < p) and DNA coils are realized for the dilute solution, whereas a situation similar to that of the globule is realized for the semidilute solution (18) of DNA.

We further assume that protein can be nonspecifically adsorbed at any site on the DNA, and that nonspecific adsorption energy , or the corresponding constant is the same everywhere on the DNA and does not depend on the DNA sequence. We also assume that every protein molecule has just one site capable of adsorbing to the DNA. (Although there are proteins with two such sites able to adsorb to two separate pieces of DNA at the same time and thus serve as a cross-linker for the DNA itself, we do not consider this possibility.)

We assume that nonspecifically bound protein can diffuse (slide) along DNA with the diffusion coefficient D₁, while protein dissolved in surrounding water diffuses in three dimensions with diffusion constant D₃. Thus, we have a unitless parameter related to the diffusion coefficients, d = D₁/D₃. In the simpler version of the theory (which we shall consider first), we assume D₁ = D₃, or d = 1. For simplicity, we assume that while protein is diffusing, either in three dimensions or along the DNA, the DNA itself remains immobile.

The quantity of our interest is the time needed for the target site to be found by a protein (consider, e.g., a restriction enzyme attacking a viral DNA intruder). One should imagine a certain concentration c of proteins randomly introduced into the system, and ask about the time needed for the first of these proteins to arrive at the target site. In this article, we will only address the mean time, averaged over both thermal noise and DNA conformation. For this averaged quantity, since the DNA is assumed immobile, the problem can be addressed in a simple way, by looking at the stationary rate. Namely, we should consider that there is a sink of proteins in the place of the specific target site, and that it consumes proteins with the rate J proportional to concentration c, which should be supported on a constant level by an influx to maintain stationarity. Obviously then, the averaged time is just 1/J. At the end of the article, we show how to rederive all our results in terms of a single protein, thus avoiding an artificial assumption that there is a sink of proteins at the place of the target.

In this article, we calculate the rate J by assuming that concentration c is an arbitrary constant. To compare the predicted rate to the Smoluchowski rate J_s = 4 D₃cb (see Appendix A for a simple derivation of Smoluchowski formula), we shall mainly look at the ratio

(1)

which characterizes the acceleration of the reaction rate achieved due to the sliding along the DNA.

We will be mainly interested in the scaling dependence of the rate J or acceleration J/J_s on major system parameters, such as y, L, and . In this context, we will use the symbol to mean equal up to a numerical coefficient of order 1, and the symbols > and < are to mean >> and <<, respectively.

Along with dropping out all numerical coefficients in our scaling estimates, we also make several assumptions driven by a pure desire to simplify formulas and to clarify major physical ideas. We assume that all the microscopic length scales are of the same order, namely, approximately target size b: protein diameter, double-helical DNA diameter, and the distance from DNA at which nonspecific adsorption takes place. These assumptions are easy to relax, but in this article we shall touch neither of these issues, guided by the prejudice that simple questions should be addressed first.

The plan of the article is as follows. First, we consider the relatively simple cases when DNA is a Gaussian coil and one-dimensional sliding of proteins along DNA involves only a small part of DNA length. Already in this situation we will be able to explain the effect of correlated readsorption and arrive at a number of new results, such as, for instance, possible asymmetric character of the maximum on the curve of the rate as a function of adsorption strength. These results are also derived through the electrostatic analogy in the Appendix B. Second, we present a summary of all possible scaling regimes and then discuss them in more detail. We start this by looking at the rate saturation when one-dimensional sliding involves entire DNA length. Third, we consider a delicate case when DNA as a whole is a globule; in this case, we found that even the three-dimensional transport of proteins is in many cases realized through the sliding of adsorbed proteins along DNA and using DNA as a network of one-dimensional transport ways. We continue by looking at the situations when diffusion coefficient of the proteins along DNA is either smaller or larger than their diffusion coefficient in the surrounding bulk water. Fourth, we rederive all our major results using the language of single protein search time instead of a stationary process and flux. Finally, we conclude with comparison of our results to those of earlier works and the discussion of possible further implications of our work.

