Understanding the Role of DNA Topology in Target Search Dynamics

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Understanding the Role of DNA Topology in Target Search Dynamics of Proteins Anupam Mondal, and Arnab Bhattacherjee J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b08199 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 19, 2017

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The Journal of Physical Chemistry

Understanding the Role of DNA Topology in Target Search Dynamics of Proteins

Anupam Mondal and Arnab Bhattacherjee* School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi110067, India *E-mail: [email protected]

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Abstract A wide range of cellular processes initiates upon recognising and binding of proteins to specific DNA sites. Typically, the recognition process is incredibly fast owing to a complex mechanism that combines different 3D and 1D modes of translocation of the protein. While few studies performed on selected DNA topologies suggested that the DNA topology might alter the balance between these two modes and therefore the target search kinetics, its detail role in target search mechanism remains unclear. Here, we present a discrete-state stochastic approach that allows us to incorporate the topological information of DNA molecule explicitly and predict its role during the process when proteins search for their specific binding sites on DNA. Applying the theory to the closed loop and different supercoiled DNA topologies, we find that the target search efficiency of the protein is strongly influenced by the DNA topology. Furthermore, if the topology is such that it promotes juxtaposition of distant DNA sites, the number, position and relative distances between such juxtaposition sites play a crucial role in facilitating the search process by providing additional routes to approach the target site. Our predictions are validated through extensive Monte Carlo simulations.

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Introduction DNA binding proteins (DBPs) search and bind to specific DNA sequences to initiate a cascade of biological processes relevant for normal functioning of cells1. Although these specific DNA sequences encode the required genetic information, it is the three-dimensional structure of DNA molecule that controls how the information will be deciphered2-3. The important fact is that on most occasions, the conformations of cellular DNAs are topologically closed as the ends are firmly anchored to the membrane in bacteria or tethered to chromosome scaffold in eukaryotes3-6. Such topologically constrained DNA molecules feature an important relation between local and global DNA conformations in accord to: Lk=Tw+Wr The physical meaning of this expression is that in a topologically closed DNA domain (with fixed linking number, Lk), any change in local secondary structure (twist, Tw) is immediately compensated by an adjustment in the overall shape (writhe, Wr) of the molecule. For example, alteration in local DNA helical turn is readily reflected by a change in supercoiling of the whole molecule, suggesting the supercoiled topology ideal for communication between distant DNA sites7. The similar argument explains how in transcriptional initiation, co-regulator proteins can interact with DNA segments separated by even thousands of DNA base pairs8-13. Furthermore, DNA supercoiling is also known to enhance the probability of juxtaposition, and thereby stimulate the reactions

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that involve the juxtaposition of distant DNA sites such as site-specific recombination and DNA cleavage by the type-II restriction enzymes14-16. While the above arguments suggest that DNA topology is crucial17-23 for a wide range of cellular processes, little is known about how it affects protein-DNA interactions. Previous experimental2429

, theoretical30-35 and computational36-40 studies have probed the issue, in particular, the target

search phenomenon of proteins for linear DNA fragments. It was suggested that proteins recognise their specific sites on a linear DNA by balancing between three-dimensional diffusion in bulk solution, and one-dimensional random walk along the DNA contour often referred as ‘sliding dynamics’28, 41-45. The diffusivity in bulk solution can exhibit anomalous behaviour in the presence of a large number of crowding agents and therefore, may impede or facilitate the target search process46. In contrary, the diffusivity in sliding dynamics typically slows down if the residues of the searching protein interact specifically with the DNA base pairs47-49. Furthermore, the DNA sequence is also known to determine its shape50-51, the impact of which on the binding of protein has been realised only partially through a series of computational investigations52-53. For example, the role of DNA coiling and effective coil density were found to enhance the target search diffusion of a searching protein54-55. In parallel, Shvets et. al. has probed the target search process of multisite proteins upon loop formation using a theoretical approach56. Along the line, our previous studies based on extensive computer simulations have also illustrated the target search dynamics of single and multidomain proteins on relaxed circular DNA21-22, 40. Despite some of these significant efforts, the possibility of the different supercoiled DNA topology has not been taken into account and therefore, how it might affect the mechanistic details of target search dynamics remains largely unanswered.

