INTRODUCTION

TOP ABSTRACT INTRODUCTION APPENDIX A APPENDIX B REFERENCES

The cooperative binding of proteins to DNA plays a significant role in the regulation of gene expression (Owen-Hughes and Workman, 1994) because it allows a sensitive response to small changes in protein concentration. In particular, it is well known that transcription factor proteins (Lodish et al., 1995) exhibit a significant level of cooperativity (Sun et al., 1997). The structural basis of the cooperativity is not fully understood (Sun et al., 1997), but it is known that long-range cooperativity is possible through loops (Schlief, 1992), formed as the result of association between two DNA-binding proteins. Looping is also believed to play an important role in gene access control. Cooperativity at shorter distances may be related to specific protein-protein interactions or to a generic cooperativity resulting from structural distortions induced by the binding of a protein to DNA (Lilley, 1995; Nelson, 1995). For example, the binding of transcription regulation proteins such as the important TATA-box promoters (TPB) involves amino acid intercalation into the stack of basepairs (Kim et al., 1993; Werner et al., 1996). The result is that kinks are produced in the form of sharp local bending angles in the DNA strand. This deformation may permit a better fit for other DNA-associating proteins, such as the polymerases. Disruptions of the basepair stacking sequence have no effect beyond about half a turn of the double helix (Kim et al., 1993; Werner et al., 1996), so it is expected that this form of cooperativity is restricted to the immediate neighborhood of the primary binding protein. Protein-induced deformations of the DNA strand are not restricted to transcription factors. DNA may wrap itself once or more around a protein, as happens in the case of complexation of DNA with nucleosomes, the gyrase enzyme, or bacterial RNA. Interestingly, nucleosome-binding appears to be cooperative with transcription factors (Owen-Hughes and Workman, 1994). DNA-deforming proteins will be referred to below as "architectural" proteins.

The aim of this article is to demonstrate the possibility of a variable-range form of cooperative DNA binding of architectural proteins with a range and strength that is regulated by the tension along the DNA strand. The cooperative interaction between proteins that are not in physical contact is mediated by the deformation of the intervening DNA strand. In the absence of tension, the proposed mechanism is absent. Our demonstration of the possibility of tension-controlled cooperativity is based on an analysis of the "worm-like chain" (WLC) model of DNA elasticity (Hagerman, 1981; Kam et al., 1981), which has been used in studies of single protein binding to DNA (Marko and Siggia, 1997). The WLC model is characterized by a single parameter, a length-scale _p, known as the persistence length. It is the distance over which a (tensionless) WLC maintains orientational order in the presence of thermal fluctuations. That is to say, the autocorrelation between orientational order at two different locations of the chain falls off with distance, , as e^/_p. Fitting the results of micromechanical in vitro studies of protein-free DNA chains to the predictions of the WLC model yields good results for a persistence length of ~50 nm (Bustamante et al., 1994; Bensimon et al., 1994), although longer persistence lengths have been reported by different methods (Bednar et al., 1995). Further studies of the elastic properties of DNA in in vitro conditions can be found in Strick et al., 1996; Cluzel et al., 1996; Larson et al., 1997.

We employ the WLC model only to evaluate binding cooperativity due to deformations produced by architectural proteins in sections of the DNA strand that are not in the immediate neighborhood of the binding proteins themselves. The WLC model will not apply reliably when the two proteins are so close together that details of basepair action (i.e., roll, slide, and twist) play an important role in the mediation of their interaction. The primary binding of the protein itself is characterized by two phenomenological parameters: the single protein, zero-tension specific binding free energy E⁽¹⁾(s), and the DNA bending angle ; see Fig. 1. The specific binding energy E⁽¹⁾(s) is a sequence-sensitive quantity that depends on the location of the protein along the chain; the index s refers to the position of the protein along the DNA strand. It is usually in the range of 10-30 k_BT, with k_B Boltzmann's constant and T the temperature in Kelvin. At room temperature k_BT  0.6 kcal/mol. By comparison, the individual ionic bonds between basepairs in DNA are in the range 2-3 kcal/mol, and typical covalent bounds are the order of 60-120 kcal/mol. The protein-DNA complex will be assumed to be rigid for the tension levels envisioned, so the bending angle  is independent of tension. The values of E⁽¹⁾(s) and  must be obtained either experimentally or by detailed structural modeling of DNA-protein interactions.