	SIMPLE CASE: STRAIGHT ANTENNA VERSUS GAUSSIAN COIL ANTENNA

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

 
The reason why nonspecific adsorption on DNA can speed up the finding of the target is illustrated in Fig. 1, a and b: it is because DNA forms a kind of antenna around the target, thus increasing the size of the effective target. How should we determine the size of this antenna? The simplest argument is this. Suppose antenna size is and contour length of DNA inside antenna is . It is worthwhile to emphasize that and do not define any sharp border, but rather a smooth crossover, such that transport outside the antenna is mainly due to the three-dimensional diffusion, while transport inside the antenna is dominated by the sliding, or one-dimensional diffusion along DNA. The advantage of thinking about stationary process is that under stationary conditions, the flux of particles delivered by the three-dimensional diffusion into the -sphere of antenna must be equal to the flux of particles delivered by one-dimensional diffusion into the target. The former rate is given by the Smoluchowski formula (see Appendix A) for the target size ; for the concentration of free (i.e., not adsorbed) proteins c_free, it is D₃c_free. To estimate the latter rate, we note that the time of one-dimensional diffusion into the target site from a distance of order is ²/D₁; therefore, the rate can be written as (c_ads)(²/D₁), where c_ads is the number of proteins nonspecifically adsorbed on the piece of DNA of the length . Thus, our main balance equation for the rate J reads

(2)

Formally, this equation follows from the continuity equation, which says that divergence of flux must vanish everywhere for the stationary process and that flux must be a potential field.

View larger version (42K): [in this window] [in a new window]   FIGURE 1  Antennae in a variety of cases. The upper part of every figure represents a poor man's idea of a prokaryotic cell. In panels a and b, DNA in the cell is a coil, because coil size R is smaller than the cell dimension; alternatively, one can think of a dilute solution of DNA in which R is much smaller than the distance to other coils (not shown). In panel c, the amount of DNA is so large that the coil size would have exceeded the cell diameter, and so DNA is a globule; alternatively, one can think of a semidilute solution (18) of strongly overlapping DNA coils. The lower figures represent a blown-up view of the region around the target site on DNA. The antenna part of DNA around the target is shown in lighter color than the rest of DNA. The space region below the crossover length scale is shadowed. This space region is roughly spherical in cases a and b; it is sausage-shaped in case c. Panel a also shows the averaged flow lines of the diffusion, which go in three dimensions far away from the target and go mostly along DNA within the antenna length scale (they are equivalent to electric field lines in terms of electrostatic analogy; see Appendix B). In panels b and c, flow lines are not shown, simply because it is difficult to draw them. In panel c, we see that DNA globule locally looks like a temporal network, with the mesh size r. In this case, the antenna might be much longer than one mesh. In the figure, the mesh size is not larger than the persistence length, so the length of DNA in the mesh g is approximately the same as r; at lesser density, mesh size might be longer, and then DNA in the mesh would be wiggly, with g >> r.

To determine the one-dimensional concentration of nonspecifically adsorbed proteins, c_ads, and the concentration of proteins remaining free in solution, c_free, we now argue that as long as the antenna is only a small part of the DNA present, every protein in the system will adsorb and desorb many times on the DNA before it locates the target; therefore, there is statistical equilibrium between adsorbed and desorbed proteins. Assuming that we know the adsorption energy or the corresponding constant and remembering that adsorbed proteins are confined within distance of order b from the DNA, we can write down the equilibrium condition as

(3)

which must be complemented by the particle-counting condition

(4)

Since volume fraction of DNA is always small, Lb² << , standard algebra then yields

(5)

Note that at the length scales smaller than persistence length p, the DNA double helix is practically straight, while on the length scales greater than p, the double helix as a whole is a Gaussian coil. This means that, if we take a piece of double helix of the contour length , its size in space scales as

(6)

Substituting this result into the balance equation (Eq. 2), we can determine the antenna size and then, automatically, the rate—the latter being either side of the balance equation. We have to be careful, because we see that there are already as many as four different scaling regimes, due to Eqs. 5 and 6:

Regime A: Antenna is straight (upper expression of Eq. 6), adsorption is relatively weak (upper expressions in Eq. 5).

Regime B: Antenna is Gaussian (lower expression in Eq. 6), but adsorption is still relatively weak.

Regime C: Antenna is Gaussian and adsorption is relatively strong (lower expressions in Eq. 5).

Regime D: Straight antenna and strong adsorption.

To begin with, suppose the antenna is straight ( < p, so , see Fig. 1 a) and the nonspecific adsorption relatively weak (y < /Lb², so c_ads cyb²). In this case, the balance equation yields b(yd)^1/2, or the rate

(7)

In other words, for the ratio of this rate to the Smoluchowski rate J_s D₃cb, we obtain

(8)

This result remains correct as long as antenna remains shorter than persistence length, and since we know , we obtain this condition explicitly: y < p²/b²d.

