Cooperative kinetics of ligand binding to linear polymers.

Ligands change the chemical and mechanical properties of polymers. In particular, single strand binding protein (SSB) non-specifically bounds to single-stranded DNA (ssDNA), modifying the ssDNA stiffness and the DNA replication rate, as recently measured with single-molecule techniques. SSB is a large ligand presenting cooperativity in some of its binding modes. We aim to develop an accurate kinetic model for the cooperative binding kinetics of large ligands. Cooperativity accounts for the changes in the affinity of a ligand to the polymer due to the presence of another bound ligand. Large ligands, attaching to several binding sites, require a detailed counting of the available binding possibilities. This counting has been done by McGhee and von Hippel to obtain the equilibrium state of the ligands-polymer complex. The same procedure allows to obtain the kinetic equations for the cooperative binding of ligands to long polymers, for all ligand sizes. Here, we also derive approximate cooperative kinetic equations in the large ligand limit, at the leading and next-to-leading orders. We found cooperativity is negligible at the leading-order, and appears at the next-to-leading order. Positive cooperativity (increased affinity) can be originated by increased binding affinity or by decreased release affinity, implying different kinetics. Nevertheless, the equilibrium state is independent of the origin of cooperativity and only depends on the overall increase in affinity. Next-to-leading approximation is found to be accurate, particularly for small cooperativity. These results allow to understand and characterize relevant ligand binding processes, as the binding kinetics of SSB to ssDNA, which has been reported to affect the DNA replication rate for several SSB-polymerase pairs.

Introduction

Understanding the kinetics of binding of large ligands to long polymers is a very relevant biological problem. The ligand binding to polymers changes their mechanical and chemical properties, plays a relevant role regulating biological functions, shrinks or expands the binding regions of the polymer (by stiffening these regions) and passivates polymer regions for the binding of other units of the ligand, among other effects. The binding may be specific or non-specific depending on whether the ligand binds to the polymer with or without sequence preference. The binding is specific between a basic polypeptide and a polynucleotide, and between an oppositely charged ionic surfactant and a linear polyelectrolyte, among other systems. One of the more attractive problems when the binding is specific involves protein molecules as ligand and DNA as polymer [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [43].

Some ligands bind to few sites, such as Thiocoraline binding to dsDNA (double-stranded DNA), human polymerase β binding to ssDNA (single-stranded DNA), binding of lysozyme to chitosans [14], [15], [16]. In contrast, there are relevant cases where the ligand is large and binds to many monomers, like E. Coli and human mitochondrial SSB that bind 30–70 nucleotides of ssDNA [2], [5], [17]. These ligand bindings have relevant implications, as degrade the polymer (chitosans) or stimulate DNA replication [10], [18].

There are relevant systems that exhibit either attractive or repulsive interactions between bound ligands (cooperativity), which perturb the binding process. They include: The binding of ionic surfactants to the charged linear polyelectrolyte changing its physical and chemical properties [19]. The binding of lysozyme to chitosans mediating its degradation [14]. The binding of a variety of proteins and intercalating agents to nucleic acids, relevant for their maintenance [1], [9], [17], [20], [21], [22], [23], [24], [25]. In particular, a binding mode of E. Coli SSB to DNA relevant for DNA replication [1], [2], [26]. In contrast, there are systems in which the ligands do not have mutual interactions (non-cooperative binding), like the binding of oligolisines to nucleic acids, mono- and multivalent ions to polyelectrolytes, or polypeptides to polynucleotides; which change the polymer physical and chemical properties [27], [28], [29]. Here, we generalize a non-cooperative kinetic model to the cooperative case.

Single-molecule measurements allow very detailed tracking of binding processes, calling for more accurate models. In particular, recently, the binding equilibrium and kinetics of human mitochondrial and E. Coli SSBs to ssDNA have been studied using optical tweezers [5], [26]. As SSB is one of the main components of the replisome, the characterization of the SSB to ssDNA is key to understanding the DNA replication [10]. These studies call for accurate kinetic models for high coverage situations. Although the equilibrium for binding large ligands to long polymers has been studied in detail, it is not the case for the kinetics, as discussed below.