View larger version (52K): [in this window] [in a new window]   FIGURE 1   Strand of DNA containing two proteins. The proteins induce "kinking" on the DNA strand. Illustrated are the case of antisymmetric and symmetric configurations of the two bound proteins. The distance between proteins is expressed in terms of the locations, s1 and s2, of each of them on the strand.

We will focus on the results of analytical and numerical studies that utilize the WLC model to compute the deformation energy of a DNA chain under tension with two identical architectural proteins attached and separated by a distance l. The two proteins are assumed to induce a bend into the DNA strand without any twisting. We find that there is, in general, both an enthalpic and an entropic contribution to the binding cooperativity proposed here (Wang and Giaever, 1988). The enthalpic contribution to the energy is associated with the configuration of DNA with attached proteins that minimizes the total classical energy of the system, while the entropic contribution is generated by fluctuations about that minimum energy configuration. We find that the enthalpic energy of cooperativity results from the deformation, here the kinking, of the DNA due to the presence of an attached protein, while the entropic contribution to the cooperative energy arises from the modification of the DNA's bending modulus induced by the proteins.

The enthalpic cooperative correction to the binding energy E⁽²⁾ between the two proteins (denoted by 1 and 2) that are separated by a distance l = |s₁  s₂| assumed large compared to the characteristic dimensions of the protein is given by

(1)

This formula, derived in Appendix A, holds when the bending angle  is small compared to /2. The case of large bending angle is discussed later. The plus sign in Eq. 1 refers to the "symmetric" case with the two proteins bound on the same side of DNA while the minus sign refers to the "antisymmetric" case with the two proteins bound on opposite sides of the DNA (see Fig. 1). The chain-tension F in Eq. 1 is variable but assumed to be in the range of 10² to 10 pN. The tension-dependent length-scale (F) in Eq. 1 is of key importance; it sets the range of the binding cooperativity. This quantity is defined by:

(2)

For a 1-pN tension, (F) is ~7 nm, while for a 10² pN force it is ~70 nm. However, Eq. 1 is only valid as long as the "tension-length" (F) is less than the persistence length _p. For larger tensions (F) is small compared to the persistence length _p  50 nm. Note that all parameters in Eq. 1 can, in principle, be determined experimentally.

According to Eq. 1, there is no bending-induced cooperativity between two proteins separated by a distance large compared to the tension length (F). In that limit, the only effect of tension is to reduce the single protein binding energy by an amount per protein E_enth = ² (using Eq. 2 to eliminate (F) in Eq. 1). This tension-induced reduction of the protein binding energy is discussed (Marko and Siggia, 1997) for the case of single-protein binding to DNA. For a 1-pN tension, the binding energy reduction is significant: ~7²k_BT. With increasing tension, the protein will be released from the DNA chain when E_enth starts to approach the protein-binding free energy E⁽¹⁾. [If the DNA-protein interaction involves a number of turns of the DNA around the protein, then we must add the quantity LF to E_enth, with L the excess DNA length wound around the protein. For low tensions F, this correction is small compared to E_enth, but for a 1-pN tension, it is comparable.]

A summary of expected values for the cooperative interaction between two bound proteins is displayed in Table 1. It is assumed that the bending angle that each enforces is 45°. Other quantities are appropriate to DNA at room temperature. These values should be compared to, for instance, the cooperative interaction energies between repressor molecules bound to adjacent sites of DNA, which are the order of 1.3 kcal/mol (Hyde and Spicer, 1995).