Let us now suppose that nonspecific adsorption is still relatively weak (y < /Lb², so c_ads cyb²), but it is strong enough such that the antenna is longer than persistence length ( > p, so that see Fig. 1 b). Then our balance equation yields (yd)^2/3 p^–1/3b^4/3 or

(9)

One should check that this new result for implies that > p at y > p²/b²d, and so y p²/b²d is the crossover line between the two regimes, A and B. In both regimes, and as expected, the rate grows with the strength of nonspecific adsorption, y, because increasing y increases the size of the antenna. However, the functional scaling dependence of the rate on y is significantly different, reflecting the difference in DNA fractality at different length scales.

Before we proceed with analysis of other scaling regimes, it is useful to make the following comment. The balance equation (Eq. 2) describes the fact that every protein going through the three-dimensional diffusion far away must then also go through the one-dimensional diffusion closer to the target. In other words, the balance equation (Eq. 2) describes the self-establishing match between the three-dimensional and one-dimensional parts of the process. But we can also look at the situation differently: suppose that one particular protein is adsorbed on DNA in a random place, and let us estimate the distance it can diffuse along DNA before it desorbs due to a thermal fluctuation. Since probability of thermally activated desorption is proportional to the time that protein spends adsorbed must be b²y/D₃. During this time, protein diffuses along DNA by the distance of This distance was first estimated in Richter and Eigen (8); following Coppey et al. (14) and Halford and Marko (16), we call it sliding distance. We see, therefore, that antenna length is just about sliding distance for the straight DNA, but >> _slide for the coiled DNA. At first glance, this seems like a very weird result: how can the antenna possibly be longer than the distance over which protein can slide? In fact, the antenna does become longer than the bare sliding distance, and this happens because, for the coiled DNA, every protein, desorbed after sliding the distance of the order of _slide, has a significant chance to readsorb nearby. Such correlated readsorption gets more likely as we consider more and more crumpled conformations of DNA. Indeed, if, in general, we assume that , then the balance equation yields y^1/(1+), which means that grows with y faster than _slide y^1/2 at every < 1. This growth of with y gets increasingly fast as decreases, which corresponds to more crumpled conformations. We should emphasize that this mechanism of correlated readsorption is impossible to see as long as polymeric and fractal properties of DNA are not considered explicitly, which is why this mechanism has been overlooked in previous works.

With further increase of either nonspecific adsorption strength y or DNA overall length L, we ran into the situation when most of the proteins are adsorbed on the DNA. In other words, if one prefers to think in terms of a single protein diffusion, then this single protein molecule spends most of the time adsorbed on DNA far away from the target. For this case, we have to use the lower lines of Eq. 5 and substitute it into the balance equation (Eq. 2). Since equilibrium condition, Eq. 3, is still satisfied, the result ydb² remains unchanged. Depending on whether antenna length is longer or shorter than persistence length, we obtain regimes C and D.

For regime C, we have > p; the antenna is a Gaussian coil; yielding (yd)^2/3p^–1/3b^4/3; and

(10)

Given our expression for , the condition > p implies the familiar y > p²/b²d, and another condition for this regime is that most proteins are adsorbed, or y > /Lb² (see Eq. 5).

For regime D, the antenna is straight, so , and we get b(yd)^1/2, just as in regime A. For the rate, however, substitution of the lower-line terms of Eq. 5 into the balance equation (Eq. 2) yields

(11)

According to our discussion, this regime should exist when y < p²/b²d and y > /Lb². As we shall see later, these two conditions can be met together and the room for this regime exists only if d < 1, which means that one-dimensional diffusion along DNA is slower than three-dimensional diffusion in space.

In both regimes C and D, overall rate decreases with the increase of nonspecific adsorption, y, because three-dimensional transport to the antenna is slowed down by the lack of free proteins.

We have so far discussed four of the scaling regimes; our results are Eqs. 8–11. Already at this stage, we gained simple understanding of the nonmonotonic dependence of the rate on y—a phenomenon formally predicted in Berg et al. (11) and observed in Winter et al. (12), but previously not explained qualitatively: at the beginning, increasing y helps the process because it leads to increasing antenna length; further increase of y is detrimental for the rate because it leads to an unproductive adsorption of most of the proteins. We have also obtained a new feature, absent in previous works: the shape of the maximum on the J(y) curve is asymmetric, at least if DNA is not too long: in regimes B and C, rate grows as y^1/3 and then falls off as y^–2/3.

