Theoretical Formulation of DNA Renaturation Kinetics

At a coarse grained level one can consider the c-ssDNA polymer as a chain of nitrogen base beads with an average bond length of 1 base. The process of renaturation essentially involves collisions between the clusters of nitrogen base beads corresponding to c-ssDNA strands. The basic steps involved in the process of renaturation of c-ssDNA strands viz. (a) formation of nonspecific contact (b) nucleation or correct contact formation and (c) zippering can be well represented by Scheme III of Fig 2. Here 3D diffusion controlled collisions between the nitrogen base beads of c-ssDNA strands lead to the formation non-specific contacts between them that results in the formation of cn-ssDNA [Figs 2A and 3A]. Under non-specifically bound condition, the probe strand of cn-ssDNA searches over the template strand for the correct contact via a combination of 1D and 3D diffusion. Here the 1D diffusion comprises of facilitating processes such as slithering and internal displacement dynamics. Slithering dynamics involves [Fig 3B] a random search with unit base step size whereas internal displacement mechanism involves an inchworm type movement of the probe strand over the template strand with a step size of few hundreds to thousand bases [Fig 3C]. One should note that internal displacement mainly depends on the 3D conformational state of the template c-ssDNA strand.

Condensed conformational state of template c-ssDNA always favours internal displacements while relaxed conformational state favours the slithering dynamics. Here slithering and internal displacement mechanism are similar to that of the sliding and intersegmental transfer dynamics as in case of site-specific DNA-protein interactions [Fig 3D]. Intersegmental transfer occurs whenever two distal segments of the same dsDNA chain come close by over 3D space via ring closure events. Under such conditions the non-specifically bound DNA binding protein molecules can move from one segment of DNA to another segment of the same DNA without dissociation. Analogous to intersegmental transfer, internal displacement occurs whenever the overhang parts of the probe strand of cn-ssDNA make contact with some other region of the template strand of cn-ssDNA without dissociation of the previous nonspecific contact. In this way the probe strand of cn-ssDNA can perform an inchworm type movement over the template strand of cn-ssDNA without physical dissociation [Fig 3C]. One should note that the probe strand of cn-ssDNA may dissociate after searching n numbers of nonspecific sites on the template strand where n is a random variable. These 1D random search processes in combination with several rounds of nonspecific association and dissociation will result in the finding of the correct-contact on the template strand of cn-ssDNA by the probe strand that in turn results in the formation of cc-ssDNA.

Upon formation of cc-ssDNA, the zipping reaction needs to progress against the entropic barrier associated with the single strand overhangs of cc-ssDNA. Nucleation occurs whenever the number of correct contacts in cc-ssDNA exceeds certain critical value (N) against the entropic barrier imposed by the freely moving single stranded overhangs [27]. Since the time that is required to locate the correct-contact is a random variable, nucleation rate also will be a random variable. Nucleation will be followed by zipping of cc-ssDNA into a completely renatured dsDNA. Here one should note that both the nucleation and zipping are parts of a continuous process after the formation of cc-ssDNA i.e. there is no clear cut timescale separation or boundary between them. Zipping step will also be a stochastic process since at each step of zipping there is a finite nonzero probability of dissociation of cc-ssDNA. When the c-ssDNA strands are repetitive then the zipping reacting can progress in parallel. As the zipping reaction progresses towards completion, in case of nonrepetitive c-ssDNA strands the stability of cc-ssDNA gradually increases and as a result the probability of dissociation of cc-ssDNA decreases.

In Scheme III of Fig 2 [S] and [S’] are the concentrations (mol/lit, M) of the colliding c-ssDNAs whose lengths are L bps (template) and l bps (probe) respectively where by definition L ≥ l, [X] is the concentration of c-ssDNAs with nonspecific contacts (cn-ssDNA), [Y_N] is the concentration of c-ssDNAs with correct-contact (cc-ssDNA) and nucleation, and [H] is the concentration of completely renatured dsDNA form. Here one should note that [Y_N] is a special form of [H] since the complementary strands are already nucleated and aligned in [Y_N]. Since it is very difficult to identify and quantity [Y_N] in experiments we consider [H] as the main product of renaturation in this paper. Further k_fQ (M^-1s^-1) and k_r (s^-1) are the forward second-order and reverse first-order rate constants associated with the formation of nonspecific contacts between colliding c-ssDNAs, k_N (s^-1) is the nucleation rate constant and k_Z (s^-1) is the zippering rate constant. Since [Y_N] is a hidden intermediate we consider the overall rate constant associated with both nucleation and subsequent spontaneous zipping as k_NZ which is the inverse of total time required for nucleation and zipping processes together i.e. k_NZ = 1/ (1/k_Z +1/k_N). Typical values of the size of nuclei [6] seems to be N ~ 4–7 bases. Here we set the subscript Q = C for condensed conformational state of c-ssDNA and Q = R for relaxed conformational state of c-ssDNA throughout this paper. The set of differential rate equations associated with Scheme III of Fig 2 can be written as follows.

(1)

Since nucleation step involves several rounds of 1D slithering and internal displacement dynamics of one of the cn-ssDNA strand over the other in combination with nonspecific association and dissociation events, k_NZ will be a function of n. Here n is the number of nonspecific sites scanned by cn-ssDNA strands on each other before dissociation. With this background one can define the overall second order rate constant as where we have defined k_HQ = k_fQk_NZ/(k_NZ + k_r) and p(n) is the probability density function associated with the random variable n. We will derive the explicit expressions for k_NZ and p(n) in subsequent sections. Various parameters and variables defined throughout this paper are summarized in Table 1. While deriving the expression for we have assumed that the nucleation and zipping are the rate limiting ones so that d[x]/dt ≃ 0 in the timescales of nucleation and zipping.