The equilibrium state of large ligands bound to macromolecules has been addressed in detail. Many equilibrium models are performed in the framework of statistical mechanical lattice models. In this type of models, the macromolecule is commonly considered as a one-dimensional lattice with a given number of potential binding sites (homogeneously or heterogeneously distributed), and constituted by elementary units (e.g., monomers for polymer, basepairs for DNA), and any bound ligand molecule occupies a certain number of elementary units. Several methods of constructing lattice models of ligand binding to macromolecules have been formulated. There are combinatorial methods that use binomial formulae to derive analytical expressions for the numbers of possible rearrangements of ligands along the macromolecule [16], [23], [19], [20], [21], [30], [31], [32]. In other type of methods, the system is characterized by a mathematical expression related to the partition function. The elementary units of the macromolecule are associated with different states (e.g., bound/unbound). The states of the whole system can be calculated as different combinations of states of the elementary units. The binding probabilities calculated for the lattice models are then equal to the fraction of molecules in a given state within the statistical ensemble consisting of many identical systems. In a statistical ensemble, each state of the system is characterized by a weight. The partition function is equal to the sum of the weights of all possible states [24], [33], [34], [35]. The transfer matrix method is based on the construction of transfer matrices for each elementary unit, whose elements contain the probabilities to find the lattice unit in a given state [24], [36], [37].

These equilibrium studies can be extended to kinetics models [38], [39], using using the McGhee and von Hippel detailed binding site counting procedure [30]. Recently, in Ref. [38], we clarified the non-cooperative kinetic equation, equilibrium coverage (McGhee and von Hippel result [30]) and its chemical potential (which matches the Tonks gas chemical potential [40], [41]). We also showed the limited validity of a simple coarse-grained kinetic model and of the ideal gas kinetics model [4], which are only valid for low and very low coverages, respectively. Here, we derive the cooperative binding kinetic equation, extending our approach [38] to the cooperative case (where the ligand affinity is modified by the presence of other ligands bound to the polymer), and deriving the approximate kinetic equations at the large ligand size limit at the leading and next-to-leading order.

In Subsection II.A, we briefly summarize the main results of the model for non-cooperative ligand binding [38]. In addition, for non-cooperative binding, we compute the next to leading order in the large ligand approximation (i.e., large number of binding sites per ligand), which is useful for comparison with the cooperative results. In Section II.B, we extend the model for cooperative ligand binding. In section II.C, we extend the results for the more general case in which the cooperativity can affect both the binding and the release rates. Finally, Section III discusses the results.

Ligand binding kinetic equation

We derive the kinetic equation for the binding of large ligands to a long polymer using the procedure proposed by McGhee and von Hippel to count the possible binding sites [30]. The polymer is represented by a linear array of N identical binding sites, and when the ligand molecule binds to the polymer covers m sites (i.e., make inaccessible to another ligand). In addition, the polymer is assumed to be infinitely long, i.e., . The fraction of polymer binding sites covered by the ligand is given by the coverage, , where is the number of ligands bound to the polymer.

Non-cooperative binding

This subsection reviews previous results for non-cooperative binding [30], [38]. We additionally derive the next-to-leading order in large ligand approximation (large ). All these results are useful for comparison with the cooperative binding results presented in the next subsections.

The kinetic equation describing the time variation in the number of ligands bound to the polymer is [38]

where is the average number of free binding sites per gap, the number of gaps, is the binding kinetic constant, and is the release kinetic constant. The binding kinetic constant generally depends on the free ligand concentration , and usually linearly , where K represents the association constant. A gap (of size ) is a set of consecutive free ligand binding sites (free lattice residues). Ligands cannot bind to gaps smaller than its size, .