Since there are quite a few more scaling regimes, it is easier to understand them if we now pause to offer the summary of all regimes as presented in Fig. 2 and Table 1.

View larger version (23K): [in this window] [in a new window]   FIGURE 2  Diagram of scaling regimes for the case d = 1, when diffusion along the DNA has the same diffusion constant as diffusion in surrounding water. Both L and y axes are in the logarithmic scale. When DNA is shorter than the persistence length (b < L < p) it is essentially a rod, and if longer than the persistence length, is a Gaussian coil. However, coil size is smaller than the restriction volume (p < L < 2/3/p), DNA is globular at L > 2/3/p, and we only consider L up to /pb, because at larger L DNA, segments start forming a liquid-crystalline order. Summary of the rates for each regime is found in Table 1. Here, as well as in the other figures, to make formulas look shorter, all lengths are measured in the units of b, meaning that L, p, and stand for L/b, p/b, and /b3.

	SUMMARY OF THE RESULTS: SCALING REGIMES

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

To be systematic, let us start our review of scaling regimes from the two trivial cases, which correspond to the axes in Fig. 2. When y 1, there is no nonspecific binding of proteins to the DNA, and no sliding along the DNA. Proteins find their specific target at a rate equal to the Smoluchowski rate, or J/J_s = 1. Similarly, if the DNA is very short, as short as the specific target site itself, or L b, then once again J/J_s = 1, for trivial reasons. Since we assume that there is some nonspecific adsorption, or y 1, and since DNA length is obviously always greater than the target size b, our diagram in Fig. 2 presents only the y > 1 and L/b > 1 regions, which is why the pure Smoluchowski regime is seen only on the axes.

If we increase y and consider the y > 1 situation, then we have significant nonspecific adsorption of proteins on DNA—which increases the rate due to the antenna effect. If y remains moderate, the antenna is shorter than DNA persistence length, and it is straight. This is the regime labeled A in Fig. 2 and described by Eq. 8. With further increase of y, when y > p²/b²d, we cross over into the regime labeled B and described by Eq. 9; in this regime, the antenna is so long that it is a Gaussian coil. From regime B, we can cross over the line y = /Lb² and get into the regime labeled C and described by Eq. 10. One can cross over into regime C by either increasing y or increasing L, because increasing either of these variables promotes unproductive nonspecific adsorption of proteins on faraway pieces of DNA and thus slows down the transport to the specific target.

From regime A, we can also cross over the line y = /Lb², but as long as d = 1, this does not bring us to regime D; instead, we get to the new regime labeled I, which we will explain below.

To understand all other scaling regimes, we have to remember that our previous consideration was restricted in two respects. First, we assumed that the entire DNA in the form of Gaussian coil fits within volume —which is true only as long as L < ^1/3 and where ^1/3 stands for the linear dimension of the restriction volume. To relax this assumption, we will have to consider a long DNA, which is many times reflected by the walls of volume , and inside volume represents a globule, locally looking like a semidilute solution of separate DNA pieces, as illustrated in Fig. 1 c. For such long DNA, we shall find two more regimes, labeled H and I in Fig. 2. Second, we assumed that the antenna length was smaller than full DNA length L; the consequence of this was our statement, Eq. 3, that there is equilibrium between adsorbed and dissolved proteins. Relaxing this assumption, we will have to discuss regimes labeled E, F, and G in Fig. 2.

In Fig. 3, we present a schematic y-dependence of the rate for a number of values of DNA lengths L. Each curve is labeled with the corresponding value of L. To be specific, we have chosen the lengths that correspond to various crossovers and are marked on the scaling regimes diagram (Fig. 2). Note that in many cases our result for the rate exhibits a maximum and saturation beyond the maximum—features first described in Berg et al. (11). Unlike Berg et al., we find that the maximum is asymmetric and, even more importantly, J/J_s can become much smaller than unity, i.e., one can observe deceleration in comparison with Smoluchowski rate. We also find a number of other features, such as the specific power law scaling behavior of the rate.