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Table 1. Various symbols and their definitions

https://doi.org/10.1371/journal.pone.0153172.t001

Essentially the first reaction in Scheme III of Fig 2 can be thought as collisions between the spatially distributed clusters of nitrogen bases corresponding to the two reacting c-ssDNAs as in case of a mean field approach [27]. As a result, the overall bimolecular rate associated with the formation of nonspecific contacts between spatially distributed base-clusters of c-ssDNAs is proportional of the product of concentrations of the total nitrogen bases in c-ssDNA molecules. The cylindrical surface area C_M ~ 2πr_DM of a c-ssDNA molecule will be confined within the spherical solvent shell with surface area (M = L for template and M = l for probe) where r_Q is the radius of gyration of the respective c-ssDNA molecule. Under strongly condensed conformational state of c-ssDNA one finds that V_M < C_M and when the DNA polymer is in a relaxed conformational state then one find that V_M > C_M. At a coarse grained level one can model the bases of c-ssDNA as a chain of spherical beads with radius r_D. Under relaxed conformational state all these nitrogen base beads are distributed on the surface of the spherical solvent shell that covers the entire c-ssDNA molecule [Fig 2B]. Whereas under condensed conformational state of template c-ssDNA significant fraction of nitrogen base beads will be inaccessible to the inflowing probe c-ssDNA molecules since they are buried inside the matrix of condensed c-ssDNA polymer [Fig 2C].

Calculation of the Nonspecific-Contact Formation Rate

Actually most of the theoretical models derived the scaling over the length of c-ssDNAs mainly from the fact that the radius of gyration (measured in bps) associated with reacting c-ssDNAs scales with their length as r_M ∝ M^α (M = L and l for the template and probe strands respectively). Here the value of the exponent 0 < α < 1 varies depending on the type of polymer and solvent conditions. For an ideal Gaussian chain polymer that is immersed in a theta solvent one finds that α ~ ½. For a linear chain polymer one finds that (M = L, l) where l_p is the average bond length [32, 33]. For c-ssDNA we have l_p ~ 1 base between nitrogen base beads. Noting these facts the Smolochowski type limiting rate for a diffusion controlled nonspecific contact forming step can be given as follows.

(2)

Here J_Q is the inflowing flux of c-ssDNA molecules, B_Q is the exposed total surface area on which the influx of c-ssDNA molecules J_Q is acting, D_S is the three-dimensional (3D) diffusion coefficient associated with the colliding c-ssDNAs and r_Q is the reaction radius that depends on the conformational state of c-ssDNA. When c-ssDNA strands are relaxed, then all the base beads will be distributed over the surface of the solvent shell that covers the entire c-ssDNA polymer. Since V_M > C_M for M = {L, l} there will be several patches on the solvent shell surface without base beads. Noting that a nonspecific contact can be formed only upon collision between probe and base beads of template, one needs to integrate over the surface of the template c-ssDNA polymer that is spread over the surface of spherical solvent shell rather than the entire surface of spherical solvent shell [Fig 2B]. In this situation one finds that B_R ≃ 2πr_RL where r_R = r_D + r_l is the reaction radius associated with the collision of the probe c-ssDNA strand on a nitrogen base bead of template c-ssDNA strand. On the other hand, when the c-ssDNA polymer is highly condensed then significant fraction of the nitrogen base beads will be buried inside the matrix of the DNA condensate [Fig 2C].

Unlike the relaxed conformational state, in case of condensed conformational state the surface of the spherical solvent shell that covers entire c-ssDNA strands will be filled with nitrogen base beads. This means that collision between base bead clusters of c-ssDNA strands always result in the formation of nonspecific contacts under condensed conformational state. In this condition one finds that . Here r_C = r_L + r_l is the reaction radius associated with the collision between the condensed nitrogen base bead clusters of template and probe c-ssDNA strands. Therefore depending on the conformational state of the colliding c-ssDNA molecules the bimolecular collision rate associated with the formation of nonspecific contact between the template and probe molecules can be written as follows.

Case I: Relaxed conformational state of c-ssDNA (3)

Case II: Condensed conformational state of c-ssDNA (4)

Here k_t ≃ (8k_BT/3η) is the Smolochowski type 3D diffusion controlled collision rate limit (M^-1s^-1) associated with the bimolecular collisions between the c-ssDNA molecules. Here k_B is the Boltzmann constant, η is the viscosity of the reaction medium and T is the absolute temperature in degrees K. Further the colliding molecules are assumed to be same in size with no charge on them. In general k_fQ ≃ k_tδ_Q where δ_R ≃ L(1/r_l + 1/r_L)/8 and δ_c ≃ (r_L + r_l)²/4r_Lr_l. When the colliding c-ssDNA strands are same in size then δ_R ≃ L/4r_L and δ_c ≃ 1. Noting that [32–33] for a linear chain polymer one obtains .

Role of Electrostatic Repulsions at the DNA-DNA Interface

While deriving Eqs 3 and 4 we have not considered the electrostatic repulsions between the negatively charged phosphate backbones of c-ssDNA chains and shielding effects of solvent and other ion molecules present at the DNA-DNA interface of c-ssDNA molecules. Upon considering this fact and following the detailed works of Montroll in Ref. [34] we find the expression for the modified bimolecular rate constants as follows.

(5)

Here κ is the Onsager radius which is defined as the distance between negatively charged phosphate backbones of colliding c-ssDNA chains at which the electrostatic repulsive energy is same as that of the thermal energy (~k_BT), ζ_Se and ζ_S’e are the overall charges on the respective c-ssDNA molecules. Since |κ| ≥ r_Q by definition and κ > 0 in the present context, one finds that ψ_Q ~ |κ|/r_Q for large values of |κ| and for |κ| = r_Q one obtains ψ_Q ≃ 0.58. For a typical value of |κ| ≃ 10r_Q we find that ψ_Q ≃ 10⁻⁴. When κ < 0 as in case of site-specific DNA-protein interactions we obtain . One should also note that ψ_Q ≃ 1 only when we have |κ| = 0. While deriving Eq 5 we have not considered the shielding effects of solvent ions present at the DNA-DNA interface over the electrostatic repulsive forces between the phosphate backbones of c-ssDNA strands. Upon following the Debye theory of kinetic salt effects over diffusion controlled collision rate processes [35], one can rewrite the modified bimolecular rate constant in the presence of other ions in the solvent as follows.