Following the McGhee and von Hippel reasoning, the average number of free binding sites per gap is

where P is the probability that any particular gap between two bound ligands is exactly g free polymer residues long, which is given by [30]

(See Appendix for the derivation). For an infinite polymer, , the average number of free ligand binding sites per polymer [using Eqs. (2), (3)] turns out to be

where we have assumed that (reasonable for an infinite polymer). This result allows to write the kinetic equation in terms of the coverage as

This equation implies that, if we start with a naked polymer, the coverage will increase until it reaches an equilibrium value where the binding and release terms balance each other. See Fig. 1(Left). The equilibrium coverage can be obtained numerically from Eq. (5) imposing . It is found that it increases as a function of the ratio between the binding and release rates, . See Fig. 2. On the other hand, the dissociation kinetics is modelled using Eq. (5) by setting (simulating setting the ligand concentration in solution to zero). Fig. 1(Right) shows the dissociation kinetics starting with a polymer occupied with the corresponding equilibrium state coverage in each case.

Fig. 1

Attachment and detachment kinetics. Polymer coverage as a function of time compared for non-cooperative () and cooperative () cases. (Left) Starting with a naked polymer, ligands binding to m = 30 sites, binding rate kb = 0.8 s−1, and release rate kr = 0.06 s−1. (Typical values of SSB binding to ssDNA [5], [10], [12]). (Right) Starting with the previous equilibrium coverages the detachment dynamics is induced setting , simulating setting ligand concentration in solution to zero. The non-cooperative (ω = 1) dynamics is computed from Eq. (5). The cooperative (attractive ω = 10, and repulsive ω = 0.1) dynamics are computed from Eq. (20), and from Eq. (40) for the extreme cases of cooperativity only affecting the binding (α = 1) or only affecting the release ). The activation state parameter measures the impact of cooperativity on the activation energy of the binding process, as introduced below in Eq. (31) of Section II.C.

Fig. 2

Equilibrium coverage as a function of the ratio of the binding and release rates, , compared for the non-cooperative and cooperative cases for a ligand binding to m = 30 sites. The non-cooperative (ω = 1) equilibrium is computed from the stationary state of Eq. (5), while the cooperative (attractive ω = 10, and repulsive ω = 0.1) equilibria are computed from the stationary state of Eq. (20).

Small ligand binding

Single binding ligand (m = 1)

When the ligand binds to a single site the kinetic equation, Eq. (5), simplifies to

The first term provides a coverage increase proportional to the non-covered fraction of the polymer, and the second term decreases the coverage proportionally to the coverage. This results in an equilibrium coverage, , obtained for

or more explicitly,

where the different ways of expressing the result are useful for comparison with other results. Please, note that Eq. (8) is equivalent to classical Langmuir absorption model [42].

Large ligand binding (m ≫ 1)

There are relevant cases where the ligand is large and binds to many binding sites, like E. Coli and human mitochondrial SSB that bind 30–70 nucleotides of ssDNA [2], [5], [17]. For applications to these cases, the large ligand approximation is useful, , which accounts to consider that the ligand binds to many binding sites, further simplifying the kinetic equations.

Leading order for large ligands (m ≫ 1)

At leading order in the large limit

Then, the kinetic equation becomes

which shows that the binding is inhibited by the additional exponential factor (not present in the case). This factor accounts for the reduction in available binding possibilities, due to the random location of the large ligands that decrease the gaps of size greater than .

The previous kinetic equation leads to the equilibrium coverage, ,

or equivalently,

where W(z) is the so-called Lambert function, which is the solution of the equation , where z is a complex number. As Lambert function is an increasing function on its argument, this implies that coverage increases for increasing binding mode provide the ligand (affinity) stays constant. See Fig. 3. While if the affinity is per site, we have that the affinity for the mode , , is related to the affinity of the mode , , by . In this later case, the coverage increases with , when (as derived from in the large limit, while decreases with , if ).

Fig. 3

Comparison of exact and large m approximation results. (Left) Equilibrium coverages as a function of the ligand size m. (Right) Relative error isolines of the next-to-leading order in the large m approximation. The results for the non-cooperative case are computed from the stationary state of Eq. (5) (ω = 1, non-cooperative), Eq. (14) (ω = 1, next-to-leading), and Eq. (10) (ω = 1, leading). The results for the cooperative cases (ω = 5 and ω = 2) are computed from the stationary state of Eq. (40), and their next-to-leading approach from Eq. (53), while their leading approach would coincide with the non-cooperative leading approach results (ω = 1, leading), as derived in the text. The results in this figure are computed using K = 1.