View larger version (16K): [in this window] [in a new window]   FIGURE 3  Schematic representation of rate dependence on y. Both the rate J and y are given in logarithmic scale. The fraction next to each curve shows its slope, which is the power of J(y) dependence. Each curve corresponds to a specified value of DNA length. Also indicated in Fig. 2, the length is shown above the right end of each curve. Experimentally, the value of y can be controlled through the salt concentration, because nonspecific adsorption of proteins is controlled by Coulomb interaction between negative DNA and positive patch on the protein surface; for instance, if the salt is KCl, then it is believed (12,15) that lg y = 10 lg [KCl] + 2.5, where [KCl] is the molar concentration of the salt. Note that we recover the possibility, first indicated in Berg et al. (11), that the rate goes through the maximum and then saturates; but in our case, the maximum is often asymmetric, and at large y the rate becomes very small J/Js << 1, particularly for long DNA. Here, as well as in the other figures, to make formulas look shorter, all lengths are measured in the units of b, meaning that L, p, and stand for L/b, p/b, and /b3.

	SYSTEMATIC CONSIDERATION OF SCALING REGIMES

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

 
DNA is not long enough for a full antenna
If DNA is too short for the antenna, then proteins already adsorbed on DNA can find their target faster than new proteins can be delivered to the DNA from solution. There is no longer any adsorption equilibrium, and instead of Eq. 3 we can only claim that c_ads < yc_freeb². Therefore, the amount of adsorbed proteins under stationary conditions is physically determined by the stationarity itself, meaning that we have to look at Eq. 2 as two equations. In doing so, we have to replace in the right-hand side (one-dimensional rate) by L, because we do not have more DNA than L; and we have to replace in the left-hand side, which is the antenna size for three-dimensional transport, by R, which is the overall size of the DNA coil. Of course, particle-counting Eq. 4 is still valid here, and it is the third equation. Thus, our equations read:

(12)

From here, we find

(13)

We can now easily address all possible scaling regimes in which the antenna is longer than the DNA.

To begin with, it is possible that DNA length is shorter than DNA persistence length L < p, such that the entire DNA is essentially straight, and then R L. Assuming also L³ < , we arrive at the scaling regime labeled E in Fig. 2, in this regime

(14)

The borderline of this regime can be established from the condition that since the entire DNA is smaller than the equilibrium antenna, we must expect that c_ads is smaller than its equilibrium value, or c_ads/c_freeb² y. Since, according to the second expression in Eq. 12 we have c_ads/c_free = LR/d, we then have the condition LR/d < yb²; at L < p this yields y > L²/b²d. At the same condition we can also arrive from the other side of the crossover, by noting that regime A continues as long as the antenna is shorter than the entire DNA, < L; and by using our result for for regime A, this produces the same crossover line between regimes A and E.

For longer DNA, when L > p, the entire DNA is a Gaussian coil, and its size is R (Lp)^1/2. Still assuming that the second term dominates in the denominator in Eq. 13, we arrive at

(15)

This regime is labeled F in Fig. 2. Its border line with regime E is obviously the vertical line L = p. In regard to crossover to regime B, once again it can be established either from c_ads/c_free = LR/d < y for regime F or from < L for regime B. In either way, we arrive at the crossover condition y = L^3/2p^1/2/b²d.

For even longer DNA, the antenna length becomes equal to the length of the entire DNA only at such large y that the system is already in regime C, with the rate falling down with increasing y because of the unproductive adsorption of proteins. Since antenna length in regime C is given by the same formula as in regime B, so the upper border line of regime C is the continuation of the corresponding line bordering regime B: y = L^3/2p^1/2/b²d. However, when we cross this line upwards from regime C, we arrive at the new situation, because now the first term dominates in the denominator of the Eq. 13, meaning that most of the proteins are adsorbed on DNA, such that we obtain

(16)

The crossover between this regime and regime F is the vertical line at which both terms are comparable in the denominator of Eq. 13: L = (d)^2/5/p^1/5. Crossover line with regime C can once again be established from the condition c_ads/c_free = LR/d < y.

In all regimes E, F, and G the rate saturates with increasing y. For regimes E and F this happens after just initial growth of the rate; for regime G, saturation occurs after the rate goes through the maximum and starts decreasing. In all cases saturation is due to the fact that increasing adsorption strength does not lead to any increase of the antenna size, because the entire DNA is already employed as antenna, and the antenna has nowhere to grow.

Cell is not big enough to house the DNA Gaussian coil
When DNA is very long for a given volume, that is, when (Lp)^1/2 > ^1/3, DNA cannot remain just a coil; it must be a globule, as it is forced to return many times back into the volume after touching the walls (see, for example, (19)).

For the purposes of this work, it is sufficient to keep assuming that the excluded volume of DNA is not important, because the volume fraction of DNA within the confinement volume is still small, and is small even when compared at b/p. Nevertheless, locally the system looks like a so-called semidilute solution of DNA, or a transient network with a certain mesh size (see Fig. 1 c).

