Cooperative binding of n-mers with steric hindrance to finite and infinite one-dimensional lattices

Cooperative ligand binding to a double-stranded Ising lattice-Application to cofilin binding to actin filaments

Cooperative ligand binding to linear polymers is fundamental in many scientific disciplines, particularly biological and chemical physics and engineering. Such ligand binding interactions have been widely modeled using infinite one-dimensional (1D) Ising models even in cases where the linear polymers are more complex (e.g. actin filaments and other double-stranded linear polymers). Here, we use sequence-generating and transfer matrix methods to obtain an analytical method for cooperative equilibrium ligand binding to double-stranded Ising lattices. We use this exact solution to evaluate binding properties and features and analyze experimental binding data of cooperative binding of the regulatory protein, cofilin, to actin filaments. This analysis, with additional experimental information about the observed bound cofilin cluster sizes and filament structure, reveals that a bound cofilin promotes cooperative binding to its longitudinal nearest-neighbors but has very modest effects on lateral nearest-neighbors. The bound cofilin cluster sizes calculated from the best fit parameters from the double-stranded model are considerably larger than when calculated with the 1D model, consistent with experimental observations made by electron microscopy and fluorescence imaging. The exact solution obtained and the method for using the solution developed here can be widely used for analysis of variety of multistranded lattice systems.

Competition between Tropomyosin, Fimbrin, and ADF/Cofilin drives their sorting to distinct actin filament networks

The fission yeast actin cytoskeleton is an ideal, simplified system to investigate fundamental mechanisms behind cellular self-organization. By focusing on the stabilizing protein tropomyosin Cdc8, bundling protein fimbrin Fim1, and severing protein coffin Adf1, we examined how their pairwise and collective interactions with actin filaments regulate their activity and segregation to functionally diverse F-actin networks. Utilizing multi-color TIRF microscopy of in vitro reconstituted F-actin networks, we observed and characterized two distinct Cdc8 cables loading and spreading cooperatively on individual actin filaments. Furthermore, Cdc8, Fim1, and Adf1 all compete for association with F-actin by different mechanisms, and their cooperative association with actin filaments affects their ability to compete. Finally, competition between Fim1 and Adf1 for F-actin synergizes their activities, promoting rapid displacement of Cdc8 from a dense F-actin network. Our findings reveal that competitive and cooperative interactions between actin binding proteins help define their associations with different F-actin networks.

DOI:http://dx.doi.org/10.7554/eLife.23152.001

Cells use a protein called actin to provide shape, to generate the forces needed for cells to divide, and for many other essential processes. Inside a cell, individual actin proteins join up to form long filaments. These actin filaments are organized in different ways to make networks that have distinct properties, each tailored for a specific process. For instance, bundles of straight actin filaments help a cell to divide, whereas a network of branched actin filaments allows cells to move.

The different proteins that bind to actin filaments influence how quickly actin filaments are assembled and organized into networks. Therefore, many of the properties of an actin filament network are due to the actin binding proteins that are associated with it. Two actin binding proteins called fimbrin and cofilin associate with a type of actin filament network known as the actin patch. A third actin binding protein called tropomyosin associates with a different network that forms a ring. It is not known how particular actin binding proteins choose to associate with one actin network instead of another.

Christensen et al. used a fluorescence microscopy technique to study how fimbrin, cofilin and tropomyosin associate with different actin networks in a single-celled organism called fission yeast. This technique involved incubating actin and actin binding proteins together in a microscope chamber. The experiments show that some actin binding proteins, like tropomyosin, cooperate to bind to actin. Individual tropomyosin molecules find it difficult to bind actin filaments on their own, but once one tropomyosin molecule is attached to the filament, others rapidly join to coat the filament.

On the other hand, some actin-binding proteins compete for binding to filaments. For example, the binding of fimbrin to actin filaments causes tropomyosin to be removed from the actin network. Further experiments revealed that fimbrin and cofilin work with each other to rapidly generate a dense actin network and displace tropomyosin. Together, the findings of Christensen et al. suggest that competitions between actin binding proteins determine which actin binding proteins are associated with an actin network.