Next-to-leading-order for large ligands (m ≫ 1)

Improving the approximation going to next-to-leading order in large gives a useful expression for later comparison. We have

Then, the kinetic equation for the coverage becomes

The equilibrium coverage can be obtained from Eq. (14) imposing ,

The last term provides the next-to-leading order correction in the 1/m expansion to the equilibrium coverage. We can compute this correction perturbatively substituting in Eq. (15) the expression

is the solution for the kinetic equation in a non-cooperative system at the leading order in large , Eq. (11). is the first-order correction in , whose solution is

This leads to an equilibrium coverage of

where the second bracket gives the next-to-leading order correction in the large expansion. As the Lambert function is positive (for positive arguments) the next-to-leading order correction increases the estimate of the equilibrium coverage in a relative amount of the order .

Cooperative binding case

Cooperative effects are present in many systems, where the (direct or indirect) interactions between ligands bound to the polymer are non-negligible [33], [36], [22], [23], [24]. Cooperative effects are known to be present in one of the two main binding modes of E. Coli SSB to DNA [2], [17].

To analyze the cooperative effects we follow here the McGhee and von Hippel approach [30], [38]. The interaction between ligands is allowed between nearest neighbors when they are bound without intervening free lattice residues. This restriction results in three different types of ligand binding sites: (i) an isolated site to which a ligand binds with a binding constant, k; (ii) a singly contiguous site to which a ligand binds with binding constant k ω; and (iii) a doubly contiguous site to which a ligand binds with binding constant k ω2 (see Fig. 4). The cooperativity parameter ω is the equilibrium constant for the process of moving a bound ligand from an isolated site to a singly contiguous site, or from a singly contiguous site to a doubly contiguous site. (See Appendix). For ω > 1, the ligands attract each other, and the binding is positively cooperative; for ω < 1, the ligands repel each other, and the binding is negatively cooperative; for ω = 1, the binding is non-cooperative.

Fig. 4

Binding and release processes to isolated, singly contiguous (sc), and doubly contiguous (dc) sites (respectively).

In a cooperative-ligand system, the kinetic equation describing the time variation in the number of ligands bound to the polymer can be written as

where, as previously noted, we have assumed that for an infinite polymer. We defined as the average number of free isolated binding sites per gap, as the average number of free singly contiguous binding sites per gap, and as the average number of free doubly contiguous binding sites per gap. (See Appendix for the derivation of the corresponding average number of sites). Following McGhee and von Hippel reasoning, the equation describing the time variation in the coverage becomes

where the function R is defined as follows

It is noted that the Eq. (20) cannot be applied directly in the case of non-cooperative binding because it diverges when the cooperativity parameter equals unity. However, we verified that, using L'Hôpital's rule and the properties of limits, Eq. (20) becomes Eq. (5) in the limit ω → 1, as expected.

Fig. 1 shows the evolution of the coverage in a cooperative-ligand system, which has been calculated numerically using Eq. (20) and compares it with the non-cooperative ligand system.

From Eq. (20), when  = 1 the kinetic equation describing the time variation in the coverage turns out to be

where we see that increasing the cooperativity parameter increases the binding affinity of the ligand. The equilibrium coverages can be obtained from Eq. (22), by imposing , as

The equilibrium coverage equation for the non-cooperative case, Eq. (7), is consistently recovered for .

As previously commented, a large ligand can bind to many binding sites (e.g., SSB to ssDNA). This enables us to approximate the expression assuming , for the factors in Eq. (20). We have that

and

These approximations lead to an approximate kinetic equation for the coverage with large ligands (large )

When only the leading order terms in large are retained, the cooperative kinetic equation [Eq. (26)] reduces to the non-cooperative leading order equation, Eq. (10). This result implies that the equilibrium coverage is also the same at leading order in large . Cooperative effects are only present at next-to-leading order in large . This indicates that isolated binding sites are the dominant contribution for large ligands.