We should recall to mind some basic facts regarding the semidilute solution, or a transient network (18,19). Let us denote r as the characteristic length scale of a mesh in the network—which, in the scaling sense, is the same as the characteristic radius of density-density correlation (see Fig. 1 c). Let us further denote g as the characteristic length along the polymer corresponding to the spatial distance r. Quantities r and g can be estimated from the following physical argument (18,19): Consider a piece of polymer of the length g starting from some particular monomer; it occupies region r³ and makes density g/r³. This density must be approximately the overall average density, which for our system is of the order of L/. Thus, g/r³ L/. The second relation between g and r is similar to Eq. 6, which depends upon whether the mesh size is bigger or smaller than persistence length p:

(17)

Accordingly, after some algebra, we obtain

(18)

The upper line of the expression above corresponds to a network so dense that every mesh is shorter than persistence length, and the polymer is essentially straight within each mesh. The lower line of the expression describes a much less concentrated network, in which every mesh is represented by a small Gaussian coil.

Returning to our problem, we should realize that the antenna length can in fact be longer than the mesh size g, as illustrated in Fig. 1 c. To estimate the antenna size for this case, we should remember that desorption from the antenna does not necessarily completely break the sliding along DNA, because protein can still readsorb on a nearby place on the DNA, or, more generally, on a correlated place on the DNA. To account for this, let us imagine that the antenna part of DNA is decorated by a tube of the radius r. Since r is the correlation length in the DNA solution, the protein remains correlated with the antenna as long as it remains within this tube around the antenna. Accordingly, our main balance equation, Eq. 2, must be modified to account for the fact that three-dimensional transport on scales larger than r is now realized through the DNA network and, therefore, the task of regular three-dimensional diffusion is only to deliver proteins over the length-scale of the order of one mesh size r, into any one of the /g network meshes along the antenna. The rate of delivery into one such mesh would be D₃c_freer, so that the overall delivery rate into the antenna tube scales as D₃c_freer/g. As usual, this must be equal to the rate of one-dimensional delivery along the antenna into the specific target, so instead of Eq. 2 we finally get

(19)

As long as the antenna is shorter than the entire DNA, the relation between c_free and c_ads equilibrates and obeys Eqs. 3–5, so we finally get

(20)

and

(21)

What is nice about this formula is that it remains correct in a variety of circumstances—when the antenna is straight ( < p); or the antenna is Gaussian (p < < ^2/3/p); or the antenna is a globule ( > ^2/3/p).

Taking r and g from Eq. 18, we finally obtain two new regimes. When every mesh is Gaussian,

(22)

This regime borders regime C along the line where the antenna size is equal to the mesh size, = g, which reads y = ³/(L³p⁴b²d). Regime H also borders regime G along the line where the antenna size is as long as the entire DNA, = L, or y = L³p²/b²d. Finally, regime H also borders another regime I along the vertical line L = /p², which corresponds to the DNA within every mesh becoming straight (shorter than persistence length). For this regime, we have to use the upper line in the expressions in Eq. 18, thus obtaining

(23)

This regime borders saturation regime G along the line y = L²/b²d where = L.

In regard to the lower border of regime I, it corresponds to the situation when the antenna becomes straight, which happens at y = /Lb²d. However, as long as d = 1, which is the case presented in Fig. 2, this line coincides with the line y = /Lb², below which most proteins are desorbed and free in solution. That is why at d = 1, there is no room for regime D, in which the antenna is straight, but most proteins adsorbed. Indeed, when d = 1, then three-dimensional transport is mostly realized by sliding along the network edges as soon as most proteins are adsorbed, which precisely means that regime A crosses over directly to regime I.

As we see, in both H and I regimes the rate J decreases with growing y, but does so more slowly than in regime C, that is, only as y^–1/2 instead of y^–2/3. This happens because adsorbed proteins are not just taken away from the process, as in regime C, but also participate in three-dimensional transport through the network (albeit this transport is still pretty slow).

This completes our scaling analysis for the d = 1 case shown in Fig. 2.

Diffusion rate along DNA is different from that in surrounding water
Let us now relax the d = 1 condition and examine the cases in which diffusion along the DNA is either slower (d < 1) or faster (d > 1) than in the surrounding water.