The next challenge is to understand how the most competitive actin-binding proteins are kept off actin networks where they do not belong. Further studies will shed light on how these interactions cause large changes in how the cell is organized.

DOI:http://dx.doi.org/10.7554/eLife.23152.002

A nucleation–growth process on the integers

We consider a Markov process with state space {0, 1}Z where Os become 1s irreversibly at rates which depend on whether none of a 0&s nearest neighbours (nucleation), its left-hand neighbour (right-hand growth), or its right-hand neighbour (left-hand growth) is in the 1-state. Furthermore, we assume that Os with both nearest neighbours in the 1-state remain in the 0-state forever and that at time 0 only the origin is in the 1-state. We determine the size distribution of the cluster (maximal sequence of 1s uninterrupted by Os) at the origin and the distribution of the time when its growth is stopped (by birth (nucleation) or competitive growth of neighbouring clusters). In the final state of the process, the spatial distribution of the (trapped) sites in the 0-state is considered. Some information on the size distribution of clusters well away from the cluster at the origin is obtained.

A nucleation–growth process on the integers

We consider a Markov process with state space {0, 1} Z where Os become 1s irreversibly at rates which depend on whether none of a 0&s nearest neighbours (nucleation), its left-hand neighbour (right-hand growth), or its right-hand neighbour (left-hand growth) is in the 1-state. Furthermore, we assume that Os with both nearest neighbours in the 1-state remain in the 0-state forever and that at time 0 only the origin is in the 1-state. We determine the size distribution of the cluster (maximal sequence of 1s uninterrupted by Os) at the origin and the distribution of the time when its growth is stopped (by birth (nucleation) or competitive growth of neighbouring clusters). In the final state of the process, the spatial distribution of the (trapped) sites in the 0-state is considered. Some information on the size distribution of clusters well away from the cluster at the origin is obtained.

Calculating transcription factor binding maps for chromatin

Current high-throughput experiments already generate enough data for retrieving the DNA sequence-dependent binding affinities of transcription factors (TF) and other chromosomal proteins throughout the complete genome. However, the reverse task of calculating binding maps in a chromatin context for a given set of concentrations and TF affinities appears to be even more challenging and computationally demanding. The problem can be addressed by considering the DNA sequence as a one-dimensional lattice with units of one or more base pairs. To calculate protein occupancies in chromatin, one needs to consider the competition of TF and histone octamers for binding sites as well as the partial unwrapping of nucleosomal DNA. Here, we consider five different classes of algorithms to compute binding maps that include the binary variable, combinatorial, sequence generating function, transfer matrix and dynamic programming approaches. The calculation time of the binary variable algorithm scales exponentially with DNA length, which limits its use to the analysis of very small genomic regions. For regulatory regions with many overlapping binding sites, potentially applicable algorithms reduce either to the transfer matrix or dynamic programming approach. In addition to the recently proposed transfer matrix formalism for TF access to the nucleosomal organized DNA, we develop here a dynamic programming algorithm that accounts for this feature. In the absence of nucleosomes, dynamic programming outperforms the transfer matrix approach, but the latter is faster when nucleosome unwrapping has to be considered. Strategies are discussed that could further facilitate calculations to allow computing genome-wide TF binding maps.

Statistical-mechanical lattice models for protein-DNA binding in chromatin

Statistical-mechanical lattice models for protein-DNA binding are well established as a method to describe complex ligand binding equilibria measured in vitro with purified DNA and protein components. Recently, a new field of applications has opened up for this approach since it has become possible to experimentally quantify genome-wide protein occupancies in relation to the DNA sequence. In particular, the organization of the eukaryotic genome by histone proteins into a nucleoprotein complex termed chromatin has been recognized as a key parameter that controls the access of transcription factors to the DNA sequence. New approaches have to be developed to derive statistical-mechanical lattice descriptions of chromatin-associated protein-DNA interactions. Here, we present the theoretical framework for lattice models of histone-DNA interactions in chromatin and investigate the (competitive) DNA binding of other chromosomal proteins and transcription factors. The results have a number of applications for quantitative models for the regulation of gene expression.