(6)

Here Ξ is the overall ionic strength of the aqueous medium and Q = (C, R) as defined in Eqs 3–5 depending on the type of conformational state of colliding c-ssDNA polymers.

Calculation of the Nucleation Time and Nucleation Rate

We learnt from recent computational studies [29] that the colliding c-ssDNA with nonspecific contacts between them undergo several trials of slithering and internal displacement dynamics before reaching the correct-contact and then nucleate the zipping process. These dynamical processes are similar to that of the facilitating 1D diffusional dynamics as in case of site specific DNA-protein interactions. Unlike the overall electrostatic attractive forces acting at the DNA-protein interface, in case of DNA renaturation there is a strong electrostatic repulsive force acting at the interface of cn-ssDNA that will be shielded by the solvent molecules present at the interface of cn-ssDNA strands. With this background one can model the slithering dynamics as 1D diffusion of one c-ssDNA molecule on the other in the process of searching for the correct-contact. To find the correct-contact on one c-ssDNA the other c-ssDNA needs to try out at least λ = L/n stretches of 1D slithering with an average size of n bases. This will ensure that the initial nonspecific contact visits all the possible positions and subsequently the correct contact is formed. The mean first passage time (τ_N) associated with the visit of all the possible positions of c-ssDNAs by the initial nonspecific-contact between them via 1D diffusion can be given as where is the average time that is required by an unbiased 1D random walker to visit n sites of a linear lattice [36–38] confined inside the interval x ∈(0, n) starting from anywhere inside the interval as shown in Appendix A.

Noting that the nucleation rate is given as k_N = 1/τ_N one can obtain k_N ≃ 12D_o/nL which means that k_N ∝ 1/L. This is a reasonable one since the probability of finding the correct contact upon each 3D diffusion mediated nonspecific collisions between c-ssDNA strands is p_cc = 1/L. One should also note that the probability of finding the correct contact upon each nonspecific collisions will be independent on the repetitiveness of c-ssDNA strands. Here D_o is the one dimensional diffusion coefficient associated with the dynamics of the probe c-ssDNA strand over the template c-ssDNA strand. One should note that D_o includes the contributions from both slithering and internal displacement dynamics which occurs within a cn-ssDNA molecule. Actually one can write down the expression for the 1D diffusion coefficient as follows.

(7)

Here D_os is the diffusion coefficient corresponds to slithering dynamics and D_oi corresponds to the internal displacement dynamics where p_f/r are the transition probabilities associated with the forward and reverse steps of the random walker with step size l_p = 1 base and w_f/r are the corresponding transition rates. Similarly in case of internal displacement dynamics l_d is the average step size and p_f/r,d are the respective forward and reverse transition probabilities and w_f/r,d are the corresponding transition rates. Here we consider the average step size since the step size associated with the inchworm type movements in the internal displacement mechanism is a random variable. For an unbiased searching via both 1D slithering and internal displacements we have p_f/r = p_f/r,d = 1/2.

Although l_d > l_p, internal displacement dynamics may not be able to contribute much to the overall searching dynamics under relaxed conformational state of the colliding c-ssDNA strands since it requires looping and segmental motion of c-ssDNA chains. Therefore in general we have w_f/r > w_f/r,d. Further one should note that slithering dynamics is a purely a local one i.e. slithering involves the local movement of bases. This means that the transition rates w_f/r and diffusion coefficient D_os will be independent of the length of c-ssDNA. Whereas internal displacement involves the movement of bulky segments [29] of the colliding c-ssDNA strands and therefore the corresponding transition rates w_f/r,d and diffusion coefficient D_oi will be dependent on the size of c-ssDNA strands. Here we can ignore the contribution of D_oi to the overall diffusion coefficient D_o only for the relaxed conformational state of c-ssDNA and we cannot ignore it for the condensed conformational state of c-ssDNA since the contribution from internal displacement mechanism will be the dominating one under such conditions.

Calculation of the Zipping Time

Formation of correct-contact will result in the nucleation of zipping process. Upon formation of a nucleation site, the subsequent stochastic zipping of cc-ssDNA can be described by the following birth-death master equation.

(8)

Here P(u,t) = P(u,t|u₀,t₀) is the probability of finding the cc-ssDNA with u numbers of correct contacts at time t starting from the nucleation at t = t₀ with u = u₀ numbers of correct contacts, k₊ (s^-1) and k_- (s^-1) are the respective average forward and reverse rate constants associated with the microscopic zipping reaction. Here the initial and boundary conditions corresponding to Eq 8 can be written as follows.

(9)

Here u = 1 is a reflecting boundary and u = β is the absorbing boundary. One can solve the difference equation Eq 8 as follows. By defining equilibrium constant as K_Z = (k_-/k₊), one can find the following expression for the overall mean first passage time associated with complete zipping of β correct contacts (u = β) of cc-ssDNA starting from the number of correct contacts u = 1 as shown in Appendix B.

(10)

From this equation we find the limits and where β = L/l_p is a dimensionless quantity which is the total number of correct-contacts between colliding c-ssDNA upon formation of dsDNA. Here L is total length of c-ssDNA in bases and l_p = 1 base (1 base ~ 3.4x10^-10 m). When the forward rate constant associated with formation of dsDNA is much higher than the reverse rate constant then we find the expression for the zipping rate constant as that k_Z = 1/τ_Z ≃ k_p/L where we have defined k_p = k₊l_p (bases/s). On the other hand when k₊ = k_- then the zipping will be a pure 1D diffusion process with the phenomenological diffusion coefficient as (base²/s) and subsequently τ_Z ≃ L²/2D_±. When the reacting c-ssDNA is repetitive with a sequence complexity of c bases (here we have c∈(1, L)), then the observed zipping rate will be proportional to the number of repeats in that template c-ssDNA strand (ρ = L/c) and one obtains i.e. the total zipping time will be directly proportional to the complexity of the reacting c-ssDNA molecules which is in line with the experimental observations [6–7]. Using these results one can write down the expression for the total time that is required for the overall nucleation and zipping processes (τ_NZ) to generate dsDNA as follows.