We extend here the previous results up to next-to-leading-order in large m. We now conserve first-order terms in (and neglect higher orders) to obtain from Eq. (26) the kinetic equation

For , we consistently recover the analogous non-cooperative result, Eq. (14). For , the cooperativity increases the effective binding rate (first term of the right-hand side of the equation), while for , the effective binding rate is reduced.

The equilibrium coverage is given by the stationary state of Eq. (27) (i.e., ),

This equation gives the equilibrium coverage for cooperative large ligands. We solve it perturbatively, using , and found

Thus, the coverage for cooperative large ligands is given by

For , the analogous non-cooperative equilibrium equations are recovered, Eq. (15), (17), (18). Positive cooperativity () increases coverage, while negative cooperativity () decreases coverage, as Eq. (30) shows. This result is consistent with the attractive effect of positive cooperativity because of the resulting increased effective binding energy. (The same argument in the opposite direction holds for negative cooperativity.)

Fig. 3 compares the previous approximations for equilibrium coverage. Interestingly, the next-to-leading order is an excellent approximation even for relatively low values of . Fig. 3 illustrates that positive cooperativity increases coverage. Note that, for intermediate cooperativity strength (e.g., ), coverage decreases first and after increases as a function of the ligand bind size . This slope change is due to the competition of two effects. On the one hand, in the absence of cooperativity, the coverage grows with ligand size . On the other hand, positive cooperativity () increases the coverage, thanks to the attractive interaction between ligands. However, this cooperativity-induced coverage increase is greater for small ligands, as there are more cooperative interactions per covered monomer (higher density of ligand–ligand interactions).

General cooperative case

In the previous subsection, we considered cooperative binding and we implicitly assumed that the cooperativity only affected the binding process. However, positive cooperativity can arrive by an enhancement of the binding or by an inhibition of the release (or even both). They correspond, respectively, to an increase in the binding rate or a decrease in the release rate (for the neighbour site). Analogously, negative cooperativity can arise as an inhibition of the binding or an enhancement of the release. In this subsection, we generalize the previous model to cooperative effects on both, binding and release processes.

General kinetic equation

A more general kinetic equation can be derived considering that cooperativity can affect the binding and release rates. Considering the case of singly contiguous ligands, Arrhenius-type equations were proposed for the binding kinetic constant, , and the release kinetic constant, ,

where is the Gibbs free energy of interaction between adjacent ligands in the polymer. The parameter , which is named as activation state parameter, measures the lowering of the energy barrier between the binding process and the release process. Dividing the kinetic constants, we obtain the equilibrium constant for the processes

Similarly, considering doubly contiguous ligands, the kinetic constants are

and the corresponding equilibrium constant is written as

Please, note that Eqs. (33), (36) are equivalent to Gibbs free energy isotherm equation. See Fig. 5 for examples of free energy landscapes giving these rates and kinetic constant relations.

Fig. 5

Free energy landscapes for singly contiguous cooperative binding processes compared to the non-cooperative reaction potential. For cooperative binding, two cases are shown, i.e., enhanced binding () and inhibited release (). In both cases the free energy of the final state is decreased by an amount . For enhanced binding the activation state also decreases its free energy by an amount ; thus, the energy barrier for the binding process is reduced, while the energy barrier for the release process remains the same. For inhibited release () the free energy of the activation state does not change; thus, the energy barrier for the binding process is the same as for the non-cooperative case, while the energy barrier for the release process is increased. Both cooperative cases have the same equilibria, as both have the same free energy in the initial and final states, but they differ in the kinetics.

As we noted before, the cooperativity parameter ω is the equilibrium constant for the process of moving a bound ligand from an isolated site to a singly contiguous site,

This allows to restate all the equilibrium constants, and binding and release rates for the cooperative kinetics in terms of their non-cooperative analogs and the model parameters

Hence, the time variation in the number of ligands bound to the polymer (kinetic equation) is now given by

where we define as the average fraction of isolated attached ligands in the polymer, as the average fraction of attached ligands with a single contiguous ligand in the polymer, and as the average fraction of attached ligands with two contiguous ligands in the polymer. (See Appendix for the derivation of the corresponding expressions) The first term of the right-hand side of Eq. (39) is related to the ligand binding process to the polymer, while the second term of the right-hand side is related to the ligand releasing process.