First, let us consider the d < 1 case, when diffusion along the DNA is slower than that in the surrounding water (D₁ < D₃); corresponding scaling regimes are summarized in the diagram Fig. 4 a. Most of the diagram is topologically similar to that in the Fig. 2, and we do not repeat the corresponding analysis. Of course, there are now powers of d in all equations, but the major qualitative novelty is that there is now room for regime D sandwiched between regimes A and I. The formal reason why this regime now exists in a separate region is because the line y = /Lb²d goes above the line y = /Lb². To understand the more meaningful physical difference, let us recall that the line y = /Lb² marks the crossover above which most of the proteins are adsorbed, but it is not enough for the sliding-along-network mechanism to dominate in the three-dimensional transport at d < 1.

View larger version (20K): [in this window] [in a new window]   FIGURE 4  Scaling regimes for the cases d < 1 (a) and d > 1 (b). In this figure, Y = yd. Also, to make formulas look shorter, all lengths are measured in the units of b, meaning that L, p, and stand for L/b, p/b, and /b3. The major difference from the d = 1 case is the presence of regime D in a and the presence of regimes K and M in b. In regime D, the majority of proteins are adsorbed, but the dominant three-dimensional transport is still the usual diffusion through the surrounding water, because sliding along the DNA is too slow (D1 < D3). In regimes K and M, the majority of proteins are not adsorbed, but the dominant three-dimensional transport mechanism is the sliding of minority proteins along the DNA network, because it is so much faster (D1 > D3). (Note that we skip J and L in labeling regimes to avoid confusion with rate J and DNA length L.)

Let us now switch to the opposite limit and consider the d > 1 case, for which the results are summarized in Fig. 4 b. This diagram is quite similar to the previously considered examples in Figs. 2 and 4 a, except there are now two new regimes labeled K and M (in alphabetical labeling of the regimes we skip J and L to avoid confusion with rate and DNA length). These regimes are both below the line y = /Lb², which means that most of the proteins are not adsorbed. However, since d > 1, the new physical feature of the situation is that adsorbed proteins, although they are in minority, can nevertheless dominate in three-dimensional transport by sliding along the DNA network, because sliding is now so fast at d > 1. Thus, regimes K and M are ones in which effective diffusion along DNA network dominates, so we have to use Eq. 19 for the rate and antenna size, while for the concentrations of free and adsorbed proteins we have to use the upper line expressions in Eq. 5. In regime K, the local concentration of DNA segments is so high that every mesh in the DNA network contains an essentially straight piece of DNA, so we have to use the upper line expression in Eq. 18, yielding (after some algebra)

(24)

Similarly, since in regime M, the mesh of the DNA network is Gaussian, we have to use the lower-line expression in Eq. 18, and this produces

(25)

Since the majority of proteins are not adsorbed, it is not surprising that the rate grows with y in both regimes K and M. Notice that the rate is given by the same formula as for regimes A and K (compare Eqs. 8 and 24). This is similar to the situation with regimes D and I, as discussed before, because even though the rate is given by the same formula, the underlying diffusion mechanism is fundamentally different. In both cases of D and I or A and K, it is possible that although scaling laws are the same, the numerical prefactors are different.

It is also interesting to note that the crossover between regimes B and M takes place on the line y = ³/p⁴L³b²d where the antenna length is equal to the DNA length in one mesh: on the side of the B regime, the antenna is shorter than one mesh, and transport to the antenna must be through water; on the side of M, the antenna is longer than one mesh, and effective transport along the DNA network is at play.

Maximal rate
To finalize our discussion of scaling regimes, it is reasonable to ask: what is the maximal possible rate? According to our results, the maximal rate is achieved on the border between regimes F and G, that is, at L (d)^2/5/p^1/5 and at y ^3/5p^1/5/b²d^2/5. Maximal possible acceleration compared to the Smoluchowski rate is (p²d/b⁵)^1/5. It is interesting to note that the optimal strategy in achieving the maximal rate at the minimal possible y requires us to have the adsorption strength y right at the level at which the probability of nonspecific adsorption for every protein is 1/2 (on the line y /Lb²).

It is interesting that the maximal possible acceleration grows with overall volume , which may seem counterintuitive. This result is due to the fact that total amount of DNA grows with increasing , and, according to our assumption, all this DNA has still just one target.