(11)

Here one should note that the zipping process can be thought as a random walk over a linear lattice [36–38] with initial and boundary conditions given in Eq 9. The phenomenological diffusion coefficient associated with such random walk is defined as where p_± are the transition probabilities associated with the forward and reverse steps in the zipping process. When the zipping is unbiased over forward or reverse steps with k₊ = k_- then p_± = 1/2 and we recover the expression for in the limit as K_Z = 1.

Calculation of the Overall Renaturation Rate

Using the expressions for the nucleation and zipping times one can define the overall bimolecular collision rate associated with the complete formation of dsDNA from c-ssDNA in Scheme III of Fig 2 as follows.

(12)

Here we have defined two important characteristic lengths and Y_E = k_p/k_r. The length Y_A describes the distance that is travelled by the initial nonspecific contact before cn-ssDNA dissociates whereas Y_E describes the distance travelled in the zipping reaction before cn-ssDNA dissociates into corresponding c-ssDNA molecules. Generally one observes that Y_A > Y_E. The probability density function connected with the 1D slithering lengths n or its weighting function p(n) can be calculated as follows. When the residence times (τ) associated with the dissociation of cn-ssDNA molecules is distributed as an exponential then one finds that and subsequently one obtains which mainly originates from the fact that within the residence time τ, the distance travelled by the nonspecific contact through 1D diffusion dynamics can be anywhere in the interval n∈(1, L) so that we obtain the transformation rule as τ = n²/12D_o. With this definition of the residence time of c-ssDNA strands in the cn-ssDNA configuration one can write down the expression for the distribution of slithering lengths n as follows [Fig 4A].

(13)

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Fig 4. Cooperative effects on DNA renaturation.

A. Probability density function associated with the one dimensional slithering length (n measured in bases) of cn-ssDNA in the process of searching for the correct-contact as given in Eq 13 for different values of the characteristic length ranging from 10 to 100 bases where D_o (base²s^-1) is the one dimensional diffusion coefficient associated with the slithering dynamics and k_r is the dissociation rate constant connected with cn-ssDNA. B. Zipping time (τ_Z, measured in seconds) in the presence of cooperative effects. Here sequence complexity (c) is same as that of the length (L) of c-ssDNA i.e. c = L. The number of correct-contacts β = L/l_p is a dimensionless quantity where l_p = 1 base and L is the length of the reacting c-ssDNA. Green solid line is calculation from Eq 18 and blue solid line is calculation from Eq C9 of Appendix C. Here we have set K_Z ~ 10⁻⁶ and k₊ ~ 1 s^-1. Red solid line is the derivative of zipping time with respect to β as in Eq 21 which shows that the value of the derivative of overall zipping time with respect to β is < 10⁻² when β > 10². These plots suggest that when K_Z tends towards zero, the overall zipping time of a non-repetitive and long c-ssDNA will be independent of the length of the reacting c-ssDNA molecules. Zipping time of a repetitive c-ssDNA with a sequence complexity of c bases increases linearly with c. C. Variation of the overall renaturation rate k_HR with respect to length and complexity of c-ssDNA under relaxed conformational state. Here settings are M^-1s^-1, Y_A = 100 bases, Y_E = 1 bases and n = 10 bases. D. Variation of the overall renaturation rate k_HR as in Eq 12 with respect to the length of reacting c-ssDNA strands L and 1D slithering distance n under relaxed conformational state. Here settings are M^-1s^-1, Y_A = 100 bases, Y_E = 1 bases and c = 10 bases. In both C and D, k_HR shows a maximum at L = L_opt. Here Lopt can be obtained by solving ∂_Lk_HR = 0 for L. Explicitly one finds that . The dotted line in (C) is which breaks down beyond L_opt.

https://doi.org/10.1371/journal.pone.0153172.g004

Using the expression of p(n) in Eq 13 and expanding k_HQ in a Maclaurin series one can obtain the following expression for the overall renaturation rate constant.

(14)

While deriving this equation without losing generality we have extended the limits of n in the integration towards infinity since both p(n) and k_HQ approach zero at this limit. Under certain conditions one can obtain the following approximation for the series in Eq 14.

(15)

Later we will show that these inequality conditions are indeed valid. When the sequence complexity of the template c-ssDNA molecule is high enough and (pY_E/Y_A) ≪ 1 then one can write down the leading zeroth order approximation (m = 0 in Eqs 14 and 15) of the overall second order rate constant associated with the renaturation of relaxed c-ssDNA chains with equal lengths L = l as follows.

(16)

Upon substituting the expression in Eq 16 one obtains that in line with the expression for the overall renaturation rate in Wetmur-Davidson model [6]. Here k_sm≃k_tε is the Smolochowski type 3D diffusion controlled collision rate limit corresponding to a situation L = c = 1 where we have defined . When the conditions given in Eq 15 are true then Eq 16 suggests that the overall bimolecular collision rate associated with the renaturation reaction is directly proportional to the square-root of the length of c-ssDNA and inversely proportional to both sequence complexity of the reacting c-ssDNA molecules and viscosity of the reaction medium in line with the experimental observations. Experiments suggest that the scaling of renaturation rate with the length of c-ssDNA molecules that is given in Eq 16 is valid [6] only in the range of L ~ 10²−10⁴ bases. In this context our model suggest that the square root scaling of the renaturation rate on length of reacting c-ssDNA strands will be valid only when the inequalities given in Eq 15 are true apart from the condition that (pY_E/Y_A) ≪ 1 which may break down beyond certain values of the copy number (ρ) in the repetitive c-ssDNA.