When the spacing between the two proteins is reduced to within a distance of order of the tension-length (F), then the binding energy increases exponentially for the antisymmetric arrangement while it decreases exponentially for the symmetric arrangement. We can interpret the last term in Eq. 1 as an effective potential energy of interaction, V_enth(l), between two proteins given by:

(3)

with l = |s₁  s₂| the interprotein spacing. The minus sign in Eq. 3 is for the antisymmetric case. In Appendix A we compute V_enth(l) for values of the bending angle  that range up to /2. The result is shown in Fig. 2. Note that both the vertical and horizontal axes are dimensionless. The energy has been expressed in units of 2k_BT(_p/(F)), the energy scale of the tension-induced binding energy reduction E_enth (see Eqs. 1 and 3), while the distance is expressed in units of the tension length, (F). It follows from Fig. 2 that an increase in tension reduces the range of the interaction, as expected from Eq. 2, while it increases the strength of the cooperativity. If the spacing l between the proteins is small compared to the tension length (F), then the effective potential V(0) cancels the E_enth term in Eq. 1. The enthalpic energy gain obtained by bringing two proteins together along the chain from a large separation is equal to twice E_enth.

View larger version (18K): [in this window] [in a new window]   FIGURE 2   The attractive interaction for an antisymmetrically oriented pair of bound proteins. Plotted is the total energy of the configuration in units of kBT, multiplied by the ratio (F)/p, where (F) is the tension-dependent length scale defined in Eq. 2, and p is the persistence length of the DNA strand. Curves are displayed for various values of the "kink angle," , as shown in Fig. 1.

For the symmetric case, the effective potential energy is repulsive. The corresponding energy plots are shown in Fig. 3. The enthalpic deformational energy now increases as the two proteins approach each other. In the limit of small bending angle , two adjacent bending proteins with  in the symmetric conformation have the same tension-induced binding energy reduction as a single protein with a double bending angle of 2. The energy scale for the cooperativity is thus in general set by E_enth. The effective interaction potential is, then, always less than the single protein binding energy, since proteins are expected to unbind when E_enth spacing approaches E⁽¹⁾(s).

View larger version (24K): [in this window] [in a new window]   FIGURE 3   The repulsive interaction between two proteins bound in a symmetric configuration. Plotted is the total energy of the configuration in units of kBT, multiplied by the ratio (F)/p, where (F) is the tension-dependent length scale defined in Eq. 2, and p is the persistence length of the DNA strand. Curves are displayed for various values of the "kink angle," , as shown in Fig. 1.

Although it would appear as if the enthalpic cooperativity depends on temperature within the WLC model (see Eqs. 1 and 3), this is not the case: _P is inversely proportional to k_BT (see Eq. A.8) so neither E_enth nor (F) depend on k_BT. There is, however, a purely entropic contribution to the cooperativity that is explicitly dependent on k_BT. In Appendix B, we obtain the following expression for the entropic correction to the cooperativity:

(4)

for two proteins of length d separated by a distance l = |s₁  s₂|. As noted previously, this energy results from the action of proteins in modifying the bending modulus of the DNA strand to which they are attached. Equation 4 follows from the assumption that the section of DNA to which a protein is attached has become infinitely stiff. The entropic contribution does not depend on the bending angle  and is the same for the symmetric and antisymmetric configurations. In the limit of protein separations that are large compared to the tension length (F), the entropic contribution E_entr = (1/2)E_entr⁽²⁾() to the single-protein binding energy is:

(5)

This is, again, a negative quantity: the local constraints imposed on the DNA chain by the two binding proteins lower the entropy of the complex as compared to a free chain, and hence reduce the binding energy. The difference can, again, be interpreted as an effective entropic potential energy of interaction, which is now entropic. It is given by

(6)

This effective entropic potential energy is always attractive. In Fig. 4 we show the entropic potential energy for two proteins of size d equal to 20 Å and for a 1-pN applied tension ((F = 1 pN) = 70 Å).

View larger version (8K): [in this window] [in a new window]   FIGURE 4   The entropic interaction potential between two identical bound proteins resulting from the effect of each of them on the bending modulus of the strand of DNA to which they are attached. The interaction is plotted in units of kBT under the assumption that the region of affected DNA is 70 Å long, and that the applied tension is 1 pN.