Allosteric conformational spread: exact results using a simple transfer matrix method

A transfer matrix method is described for the conformational spread (CS) model of allosteric cooperativity within a one-dimensional arrangement of four-state binding sites. Each such binding site can realize one of two possible conformational states. Each of these states can either bind ligand or not bind ligand. Thus, analytical expressions that are exact within the context of the CS model are derived for the grand partition function, for the mean fraction of binding sites occupied by ligand versus ligand concentration, and for the mean fraction of binding sites in a given allosteric state versus ligand concentration. The utility of our analytical results is demonstrated by least-mean-square fitting of prior experimental results obtained on the bacterial flagellar motor for the fraction of FliM/FliG/FliN complexes with CheY-P bound [V. Sourjik and H. C. Berg, Proc. Natl. Acad. Sci. U.S.A. 99, 12669 (2002)] and for the cw bias [P. Cluzel, Science 287, 1652 (2000)], which plausibly may be identified as the fraction of protomers realizing state 2. Finally, the relationships between our analytical results and the classical Monod-Wyman-Changeaux, Koshland-Nemethy-Filmer, and McGhee-Von Hippel treatments of allosteric cooperativity are elucidated, as is the connection to an earlier approximate analytical treatment of the CS model.

Cooperative binding of tropomyosin to actin

Tropomyosin molecules attach to the thin filament conjointly rather than separately, in a pattern indicating very high cooperativity. The equilibrium process drawing tropomyosins together on the actin filament can be measured by application ofa linear lattice model to bindingisotherm data and hypotheses on the mechanism of cooperativity can be tested. Each end of tropomyosin overlaps and attaches to the end ofa neighboring tropomyosin, facilitating the formation of continuous tropomyosin strands, without gaps between neighboring molecules along the thin filament. Interestingly, the overlap complexes vary greatly in size and composition among tropomyosin isoforms, despite consistently cooperative binding to actin. Also, the tendency of tropomyosin to bind to actin cooperatively rather than randomly does not correlate with the strength ofend-to-end binding.By implication, tropomyosin's actin-binding cooperativity likely involves effects on the actin filament, as well as direct interactions between adjacent tropomyosins.

A note on the quantitative properties of McGhee-von Hippel model

Exact closed-form expressions are presented for the properties of the McGhee-von Hippel model of non-specific binding of large ligands to one-dimensional homogeneous lattices. These properties include the midpoint location and the slope at the middle point of the binding isotherms (v varies with ln L plots), the location and magnitude of the maximum, as well as the location of the inflection point, in the Scatchard plots (v/L varies with v plots).

Modulation of myosin function by isoform-specific properties of Saccharomyces cerevisiae and muscle tropomyosins

Tropomyosin is an extended coiled-coil protein that influences actin function by binding longitudinally along thin filaments. The present work compares cardiac tropomyosin and the two tropomyosins from Saccharomyces cerevisiae, TPM1 and TPM2, that are much shorter than vertebrate tropomyosins. Unlike cardiac tropomyosin, the phase of the coiled-coil-forming heptad repeat of TPM2 is discontinuous; it is interrupted by a 4-residue deletion. TPM1 has two such deletions, which flank the 38-residue partial gene duplication that causes TPM1 to span five actins instead of the four of TPM2. Each of the three tropomyosin isoforms modulates actin-myosin interactions, with isoform-specific effects on cooperativity and strength of myosin binding. These different properties can be explained by a model that combines opposite effects, steric hindrance between myosin and tropomyosin when the latter is bound to a subset of its sites on actin, and also indirect, favorable interactions between tropomyosin and myosin, mediated by mutually promoted changes in actin. Both of these effects are influenced by which tropomyosin isoform is present. Finally, the tropomyosins have isoform-specific effects on in vitro sliding speed and on the myosin concentration dependence of this movement, suggesting that non-muscle tropomyosin isoforms exist, at least in part, to modulate myosin function.