Role of Cooperativity in Renaturation Kinetics

Scaling results given by Eq 16 works only for highly repetitive and relaxed conformational state of c-ssDNA (L > c) and it will break down at L = c since at this point the scaling becomes as which is not true. The main reason for this observation is that while calculating the zipping rate constant we have not considered the underlying cooperative effects over the zipping process. When c-ssDNA is highly repetitive then the zipping process can take place in parallel for all the ρ = (L/c) number of short repetitive motifs. Under such conditions the cooperative effects will not be noticeable since the enhancement of renaturation process by the parallel-zipping will dominate over the underlying cooperative effects. This means that Eq 16 will be true only for repetitive c-ssDNA.

When the reacting c-ssDNA molecules are non-repetitive and long enough then the probability of formation of an additional correct-contact in the cc-ssDNA molecule that is undergoing zipping reaction will be directly proportional to the existing number of correct-contacts (u) and the probability of breakdown of an existing correct-contact will be directly proportional to the number of overhanging single stranded stretches of cc-ssDNA (β-u). This is true since the existing correct-contacts always stabilize newly formed correct-contacts and overhang single stranded regions of cc-ssDNA always try to destabilize the newly formed correct-contacts. Here one should note that we are dealing with the cooperative effects at a mesoscopic level within an independent and single renaturing cc-ssDNA molecule rather than at macroscopic level where the descriptive parameter of the renaturation process will be the number of mol-bases in c-ssDNA (n_SS) or dsDNA (n_DS) form rather than the number of correct-contacts in cc-ssDNA (u) as in the current context.

At macroscopic level the rate of change in the number of mol-bases in ssDNA form in the process of zipping will be directly proportional to the number of mol-bases in ssDNA form as well as number of mol-bases in dsDNA form which results in a cooperative sigmoidal type time evolution of the renaturation process [16–17, 28] where the macroscopic kinetic rate equation will be written as dn_DS/dt ∝ n_cc(n₀−n_DS)n_DS. Here n₀ is the initial concentration of mol-bases of c-ssDNA molecules in the system and n_cc is the total number of cc-ssDNA molecules in the system. With this background the birth-death master Eq 8 can be rewritten to include the cooperative effects for renaturation of a single cc-ssDNA molecule as follows.

(17)

Upon solving the backward type equation corresponding to this differential difference equation for appropriate boundary conditions as shown in Eq 9 and Appendix C one obtains the following expression for the overall zipping time that is required for the formation of u = β numbers of correct-contacts starting from u = 1 in the presence of cooperative effects.

(18)

Here ₂F₁ is the hypergeometric function and we have defined various parameters as follows.

(19)

The hypergeometric function of type ₂F₁ is defined as follows.

(20)

One can simplify the complicated expression given by Eq 18 using Fokker-Planck equation especially for large values of β as shown in Appendix C. Upon defining one can derive the following approximate expression for the overall zipping time.

(21)

It seems from Eq 21 along with other computational analysis that the dependency of overall zipping time decreases with increasing β in the presence of cooperative effects which can be demonstrated by the following limiting conditions.

(22)

From Eqs 17–22 one finds that when K_Z tends towards zero then in the presence of strong cooperative effects the zipping time of a non-repetitive cc-ssDNA will be independent of the length of the reacting c-ssDNA molecules especially for large values of L as shown in Fig 4B. Based on these observations we recover the observed scaling of the overall bimolecular rate constant on the length of the colliding c-ssDNA molecules as for the renaturation of the non-repetitive c-ssDNA strands for which L = c in Eqs 15 and 16 since Y_E will be independent of sequence length and the number of copies will be ρ = 1.

Two-Step DNA Renaturation Model of Wetmur and Davidson

Understanding the mechanism of renaturation of c-ssDNA is one of the central topics in molecular biology and biological physics. Wetmur and Davidson [6] developed their model by assuming that the renaturation rate is directly proportional to the total phosphate concentration which is in turn directly proportional to the total number of mol-bases in ssDNA or dsDNA form. According to their model the overall second order rate (k₂) associated with the renaturation of repetitive c-ssDNA can be written as k₂∝k_Np where ρ = L/c is the copy number of repetitive motifs in the entire c-ssDNA polymer. Here L is the length of c-ssDNA and c is the sequence complexity and the nucleation rate was assumed to scale with L as L^-α where α = 1/2 due to the excluded volume effects associated with the interpenetration of c-ssDNA molecules that is essential for the nucleation reaction. As a result one obtains the scaling as where the proportionality constant was assumed to be the Smolochowski type bimolecular collision rate limit (k_sm) i.e..

In Wetmur-Davidson model it was assumed that k_sm = k_t. Here one can identify that and k_sm = (k_t ε) of our model Eq 16 particularly for a relaxed conformational state of c-ssDNA that also includes the contributions from the electrostatic repulsions at the interface of colliding c-ssDNA molecules. Since in this model the nucleation is combined with nonspecific-contact formation step one finds that the nucleation rate is inversely proportional to the square root of the length of c-ssDNA strands. The main arguments for this scaling result put forth by Wetmur and Davidson were viz. (a) the radius of gyration of c-ssDNA is directly proportional to its length and (b) the reaction radius associated with the collision between c-ssDNA molecules is independent of the radius of gyration of c-ssDNA chains since both these strands can interpenetrate freely upon their collision. Though the assumption (a) is a right one for Gaussian chain polymers there are several questions associated with the assumption (b) since the reaction radius always depends on the sum of the radii of gyration of reactant molecules. In this context our detailed model clarifies the origin of such scaling in the renaturation phenomenon. It is clear from our theory that the rate constant associated with the formation of initial nonspecific-contact is directly proportional to the square-root of the length of the reacting c-ssDNA molecules.