For larger bending angles, the entropic interaction is both weaker and shorter in range than the enthalpic interaction. However, because this interaction does not depend on the magnitude of the bending angle, it dominates for zero bending angles, or bending angles that are very small. An interesting special case concerns the entropic interaction between two long strings of binding proteins. If we model a polymerized string of proteins bound to DNA as a rigid section of size d, with d assumed large compared to the tension length, and with zero total bending angle, then two such strings are expected to have an effective entropic interaction potential given by:

(7)

[Formally, there was a divergence in Eq. 7 for small spacings l, but Eq. 7 should, of course, not be expected to retain validity when l approaches a basepair spacing.] The free energy of two long rigid strings separated by a small gap is lowered by an amount of order k_BT if the intervening gap is filled in either by shifting one of the two strings to close the gap or by adding additional binding proteins inside the gap. This effect provide us with a curious entropic stabilization of mechanism of polymerization of proteins along DNA strands.

When we increase the bending angle beyond /2, new physical effects appear. As shown in Figs. 5 and 6, there are in general two possible configurations for a symmetric two-protein/DNA complex. Up to now, it has been tacitly assumed that the "S" or "stretched" configuration was the appropriate one (as shown in Fig. 1) and indeed the S configuration has the lower free energy for bending angles  < /2. However, when the bending angle exceeds /2 this is no longer the case. For low tensions, the "L," or "looped" configuration has in fact the lower elastic free energy, while for higher tensions, the S configuration is more stable. The two regimes are separated by a mathematical singularity that has the character of a first-order phase transition. In Fig. 7 we show the extension, X, of the two-kink configuration for which the kink angle is greater than /2, as a function of tension, F. There is a transition from the L to the S configuration with increasing tension visible as a discontinuity of the extension at the transition point. There is no transition in the antisymmetric case.

View larger version (43K): [in this window] [in a new window]   FIGURE 5   The "loop" or L configuration of two symmetrically bound proteins, when the bend angle enforced by a bound protein is > /2. This is the preferred configuration when the applied tension is small.

View larger version (21K): [in this window] [in a new window]   FIGURE 6   The stretched, or S, configuration of two symmetrically bound proteins when the bend angle exceeds /2. This is the preferred configuration at high levels of applied tension.

View larger version (18K): [in this window] [in a new window]   FIGURE 7   The overall extension of a very long strand of DNA containing two identical symmetrically bound proteins, the bend angle, , of each of which is 2.1. The extension, X, is relative to the fully extended DNA strand, measured in units of , where F is the applied tension and K is the bending modulus. The vertical axis is the applied tension, F, in units of K/2l2, where l is the distance along the DNA backbone between proteins along the DNA strand. See Appendix A for a full discussion of the parameters utilized.

In summary, we have shown that, within the confines of the WLC model, tension can trigger cooperative binding for antisymmetrically arranged architectural proteins. The binding strength has both an enthalpic and an entropic contribution, and it has an appreciable magnitude for tensions of the order of 1-pN or higher. For larger bending angles, we find two competing configurations connected by a tension-induced phase-transition. Both the enthalpic and entropic contributions to cooperativity vanish in the limit of zero tension (see Eqs. 1 and 6). This last result is certainly not self-evident. The decrease in chain entropy imposed by two rigid sections can, for instance, be expected to depend on the spacing, even for zero tension. Interestingly, a study of the interaction between two stiff inclusions inside a two dimensional surface (such as a membrane) reports (Bruinsma et al., 1994) that in this case there is a long-range zero-tension interaction with both entropic and enthalpic contributions, both dropping off as the inverse fourth power of the spacing between the inclusions. The disappearance of tension-induced cooperativity at F = 0 thus must be related to the fact that we are dealing with a one-dimensional geometry.

Experimental in vitro tests of the proposed mechanism can be performed by preparing a bundle of DNA strands, each strand containing bacterial gene operator sequences periodically spaced by a distance of l basepairs. The associated repressor protein (such as the lac repressor) binding specifically to the operator sites would induce local kinks at the operator sites. According to the model, the logarithm of the equilibrium repressor-operator binding constant K_RO contains a contribution that depends on the operator spacing l and the tension F of the DNA bundle according to Eq. 1.