After computing these average fractions (, see Appendix), we can express the kinetic equation [Eq. (39)] in terms of the conditional probabilities,

The conditional probability is the probability that, when you are in a free binding site, finding that the next binding site is also free, while is the probability that, when you are in the binding site bound to the rightmost extreme of the ligand, the next binding site is free. See Fig. 6. These two conditional probabilities are related by

Fig. 6

Schematics of the conditional probabilities used to describe their distribution illustrated for the binding of ligands with binding mode bound to a chain of . Figure from Ref. [38].

(see Appendix), and we have

where R is given by

[Please note that the denominator of Eq. (41) never becomes zero, . is a probability between 0 and 1, and both terms never become zero simultaneously. Also note that to recover the non-cooperative case from Eq. (42) we have to follow the procedures described below Eq. (21).]

The general kinetic equation is given by Eq. (40) with the magnitudes given by Eqs. (41) to (43). The stationary state () of the kinetic equation gives the equilibrium coverage relation

where is given by Eq. (42). This implicit expression can be solved numerically to obtain the equilibrium coverage . It is very relevant to note that this result implies that the equilibrium coverage is independent of , i.e., in the modification of the height of the activation barrier for singly or doubly contiguous binding processes.

The kinetic of two cases is represented in Fig. 1. First, cooperativity due to pure binding enhancement, which corresponds to and recovers the results presented in previous subsection, Subsection II.B. Second, cooperativity due to pure release inhibition (). Pure binding enhancement has a faster binding kinetics (Fig. 1). Pure release inhibition starts with a coverage kinetics like the non-cooperative case but continues to grow until it reaches the same final equilibrium coverage, as the pure binding enhancement case with the same cooperative free energy. See also Fig. 5 for a schematic representation of the respective free energy landscapes.

For , the kinetic equation [Eq. (40)] simplifies to

with the following conditional probabilities

where the function R is given by

Analogous equations with a different notation were derived and compared with Monte Carlo simulation in Ref. [39].

The stationary state condition for the equilibrium coverage is given in terms of the conditional probability , for , as

This implicit expression can be solved analytically to obtain the equilibrium coverage , as expressed by Eq. (23).

Under the assumption of a ligand binding to many binding sites, , the first term of the right-hand side of the general kinetic equation, Eq. (40), can be written as (See Appendix for derivation)and the second term of the right-hand side of the Eq. (40) as

Leading order in the large ligand expansion (m ≫ 1)

If only the leading-order () contributions are retained from Eqs. (50), (51), the general cooperative kinetic equation, Eq. (40), becomes identical to Eq. (10), which is the non-cooperative kinetic equation at leading order in the large expansion. Therefore, when large size ligands are involved, the equation describing the time variation in the coverage is independent of both the parameter ω and the parameter . Consequently, the equilibrium coverage can also be expressed by equations (11) or (12), of the non-cooperative case. This generalizes the result presented in Section II.B.2.1, showing that also in the general cooperative case the cooperative effects are absent in the leading order in the large ligand size expansion. Cooperative effects only appear in the effective binding and release rate at the next-to-leading order as we show just below.

Next-to-leading-order in the large ligand expansion (m ≫ 1)

Retaining the next-to-leading-order terms of Eqs. (50), (51), the kinetic equation, Eq. (40), becomes

Thus, the equilibrium coverage, , in the limit of large , is given by

[See Eq. (A40) of the Appendix and the previous definitions and derivations.] The equilibrium coverage is then independent of the parameter [consistently with the general result in Eq. (44)], and the equilibrium coverage computed previously for the case of is recovered, Eq. (28) in Section II.B.2.2. Instead, the kinetic equation, Eq. (52), explicitly depends on the parameter . This implies that the binding kinetics is faster for enhanced binding, [recovering Eq. (27)], than for inhibited release, , giving

The kinetic of these two cases are compared in Fig. 1 using the full kinetic equation, Eq. (40). Fig. 1 illustrates that both reach the same equilibrium coverage in a different time interval. Faster/slower roles are inverted for detachment.