One should note that it is very difficult to identify and isolate nucleated cc-ssDNA molecules since they are indistinguishable from the zipping cc-ssDNA molecules. Therefore it is more appropriate to combine the nucleation step with the zipping step rather than with the nonspecific-contact formation step as in case of Wetmur-Davidson model. Other issues in their model are such as the breakdown of scaling at L = c arises because the underlying cooperative effects in the long and non-repetitive c-ssDNA were not considered in their model as in case of our Eq 16. Further upon extrapolating towards the limit L = 1 base (so that the sequence complexity becomes as c = 1 base) Wetmur-Davidson model predicted that k₂ = k_t. However experimental observations suggested that the extrapolated bimolecular collision rate constant associated with the renaturation reaction at L = c = 1 was ~10³ times lower than the Smolochowski type diffusion controlled bimolecular collision rate limit (k_t). On this basis they in turn discarded the possibility of diffusion control in the kinetics of renaturation of c-ssDNA molecules.

In this context Eq 16 of our model suggests an approximate expression for the extrapolation intercept as k_sm ≃ k_tε from which we can deduce that . One should note that scheme I of Fig 1 is still valid with zero nucleation and zipping times and the steric factor ε mainly accounts for the geometric constraints associated with the bond formation between nitrogen bases A-T or G-C of the colliding single nucleotides in the limit L = 1 and c = 1 apart from the electrostatic repulsions due to the negatively charged phosphate groups. Here one should note that in our model the square-root dependency of renaturation rate on the length of c-ssDNA molecules mainly originates from the fact that the radius of gyration of c-ssDNA molecules scales with length as r_L ∝ L^α where α = ½ which is valid only for a Gaussian type polymers in a theta solvent. It seems that the error introduced by this assumption in the exponent is within the experimental error range [6].

Temperature Dependency of DNA Renaturation Rate

Although the dissociation rate (here it is k_r) constant increases exponentially with temperature [39–42] there are several controversies exist on the dependency of renaturation rate constant on increasing temperature. Some experimental studies established [39–40] a decrease in the renaturation rate constant with increase in temperature and some other studies have shown an increase in the renaturation rate constant with increase in temperature [8, 41]. In general it seems that the temperature dependency of the renaturation rate constant is of non-Arrhenius one and non-monotonic type [42]. Simulation studies suggested that there exists an optimum temperature at which the renaturation rate constant is a maximum [29]. In our model the nonspecific-contact formation, nucleation and zipping steps are all influenced by the rise in temperature in a complicated way.

Actually in Eq 16, the Smolochowski type bimolecular collision rate constant depends on temperature as k_t = (8k_BT/3η) where we assume that viscosity of the medium is not changing much in the range of temperature variation and the dissociation rate scales with temperature as in line with transition state theory where ω is the free energy barrier associated with the dissociation of cn-ssDNA complex. The rate constant associated with the microscopic zipping (k_p ~ l_pk₊) is connected with the microscopic diffusion coefficient D_± ~ l_p²k₊ associated with the zipping reaction. Apart from these the dimensionless parameter χ_R corresponding to the overall electrostatic repulsions and the shielding effects of solvent ions at the interface of cn-ssDNA molecules also depends on the temperature as given in Eqs 5 and 6. It seems that the non-Arrhenius type kinetic behaviour arises as a consequence of a complicated interplay between increase in the rate of nonspecific-contact formation and combined effects of increase in the dissociation rate constant and microscopic zipping rate constant as the temperature increases from low to high values. Since one finds that will be a maximum approximately at T ~ ω/k_B as observed in the simulation studies [29].

Comparison with Experimental Data on DNA Renaturation

Under relaxed conformational state, the 3D diffusion mediated nonspecific contact formation rate scales with the size of c-ssDNA strands as . Since the negatively charged phosphate backbones of c-ssDNA repel each other the Onsager radius (κ) associated with the collision of c-ssDNA strands will be much higher than the reaction radius (r_R) under relaxed conformational state. When κ ~ 10r_R then one obtains χ_R~10⁻³. Noting that k_t~10⁹ M^-1s^-1 and l_p ~ 1base one finds that M^-1s^-1. In the calculation of k_t we have used T = 298K and viscosity coefficient η ~10⁻¹ kgm^-1s^-1 for aqueous conditions. Single molecule experiments on DNA polymer in aqueous solution suggested [43–44] a 3D diffusion coefficient as where base²s^-1 and α ~ 0.59–0.71 depending on the condition of the aqueous medium. Furthermore the 3D diffusion coefficient seems to be several orders of magnitude less under crowded cytoplasmic environment [43]. Therefore the 1D diffusion coefficient associated with the slithering dynamics will be 10−10² times slower than the 3D one since the local dynamics of individual bases which are involved in the slithering dynamics will be significantly restricted by the adjacent bases apart from the reduced degrees of freedom. In this background one can approximate the 1D diffusion coefficient as D_o ≃ 10⁹ base²s^-1 under relaxed conformational state of c-ssDNA strands. Using this one can arrive at an empirical expression for the nucleation rate as k_N ≃ (10⁹/nL).

From experimental studies [41] one finds the zipping rate as k_p ~ 10⁶ bases/s which means that k₊ ~ 10⁶ s^-1. Using these values one obtains the zipping rate as k_Z ≃ (10⁶/c) s^-1. Noting from experimental observations that ~10⁻³ and using the values of k₊ and χ_R one can obtain the value of the dissociation rate as k_r~10⁶ s^-1. Using the numerical values of (D_o, k_p and k_r) one finds that Y_A ~ 10² bases and Y_E ~ 1 bases. Computational studies on the renaturation of short fragments of c-ssDNA strands suggested [29] the most probable value for the 1D slithering length as n ~ (4–10) bases. For the purpose of calculations we use n = 10 bases. From Eq 12 one finds that the overall renaturation rate k_HR will be a maximum at L = L_opt where (Fig 4C and 4D). This can be obtained by solving ∂_Lk_HR = 0 for L. Here k_HR scales with L as whenever L<L_opt since the nonspecific contact formation step will be the dominating component under such conditions. When L > L_opt then the scaling with L becomes as since the nucleation and zipping steps will be the bottlenecks under such conditions. Upon substituting L = L_opt into k_HR one can obtain the maximum achievable bimolecular renaturation rate as for the renaturation of repetitive c-ssDNA strands. When the sequence complexity c is much higher than the characteristic length Y_E then one can deduce that and the maximum achievable renaturation rate seems to be M^-1s^-1. Upon substituting the numerical values of Y_A and Y_E into the expression for L_opt one obtains L_opt ~ 10⁴c/n. For a sequence of c-ssDNA strands with a complexity of c ~ n one finds that L_opt ~ 10⁴ bases. From Eq 12 we find that the square root scaling of the renaturation rate on L will be valid only up to ~10⁴ bases in line with the experimental observations [6, 27].