Sequence distribution and intercooperativity detection for two ligands simultaneously binding to DNA

A method for detecting and quantifying the cooperativity in the simultaneous binding of two ligands, A and B, to DNA (intercooperativity; omega(AB)) is proposed. This involves the determination of an apparent affinity constant K(app) for one of the ligands (A) in the limit of its null saturation (nu(Alpha) --> 0), in the presence of the second one (B). A definition for this constant is given and an expression is derived corresponding to a simple model of competitive binding to an unbranched three-state homogeneous polar lattice with nearest-neighbor interactions (Markov chain). The ratio between the apparent and intrinsic affinity constants of one ligand in the maximum saturation limit of the other one becomes omega(2)(AB), and thus can be graphically obtained from K(app)(A) vs nu(B) plots. All the frequencies that define the sequence distribution of ligands can be easily calculated by introducing some generalized statistical weights for the free lattice monomer in a standard sequence generating function procedure. A model of fluorescence quenching emission is obtained from such frequencies under the hypothesis of a short-range electron transfer mechanism of the deactivation; it confirms, as suggested by the binding model, an outstanding influence of the intercooperativity on the distribution.

Effects of tropomyosin internal deletions on thin filament function

Striated muscle tropomyosin spans seven actin monomers and contains seven quasi-repeating regions with loose sequence similarity. Each region contains a hypothesized actin binding motif. To examine the functions of these regions, full-length tropomyosin was compared with tropomyosin internal deletion mutants spanning either five or four actins. Actin-troponin-tropomyosin filaments lacking tropomyosin regions 2-3 exhibited calcium-sensitive regulation in in vitro motility and myosin S1 ATP hydrolysis experiments, similar to filaments with full-length tropomyosin. In contrast, filaments lacking tropomyosin regions 3-4 were inhibitory to these myosin functions. Deletion of regions 2-4, 3-5, or 4-6 had little effect on tropomyosin binding to actin in the presence of troponin or troponin-Ca(2+), or in the absence of troponin. However, all of these mutants inhibited myosin cycling. Deletion of the quasi-repeating regions diminished the prominent effect of myosin S1 on tropomyosin-actin binding. Interruption of this cooperative, myosin-tropomyosin interaction was least severe for the mutant lacking regions 2-3 and therefore correlated with inhibition of myosin cycling. Regions 3, 4, and 5 each contributed about 1.5 kcal/mol to this process, whereas regions 2 and 6 contributed much less. We suggest that a myosin-induced conformational change in actin facilitates the azimuthal repositioning of tropomyosin which is an essential part of regulation.

Ligand binding on ladder lattices

The ligand binding problems on two-dimensional ladders, which model many important binding phenomena in molecular biology, are studied in details. The model is represented by four parameters, the interactions between ligands when bound to adjacent sites on opposite legs of the ladder (tau), the interactions between bound ligands in the longitudinal direction of the ladder (sigma), the number of binding sites that are covered by a bound ligand (m), and the intrinsic binding constant (K). The partition functions of ring ladders are approached with the transfer matrix method. A general relation is derived which connects the partition function of a linear ladder with that of a ring ladder. The results obtained apply to the general situation of multivalent binding, in which m>1. Special attention is paid to the case where the ligand covers one site (m=1). In this case explicit formulas are given for the partition functions of ring and linear ladders. Closed-form expressions are obtained for various properties of the system, including the degree of binding (theta), the midpoint in the binding isotherm (1/square root(tau sigma)), the initial and end slopes of the Scatchard plots (2sigma + tau - 4 and -sigma2 tau, respectively). From these closed-form formulas, sigma and tau may be extracted from experimental data. The model reveals certain features which do not exist in one-dimensional models. Using the general method discussed in [1], the recurrence relation is found for the partition functions. The analytical solution found for this model provides test cases to verify the numerical results for more complex two-dimensional models.