	DISCUSSION

TOP ABSTRACT INTRODUCTION SIMPLE CASE: STRAIGHT ANTENNA... SUMMARY OF THE RESULTS:... SYSTEMATIC CONSIDERATION OF... DISCUSSION CONCLUSION APPENDIX A: SIMPLE SCALING... APPENDIX B: ELECTROSTATIC... ACKNOWLEDGEMENTS REFERENCES

 
Single protein view
Many of the previous theoretical works (14–17) looked at the situation in terms of a single protein molecule diffusing to its target. In this view, one should imagine that a protein molecule is initially introduced into a random place within volume , and then one should ask what is the first passage time (20) needed for the protein to arrive to the specific target site on DNA. The mean first-passage time can, of course, be found using our results for the rate J by inverting the value of the rate and assuming that, on average, there is just one protein molecule in the system at any time, i.e., However, we want to rederive all our results directly in terms of to build bridges to the works of other authors. The rederivation also turns out quite illuminating.

First let us consider that DNA is a globule, L > ^2/3/p (or semidilute solution), and look at regimes H, I, K, and M; unlike stationary diffusion approach above, in the single protein language the derivation for the globular DNA case is actually simpler. Following Slutsky and Mirny (17), we imagine that the search process for the given single protein consists of tours of one-dimensional sliding along DNA followed by diffusion in three-dimensions, followed by one-dimensional sliding, etc. If in one tour of one-dimensional sliding, protein moves some distance along the DNA, then it takes time at ²/D₁. The length here is, of course, our familiar antenna length, but we will rederive it here, so we do not assume it known. In regard to the tour of three-dimensional diffusion, it breaks the correlation of the one-dimensional sliding if it carries the protein over a distance larger or approximately the same as the correlation length in the DNA system, which is r—mesh (or blob) size. Thus, the longevity of one tour of three-dimensional diffusion is r²/D₃.

The next step of our argument is this. On its way to the target, the protein will go through a great many adsorption and desorption cycles; therefore, the ratio of time that protein spends adsorbed and desorbed should simply follow equilibrium Boltzmann statistics:

(26)

Here, we note that there is an approximation underlying our argument: one tour of correlated one-dimensional sliding does include small three-dimensional excursions of the protein into water, but they are small in the sense that they do not go beyond the crossover correlation distance and, therefore, readsorption after excursion occurs at a correlated place on the DNA. Accordingly, these excursions make only a marginal contribution to the sliding time, which is correctly estimated as ²/D₁.

The final part of the argument is most clearly formulated by Bruinsma (15): since subsequent tours of one-dimensional sliding occur over uncorrelated parts of DNA, full search requires L/ rounds. Therefore, the total search time can be written as

(27)

Equations 26 and 27 solve the problem for all regimes of globular DNA if we remember that mesh (or blob) size r is given by Eq. 18. Notice that Eq. 26 gives a new interpretation to the line y /Lb² on any of our diagrams in Fig. 2 and Fig. 4, a and b: for the parameters below this line most of the overall search time is spent in three-dimensional diffusion, while for the system with parameters above the line the major time-consuming part is the one-dimensional sliding. It is close to this line where the result of Slutsky and Mirny (17) applies and these two times are of the same order. And let us recall that it is also close to this line where the maximal possible rate is achieved (see Maximal Rate, above).

Thus, regimes H, I, K, and M result from two possibilities for r in Eq. 18 (straight or Gaussian DNA within a mesh) and two possibilities of either first- or second-term dominance in Eq. 27.

Let us now turn to regimes A, B, C, and D, in which DNA is a coil. In this case, we still essentially rely on the equations similar to Eqs. 26 and 27, except some effort is now needed to understand the time of three-dimensional diffusion.

Our argument for this case starts from our noticing that there is a crossover spatial scale , such that correlated sliding takes place inside scale , while regular three-dimensional diffusion in water occurs on a larger length scale, as it breaks correlations between desorption and subsequent readsorption. Thus, the time of one tour of three-dimensional diffusion is the mean first-passage time into any one of the L/ spheres of size (here is the contour length of DNA accommodated by one sphere of the size ; once again, we pretend that we do not know and , but will rederive them in this single-protein language). The arrival time into one such sphere is the Smoluchowski time (discussed in Appendix A) for the target of size , which is /D₃; the arrival time into any one of the L/ spheres is L/ times smaller, /D₃(L/) To present our equations for and overall search time in form similar to Eqs. 26 and 27, we define distance r_eff such that and then we obtain

(28)

and

(29)

Once again, remembering two regimes for the relation between and , Eq. 6, and having either the first or second term dominate in the total time of Eq. 29, we recover the regimes A, B, C, and D.

Finally, the results for all saturation regimes E, F, and G are recovered by replacing the antenna length