Upon substituting the numerical values of (Y_A, Y_E and k_fR) into the expression for the renaturation rate k_HR one finds that . Interestingly one should note that this functional form will behave as whenever . When L < L_opt and c > 1 then one obtains that . Remarkably this expression for the renaturation rate k_HR is much close to the experimentally obtained fitting function of Wetmur-Davidson for the overall bimolecular renaturation rate as for the experimentally measured range of L from 10² to 10⁴ bases (Eq. 20 in Ref. [6]). It is still not clear [27] whether this scaling relationship will be valid beyond this range of L or not. In this context our theory based on Scheme III predicts that the square root scaling of the overall renaturation rate on the length of c-ssDNA will break down beyond L_opt. When the inequality conditions given in Eq 15 are not true then one can also show that the average renaturation rate will be a maximum at where is the solution of for L as follows.

Here the subscript b in g_b can take (n, L). Detailed analysis suggests an approximation as where . Clearly we find that . This is because the probability density function associated with the 1D slithering lengths as defined in Eq 13 will be valid only when the sequence complexity c is much higher than the maximum possible 1D slithering length Y_A. When c < Y_A then the slithering dynamics associated with a nonspecific contact present in between the cn-ssDNA strands can progress only for c number of steps. Beyond this point either dissociation of cn-ssDNA strands or relocation of the nonspecific contact to some other position of cn-ssDNA is necessary. Therefore under such conditions one can replace the probability density with a delta function as p(n) ≃ δ(n−c). This means that we need to substitute as in our calculations whenever c < Y_A and one obtains in the present context.

Justifications for the Three-Step Model

In Scheme III the nucleation rate is inversely proportional to the length and zipping rate in inversely proportional to the complexity of the c-ssDNA strands. Therefore it is reasonable to cluster both nucleation and zipping steps together and assume that they are the rate limiting ones compared to the nonspecific contact formation step. Unlike the rate of nucleation and zipping the nonspecific contact formation rate increases with the length of c-ssDNA in a square root manner. We substantiate the three steps of renaturation by the following reasons viz. (a) the underlying microscopic processes are clearly dissimilar in cases of nonspecific contact formation, nucleation and zipping. Here nonspecific contact formation is a pure 3D diffusion mediated collision process. Whereas nucleation involves a combination of 1D and 3D diffusion. The zipping step is pure 1D diffusion like process which progresses from a stable nucleus, and (b) the scaling relationships associated with the corresponding rates on the length of reacting c-ssDNA and sequence complexity are different from each other. Clearly nonspecific contact formation, nucleation and zipping are all phenomenologically distinct processes which substantiate our three-step model. Since the nucleation and zipping and parts of a continuous process and a nucleated cc-ssDNA molecule is indistinguishable from the zipping one we have combined the nucleation step with the zipping step for the purpose of computing the rate associated with the overall nucleation-zipping. Here one should note that cn-ssDNA which is the product of nonspecific contact formation step is distinct from nucleated/zipping cc-ssDNA molecule.

Comparison with Earlier Diffusion Based Models

Since 3D diffusion based models predict the scaling of overall renaturation rate on the length and sequence complexity of c-ssDNA strands as k_HR ∝ L/c one can rule out the possibility of Scheme I. Two step models as in Scheme II correctly predict the scaling of renaturation rate on the length and complexity of the reacting c-ssDNA strands. However models based on Scheme II such as the Wetmur-Davidson model [6] fail when L = c apart from their inability to explain the discrepancy in the intercept value of the bimolecular renaturation rate at L = c = 1. Models based on transition rate theory cannot explain the inverse viscosity dependence of the overall renaturation rate.

Sikorav et.al [27] suggested a Kramer’s type expression for the bimolecular nucleation rate constant where the scaling dependency of the overall bimolecular collision rate associated with renaturation on the length of c-ssDNA mainly originates from the entropic component of free energy barrier. However in this model the reaction coordinate and origin of free energy barrier associated with the nucleation and zipping are not clearly defined. Further the exact mechanism of formation of nucleation sites is not clearly explained. On the other hand as correctly pointed out by them, one cannot explain all the experimental observations related to the entire process of renaturation of c-ssDNA molecules with purely diffusion-controlled formalism or transition state theoretical framework. From our model we can conclude that the nonspecific contact formation step is a pure three dimensional diffusion controlled collision rate processes whereas both nucleation and zipping steps involve a sequence of several microscopic crossings of free-energy barriers as well as one dimensional diffusion type slithering dynamics on a linear lattice.

Effects of Conformational State of DNA on the Renaturation Rate

Condensed conformational state of c-ssDNA polymers is one more cause for the breakdown of the scaling of renaturation rate on the length of c-ssDNA that is given in Eq 16. When the colliding c-ssDNA molecules are in condensed conformational state then the rate constant associated with the nonspecific-contact formation step will be independent of the length of c-ssDNA when L = l. Under such conditions the overall second order rate constant associated with the renaturation of repetitive c-ssDNA chains will be inversely proportional to the sequence complexity. Here one should note that while deducing these facts we have not considered the condensation of both c-ssDNA molecules together which is known to enhance the overall renaturation rate over several orders of magnitude as in case of renaturation in the phenol-water interface [45]. Under such co-condensation of both strands of c-ssDNA the rate of nonspecific-contact formation is very large and the dissociation rate will be very small and the rate limiting steps are the nucleation and zipping ones.