Theory of multivalent binding in one and two-dimensional lattices

Ligand binding to a linear lattice composed of N sites, under general conditions of cooperativity and number of sites covered upon binding, m, is approached in terms of the theory of contracted partition functions. The partition function of the system obeys a recursion relation leading to a generating function that provides an exact analytical solution for any case of interest. Site-specific properties of the lattice are derived from simple transformations of the analytical expressions. The McGhee-von Hippel model is obtained as a special case in the limit N --> infinity. The derivation is straightforward and involves no combinatorial arguments. Partition functions and site-specific properties are also derived for the case of non-cooperative binding to a two-dimensional torus of length N, containing s sites in its section for a total of sN sites. The torus provides a relevant model for ligand binding to double-stranded DNA (s = 2) or protein helices (s = 3,4). It is proved that non-cooperative binding to the two-dimensional torus can mimic cooperative binding to a one-dimensional linear lattice when m = s. The dimensional embedding of the lattice and the geometry of interaction of its sites play a crucial role in defining the binding properties of the system accessible to experimental measurements. Hence, caution must be exercised in the interpretation of Scatchard plots in terms of the one-dimensional McGhee-von Hippel model, especially when m < or = 4 and the geometry of the system is clearly two-dimensional.

Map analysis of ligand binding to a linear lattice

A one-dimensional mapping of the binding properties of a linear lattice offers an exact analytical solution for the site-specific properties of the lattice once the length N and the parameter for nearest neighbor interactions are specified. The solution is derived independent of the definition of the partition function or the transfer matrix, nor does it involve combinatorial arguments. This result provides a simple and effective way of analyzing experimental data for protein-ligand interactions and broadens our understanding of site-specific properties in biological macromolecules.

NH2-terminal truncation of skeletal muscle troponin T does not alter the Ca2+ sensitivity of thin filament assembly

To investigate how Ca2+ binding to troponin C regulates muscle contraction, the Ca(2+)-sensitive properties of thin filament assembly were studied as the tropomyosin binding, NH2-terminal region of troponin T was progressively shortened. Troponin complexes were prepared that contained skeletal muscle troponin C, troponin I, and either intact troponin T (TnT) (residues 1-259) or fragment TnT-(70-259), TnT-(151-259), or TnT-(159-259). In the absence of Ca2+ their respective affinities for pyrene-labeled tropomyosin were 2.3 x 10(7) M-1, 1.2 x 10(7) M-1, 1.9 x 10(5) M-1, and 1.9 x 10(5) M-1. Ca2+ had only a small effect on these affinities: 1.1 x 10(7) M-1 for whole troponin, 2 x 10(5) M-1 for troponin-(151-259), and 2.8 x 10(5) M-1 for troponin-(159-259). Forms of troponin that bound weakly to tropomyosin in the absence of actin increased the actin affinity of tropomyosin only 2-3-fold, even in the absence of Ca2+; weak binding of troponin to tropomyosin correlated with weak effects on tropomyosin-actin binding. In contrast, whole troponin had an approximately 500-fold effect on tropomyosin binding to actin, regardless of whether Ca2+ was present. The small effect of Ca2+ on the energetics of thin filament assembly is not attributable to the amino-terminal region of troponin T. The results suggest that Ca2+ causes the interaction between actin and the globular region of troponin to switch between two energetically similar states.

Use of binding site neighbor-effect parameters to evaluate the interactions between adjacent ligands on a linear lattice. Effects on ligand-lattice association

A method using binding site "neighbor-effect" parameters (NEPs) is introduced to evaluate the effects of interaction between adjacent ligands on their binding to an infinite linear lattice. Binding site overlap is also taken into account. This enables the conditional probability approach of McGhee & von Hippel to be extended to more complex situations. The general equation for the isotherm is v/LF = SFKF, where v is the ratio of bound ligands to lattice residues, LF is the free ligand concentration, SF is the fraction of binding sites that are free, and KF is the average association constant of a free site. Solutions are derived for three cases: symmetric ligands, and asymmetric ligands on isotropic or anisotropic lattices. For symmetric ligands there is one NEP, E, which is the ratio of the average binding affinity of a free site if the status of the lattice residue neighboring one end of the site is unspecified (left to chance) to the affinity when this residue is free (holding the other neighbor constant). Thus KF is KE2, where K is the affinity of an isolated site. If a site is n residues long, SF is f ffn-1, where f = 1 - nv is the fraction of residues that are free and ff is the conditional probability that a free residue is bordered on a given side by another free residue. The expression for ff is 1/(1 + x/E), where x is v/f, E is (1 - x + [(1 - x)2 + 4x omega]1/2)/2, and omega is the co-operativity parameter. The binding of asymmetric ligands to an isotropic lattice is described by two NEPs; the last case involves four NEPs and a bound ligand orientation parameter. For each case, the expected length distribution of clusters of bound ligands can be calculated as a function of v. When Scatchard plots with the same intercepts and initial slope are compared, it is found that ligand asymmetry lowers the isotherm (relative to the corresponding symmetric ligand isotherm), whereas lattice anisotrophy raises it.