A study of the kinetics of cooperative protein-DNA association can also be a testing ground. For instance, since the TATA box binding protein TBP is known to produce a large bending angle, the one-dimensional diffusion along the DNA of other proteins required for the RNA polymerase initiation complex, such as TFIIE, H, and J, that are nonspecifically bound to DNA, will be speeded up by bending-induced cooperativity. The reason is that the effective potential V(l) turns the random one-dimensional diffusion into a directed process. It should be noted here that the weaker nonspecific binding of DNA associating proteins will not deform the DNA strand as much as specific binding. However, studies of the dependence of nonspecifically bound 434 repressors on the DNA flexibility indicate that nonspecific bonding also involves distortions of the local DNA structure, so there still ought to be a tension-controlled interaction between specific and nonspecifically bound proteins (Hogan and Austin, 1987). We thus predict that (modest) tension will actually increase the formation rate of the RNA polymerase initiation complex. At high tension levels, the formation rate will decrease with tension for reasons discussed earlier.

Another possible area where the present theory could be applied is histone-DNA interactions. According to our model calculations, a collection of nonspecifically bound proteins ought to adopt an antisymmetric zig-zag configuration under tension. The binding of histones to DNA is reported to produce a zig-zag nucleosome structure consistent with our calculations (Thoma et al., 1979). Under tension, the zig-zag structure should thus be stabilized. It should be kept in mind, though, that the intervening linker histones may well affect the competition between different configurations (Thoma et al., 1979).

An important question for the relevance of the work presented here is whether DNA is under tension under in vivo conditions. DNA strands suspended in good solvent are in fact not under tension. We believe that this is an exceptional case. A DNA strand whose ends are fixed is, even for low extensions, typically under a tension in the range of 0.01 pN due to thermal fluctuations, as shown by the micromechanical studies (Bustamante et al., 1994; Bensimon et al., 1994). For extensions closer to one, the tension can be much larger. If a DNA strand is subject to the activity of force-transducing proteinsfor instance during mitosis, RNA transcription, or homologous recombinationthen much higher tensions can be generated. It is known that a single motor protein is able to generate a force of 10 pN or more (Yin et al., 1995). Finally, architectural proteins themselves generate tension if they attach to a DNA strand with fixed ends. When an architectural protein attaches, DNA material is required to accommodate the deformation of the DNA chain near the protein. A simple calculation shows that the self-induced tension of a DNA strand of length L whose ends are held fixed a distance X apart and which contains a line density  of bending proteins is given by:

(8)

with ² the average of the square of the bending angle. If the line density is of the order of one protein/100 Å and if ² is of the order one, then this self-generated tension is of the order 1-pN (it should be possible to verify Eq. 8 in micromechanical measurements). It thus seems reasonable to assume that DNA is under tension for in vivo conditions.

Our results were derived with DNA-protein binding in mind, but the general aspects of our conclusions are related to work in other areas. We can consider the predicted aggregation of proteins under tension as a form of stress-induced decomposition. Stress-induced phase-separation of multicomponent systems is actually a classic phenomenon in solid-state materials for the case that different constituents have different elastic moduli (Cahn, 1961), just as envisioned in the present case.

We conclude by noting that a number of technical objections can be raised against the method used in our study. It is not really reasonable to assume that DNA twist plays no role, in view of the helical nature of DNA, and that we can be allowed to restrict ourselves to a purely two-dimensional arrangement. However, it is our current belief that the introduction of the twist degree of freedom introduces qualitative, but not quantitative, modifications to the results reported here. The assumption of internal structural rigidity of proteins also is questionable. For instance, it is well known that many enzymes can undergo stress-induced structural changes. Enzymes bound to DNA also may change their structure under stress. In addition, it is true that the thermodynamic ensemble in which the tension is kept fixed is not identical to the ensemble in which the length of the DNA segment is held constant, and the condition of fixed segment length is a more reasonable assumption in many in vivo situations. However, the calculations reported here were for two proteins attached to an asymptotically long segment, and in that limit, the two ensembles yield identical results. Finally, it must be kept in mind that the WLC model neglects the possible influences of intervening structures, such as other bound proteins, on the tension-induced interaction between two bound proteins, and that looping and other manifestations of DNA self-interaction are ignored. Despite all these caveats, we feel that the basic resultstress-induced aggregation of DNA-bound proteinsis robust and should remain present in more realistic models.