A. Nucleation via 1D Diffusion over Linear Lattice

The searching of the non-specifically bound cn-ssDNA strands for the correct-contacts on each other can be modelled as 1D random walk on a linear lattice. The stochastic differential equation associated with such an unbiased random walk on a linear lattice can be written as follows [36–38].

(A1)

Here x is the position of the random walker on the linear lattice, D_o is the one dimensional diffusion coefficient associated with the dynamics of the random walker, Γ_t is the Gaussian white noise with mean and variance as given in Eq A1. The probability density function associated with the dynamics of such random walker obeys the following forward Fokker-Planck equation (FPE) with initial condition.

(A2)

The boundary conditions associated with Eq A2 are p(0,t|x_0,t₀) = p(n,t|x_0,t₀) = 0. Here p(x,t|x_0,0) is the probability of observing the random walker at position x at time t with the condition that the random walker was at position x₀ at t = t₀. Settings t₀ = 0, the general solution to Eq A2 for the appropriate initial and boundary conditions can be obtained by the method of Eigen function expansion using biorthogonal set as follows [36–38].

(A3)

The mean first passage time (MFPT) associated with the escape of a random walker obeying Eq A3 from the interval x ∈(0, n) starting from an arbitrary lattice point x inside the interval obeys the following backward type Fokker-Planck equation.

(A4)

Since the random walker can enter initially anywhere in interval x ∈(0, n) of linear lattice with equal probabilities one needs to average the computed MFPT τ_x over all the values of initial positions x. As in Eq A4 we find the initial position averaged value as . This is approximately the time that is required by a random walker to visit all the sites of a linear lattice confined inside the interval x ∈(0, n) starting from anywhere inside the interval. This is evident from the following arguments. When we introduce reflecting boundaries at x = 0 as well as x = n then [∂_xp(x,t|x_0,0)]_{x = 0} = [∂_xp(n,t|x_0,0)]_{x = n} = 0 are the corresponding boundary conditions. The probability density function associated with the dynamics of such a random walker confined inside those reflecting boundaries can be given as follows.

(A5)

From this equation one can conclude that p(x,t|x_0,0) ≃ 1/n whenever t ≥ (n²/π²D_o) which is close to the initial position averaged mean first passage time . In other words in the presence of reflecting boundaries at both the ends of the linear lattice, the probability of observing the random walker anywhere within those boundaries will be equal when .

B. Zipping of Repetitive c-ssDNA Sequences

The zipping of cc-ssDNA strands will also be a stochastic process which can be described by the following birth-death master equation.

(B1)

Here P(u, t) = P(u,t|u₀,t₀) is the probability of finding the cc-ssDNA with u numbers of correct contacts at time t starting from the nucleation at t = t₀ with u = u₀, k₊ (s^-1) and k_- (s^-1) are the respective average forward and reverse rate constants associated with the microscopic zipping reaction. Here the initial and boundary conditions corresponding to Eq B1 can be written as follows.

(B2)

The mean first passage time associated with the complete zipping of cc-ssDNA strands obeys the following backward type master equation with similar boundary conditions.

(B3)

Here u = 1 is a reflecting boundary and u = β is the absorbing boundary. One can solve the difference equation Eq B2 as follows. By defining equilibrium constant as K_Z = (k₋/k₊), Eq B2 can be rewritten in the following form.

(B4)

Upon solving [36–38] this difference equation for the boundary conditions given in Eq B3 we find the following expression for the overall mean first passage time associated with complete zipping of β correct contacts (u = β) of cc-ssDNA starting from the number of correct contacts u = 1.

(B5)

C. Cooperative Effects on Non-Repetitive c-ssDNA Sequences

In the presence of cooperative effects the probability of formation of an additional correct contact in cc-ssDNA will be directly proportional to the already exiting number of correct contacts. Similarly the probability associated with the breaking of a correct contact will be directly proportional to the already exiting single stranded overhangs of cc-ssDNA. In the background the birth-death master equation described by Eq B1 can be rewritten to include the cooperative effects for the renaturation of a nonrepetitive single cc-ssDNA as follows.

(C1)

The mean first passage time τ(u) associated with evolution of the system from correct-contact u = 1 to complete dsDNA form with correct-contacts u = β can be written as follows where U(u) and other boundary conditions are defined as in Eq B3.

(C2)

Here Ω(u) = U(u)/ϕ(u) and the function ϕ(u) is defined as follows.

(C3)

Upon solving the difference equation Eq C2 for appropriate boundary conditions one obtains the following expression for the overall zipping time that is required for the formation of u = β numbers of correct-contacts starting from u = 1 in the presence of cooperative effects.

(C4)

Here ₂F₁ is the hypergeometric function and we have defined various parameters as follows.

(C5)

The hypergeometric function of type ₂F₁ is defined as follows.

(C6)

To simplify the complicated expression for τ_Z in Eq C3 particularly for sufficiently large values of β one can approximate Eq C3 by the following continuous type Fokker-Planck equation (FPE) [36–38].

(C7)

Here the drift and diffusion coefficients can be written as follows.

(C8)

Eqs C7 and C8 suggest that in the presence of cooperative effects the phenomenological diffusion coefficient associated with the zipping dynamics (D_±) will be dependent on the number of correct-contacts. Using the backward type FPE corresponding to Eq C7 one can obtain the mean first passage time associated with the evolution of the system starting from u = 1 to u = β as follows.

(C9)

In this equation various functions and parameters are defined as follows.

(C10)

Computational analysis of Eqs C7 and C9 suggests that in the limit as K_Z tends towards zero, the overall zipping time τ_Z approximately scales with β as 1-e^-2β. Upon defining the limit as one can derive the following expression for the overall zipping time.

(C11)

This equation suggests that for a sufficiently large value of β, in the presence of cooperative effects the overall zipping time will be almost independent of β since .