Thermodynamic methods for model-independent determination of equilibrium binding isotherms for protein-DNA interactions: spectroscopic approaches to monitor binding

The measurement of equilibrium binding constants for ligand-macromolecule interactions by monitoring a change in some spectral property of the ligand or the macromolecule is a common method used to study these interactions. This is due to the high sensitivity of the spectroscopic methods and general ease in applying these experimental procedures. In addition, binding can be monitored continuously, thus facilitating kinetic measurements. The main problem with these methods results from the fact that the spectroscopic signal is an indirect measure of binding, since the relationship between the change in the spectroscopic signal and the extent of binding is unknown, a priori. A common recourse is to assume a strict proportionality between the signal change and the fractional saturation of the ligand or macromolecule; however, it is often the case that such a direct proportionality does not hold. In this chapter we have reviewed the use of methods to analyze ligand-macromolecule equilibrium titrations that are monitored by indirect spectroscopic techniques. These methods of analysis yield thermodynamically rigorous, model-independent binding isotherms, hence assumptions concerning the relationship between the signal change and the extent of binding are not required. In fact, these methods can also be used to determine quantitatively the relationship between the signal change and the average degree of binding. In addition, the approaches discussed here are general and not limited to spectroscopic signals and therefore can be used with any intensive physicochemical property that reflects binding.

A general secular equation for cooperative binding of n-mer ligands to a one-dimensional lattice

The binding of n-mer ligands to a one-dimensional lattice involving many ligand species and complex multiple-binding mechanisms is studied. We show that, when derived using the sequence-generating function method of Lifson, the secular equation of any binding system with a finite number of "elementary units" can be expressed in a matrix determinant form that is very symmetric and easy to construct. In other words, for any binding system whose elementary units are known, the secular equation of the system can be obtained readily without going through the formal derivation of the equation. We also show that the "determinant" secular equation obtained using the present procedure can be employed directly to the calculation of binding isotherms.

On the cooperative binding of large ligands to a one-dimensional homogeneous lattice: the generalized three-state lattice model

We present the general secular equation for three-state lattice models for the cooperative binding of large ligands to a one-dimensional lattice. In addition, a closed-form expression for the isotherm is also obtained, that can be used with all values of the cooperativity parameter omega(0 less than omega less than infinity) thus eliminating the need for multiple equations.

Binding of n-mers to one-dimensional lattices with longer than close-contact interactions

A general formalism is derived for the evaluation of binding isotherms of n-mers (ligands) to one-dimensional polymers in the presence of ligand-ligand interactions which extend over several binding sites with distance-dependent interaction energies (multi-parameter model). This is an extension of the usual n-mer binding theory developed by several investigators in which ligand-ligand interaction occurs only when two ligands are in close contact (one-parameter model). The difference in binding isotherms between a one-parameter model and a multi-parameter model is studied numerically using the present formalism.