Discussion

Making an accurate account of the potential binding sites, we have derived the kinetic equation for the cooperative binding of ligands to long linear polymers. We derived a general kinetic equation considering that cooperativity can affect the binding and release rates. Thus, positive cooperativity emerges by an enhancement of the binding or by an inhibition of the release. Negative cooperativity arises as an inhibition of the binding or an enhancement of the release. We found that positive and negative cooperativity gives different kinetics, but the equilibrium state is independent of the origin of cooperativity and recovers the McGhee and von Hippel’s equation for the coverage. In addition, the results show that the equilibrium coverage is increased significantly with increasing positive cooperativity (see Fig. 3). The kinetic equations presented are valid for the global coverage of long polymers. Distant regions of long polymers are nearly independent, making global coverage an effective average over realizations.

For large ligand size, the cooperativity effects are noticeably reduced, and the equilibrium coverages obtained in different scenarios are similar (see Fig. 3). In the large ligand size expansion, the cooperative effects are absent in the leading order for both the general cooperative case and McGhee and von Hippel’s approach. Thus, kinetics and equilibrium coverages are independent of cooperativity and activation state parameters. We found that cooperative effects only appear in the effective binding and release rate at the next-to-leading order, which means that isolated binding sites are the dominant contribution for large ligands. Within McGhee and von Hippel’s approach, the next-to-leading order contribution shows that positive cooperativity increases coverage, while negative cooperativity decreases coverage. Within our approach, the next-to-leading order contribution shows that the equilibrium coverage is independent of the activation state parameter, but the binding kinetics is faster for enhanced binding, than for inhibited release, .

The derived cooperative kinetic equation provides the adequate theoretical framework required for the description of the ligand binding processes observed with single-molecule techniques. In particular, recently these techniques have allowed to measure the real-time kinetics of SSB to ssDNA [5], [26], including E. Coli SSB, which has a cooperative biding mode. However, already for the non-cooperative case it has been shown [38] that previous simplified kinetic models for these systems [4] are only accurate for low coverage. The key is the detailed counting of potential binding sites. This detailed counting of binding sites was done for the non-cooperative case in Ref. [38], following Ref. [30], allowing to obtain an accurate kinetic equation. Here, we extended this detailed account to obtain an accurate cooperative kinetic equation, allowing more accurate descriptions of cooperative ligand binding processes for medium and high coverage situations. For example, for the typical biding affinities found for SSB to ssDNA, the simplified models predict the full coverage of ssDNA [5], [26], while the more accurate models generalized here predict coverage from 80% to 90% (depending on the ligand affinity), i.e., a 10% to 20% error correction. This predicted (and observed) decrease in coverage has also conceptual implications, as allow, in principle binding of other small ligands. Our results from the model presented here stresses that SSB positive cooperativity increases the ssDNA coverage at equilibrium, and during the whole dynamics (provided the cooperative mechanism is enhancement of the binding).

In the application of the present model, the ligand size is given by the number of occluded monomers. This comprises the bound monomers and the additional occluded monomers due to steric hindrance (between ligands). One ligand can bound in different modes, i.e., occluding different number of monomers, depending on its concentration and salt concentration in the solution. For example, human mitochondrial and E. Coli SSB have been reported to bound, both in a low binding mode ( nucleotides) or in high binding mode ( nucleotides), depending on the SSB and salt concentrations on solution [5], [17]. Understanding this mode selection and the involved mode competition is a relevant open question, which we plan to address in the future extending the kinetic model presented here.

Effects of conformational coiling and steric hindrance of the polymer are beyond the scope of the present study. These effects will reduce the effectively available number of gaps or decrease the binding affinity, and might be relevant for some applications.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.