Limited co-operativity in protein-nucleic acid interactions. A thermodynamic model for the interactions of Escherichia coli single strand binding protein with single-stranded nucleic acids in the "beaded", (SSB)65 mode

We present a statistical thermodynamic model ("tetramer/octamer" model) that describes the equilibrium binding of the Escherichia coli single strand binding (SSB) protein to single-stranded nucleic acids in its "beaded" binding mode, which seems to be equivalent to the high site size, (SSB)65 binding mode. The method of sequence-generating functions is used to derive the model, which accounts for the observation that clustering of bound SSB tetramers is limited to the formation of octamers, which have been observed as "beads" in the electron microscope. The model also accounts for the overlap of potential protein binding sites on the nucleic acid. The "tetramer/octamer" model is fully described by only three parameters: the site size, n; the intrinsic equilibrium constant, K; and the co-operativity parameter, omega, and we obtain exact, closed form expressions for the binding isotherm as well as the distribution of DNA-bound SSB tetramers and octamers. The closed form expressions allow one to calculate easily average binding properties and analyze experimental binding isotherms to obtain estimates of K and omega. In order to test the tetramer/octamer model, we have determined the equilibrium binding isotherm for the E. coli SSB protein-poly(U) interaction in 0.2 M-NaCl over a wide range of binding densities. These are conditions in which the low co-operativity (SSB)65 binding mode solely exists. The tetramer/octamer model provides a much better description of the experimental isotherm over the entire binding density range than a model that assumes the formation of clusters of unlimited size. A co-operativity parameter of omega = 420 +/- 80 provides a good fit to data for SSB binding to poly(dA) and poly(U), corresponding to an interaction free energy of -3.6 kcal/mol of SSB octamer formed. On the basis of this moderate value of omega, the tetramer/octamer model predicts that at low to intermediate binding densities, a significant fraction of bound SSB exists in the form of tetramers co-existing with octamers. In the case of E. coli SSB protein binding in the "beaded", (SSB)65 mode this model provides a significant improvement over previous treatments which assume unlimited nearest-neighbor interactions, since the binding parameters, K and omega, represent physically meaningful interaction constants rather than fitting parameters.

Cooperative effects on binding of proteins to DNA

Equations are derived for description of cooperative binding of large ligands to a homogeneous polynucleotide lattice for a wide variety of binding models. Both short- and long-range interactions between nearest-neighbour bound ligands are taken into account. It is shown that cooperative binding of ligand at high levels of occupancy can be described with good accuracy by the equation derived for the noncooperative binding of the same ligand with an apparent binding constant Keff. A new method is proposed for the analysis of cooperative binding isotherms. It is based on a comparison of the asymptotic behavior of cooperative and noncooperative binding isotherms in the limit when the occupancy of lattice by ligand approaches the saturation level of binding. It is demonstrated that cooperative effects mediated by direct contact between bound ligands can be divided into two classes depending on whether dimeric species or aggregates of unrestricted size are formed by bound ligands on the lattice at high levels of occupancy. These two classes can be easily distinguished on strictly empirical grounds. In particular, if interligand interactions favor the formation of dimeric species on DNA, Keff approximately a 1/2 where a is the interligand interaction constant. If interligand interactions generate aggregates of unrestricted size, Keff approximately aL + 1 where L is the size of binding site for the ligand on DNA. We also demonstrate that cooperative systems in which interligand interaction extends over two or more free polymer residues can be distinguished from systems in which only short-range interactions mediated by direct contacts between bound ligands are allowed.

Co-operativity and enzymatic activity in polymer-activated enzymes. A one-dimensional piggy-back binding model and its application to the DNA-dependent ATPase of DNA gyrase

The binding of a ligand to a one-dimensional lattice in the presence of a second ("rider") ligand, which binds only to the first ligand (piggy-back binding), is studied. A model derived from this study is used to analyze the effects of co-operativity on the reaction rates of enzymes activated by polymeric cofactors that provide multiple binding sites for the enzyme. It is found that in the presence of strong co-operativity, the steady-state reaction rates of polymer-activated enzymes can be very different from the Michaelis-Menten paradigm. By adjusting the co-operativity parameters and the binding constants of the ligands, the model can generate apparent auto-catalytic enhancement by substrates at low substrate concentrations and apparent substrate inhibition at high substrate concentrations. The model is shown to be able to explain the differences in the rates of ATP hydrolysis by DNA gyrase in the presence of long versus short DNA molecules and in the presence of long DNA molecules at different gyrase to DNA ratios.