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In this study, we present a theoretical framework that shows how the DNA topological information can be explicitly taken into account to predict and quantify their impacts on the target search process of proteins. It is based on the discrete-state stochastic approach that considers most of the physical and chemical processes by analysing first-passage events in the system. To test our theory, we select closed loop and various degree of complex supercoiled DNA topologies and analyse the target search mechanism of proteins on those DNA topologies. The theory is compared with Monte Carlo (MC) simulation results on lattice DNA model. An excellent agreement between the theory and simulation results are obtained that suggest DNA topology determines the approachability of the target site located on the DNA. If the topology favours the juxtaposition of distal DNA sites, the number, spatial distribution and relative distances between such juxtaposition sites may strongly influence the approachability of the target site further.

Theoretical Methods To begin with, we consider a closed loop DNA topology (Fig 1B), which can be assumed as a result of fusing the two free ends of an open chain DNA conformation (Fig 1A). As the target search mechanism of proteins on open chain DNA has already been captured successfully30, 33, it

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is, therefore, a natural starting point to develop a theoretical framework for protein diffusion on closed loop DNA. This allows us to compare the differences in target search mechanism and undermine the role of topological constraint present in a closed loop DNA because of fused DNA ends. Once developed, we further introduce constraint equations corresponding to complex supercoiled DNA topologies (shown in Fig 1C-E) and study how the search dynamics of proteins vary on different supercoiled DNA topologies. Our model assumes a single closed loop DNA and a searching protein (Fig 1B). The DNA molecule is composed of L sites out of which one site is considered as the target, where a protein can bind specifically. Since all the sites on a closed DNA topology are identical, we choose L-th site as the target site. The protein search always starts from the bulk solution phase, labelled as state 0. From the solution, protein can bind to any site on closed DNA with total association rate kon whereas, in the opposite reaction, the protein dissociates from the DNA to solution with a rate koff. On the DNA, the protein molecule diffuses in forward and backward directions with equal rate u. Typically these transition rates are determined from experimental measurements and one such example is lac repressor protein, which was extensively studied and the associated rates are reported Figure 1. A general scheme of the discrete-state stochastic approach for the topology-dependent pro-

tein target search on (A) open DNA, (B) closed loop DNA, (C) supercoiled DNA with one, (D) two and (E) three juxtaposition sites. The searching protein can slide along the DNA with rate u in both the directions or it can diffuse into the solution with rate koff. From the solution, protein can associate to any site on DNA with total rate kon. For supercoiled DNA, the transition rate for intercommunication between juxtaposition points and targets is kt. Black, red and blue circles correspond to the juxtaposition sites, target sites and non-specific binding sites respectively on DNA of length L.

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as u ~ 103-106 s-1, koff ~ 200-3000 s-1 and kon ~ 104-106 s-1 respectively28, 41. In the present study, we adopted these as the input parameters. To analyse the dynamic description of target search process on closed loop DNA, we define a function 𝑃" 𝑡 that denotes the probability of the protein to reach the target site on DNA at time t, if at t=0 the protein was at state n (where 𝑛 = 1,2, . . . . 𝐿 are sites on the DNA and n=0 signifies bulk solution phase). The temporal evolution of 𝑃" (𝑡) can be described by a set of backward master equations30, 33, -./ (0) -0

= 𝑢 𝑃"23 𝑡 + 𝑃"53 𝑡

+ 𝑘788 𝑃9 𝑡 − 2𝑢 + 𝑘788 𝑃" 𝑡 , 𝑓𝑜𝑟2 ≤ 𝑛 ≤ 𝐿 − 2

(1)

The physical meaning of the expression is following. The first two terms on the right-hand side represent the probabilities of the protein to move unbiasedly from site n to the adjacent sites n-1 and n+1 with 1D diffusion rate u and the third term includes the possibility of the protein to diffuse into bulk solution from site n with rate koff. The reverse mechanism in which the protein is equally likely to reach the site n from its adjacent sites and bulk solution is given by the last term and the associated rate is -(2u+koff). To this end, it should be noted that despite the wide applications of an alternative approach, namely the forward Chapman-Kolmogorov equation in describing measurable quantities directly as functions of the measured (real) time, casting the target search dynamics of DNA binding proteins using backward master equation is advantageous. This is due to the fact that the backward approach reconstructs the past events, and therefore, is preferentially used to predict the first passage times of reaching the target DNA site. Further details can be found elsewhere57-58. In a similar manner, the time evolution of the solution phase is given by,

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-.? (0) -0

=

@A/ B

B "C3 𝑃"

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(2)

𝑡 − 𝑘7" 𝑃9 (𝑡)

The equation suggests that while diffusing in the solution phase, the protein can associate to any vacant DNA site. As all sites are equivalent, the binding rate for a site is simply the total binding rate kon per site (kon/L), whereas, the protein, if already is in bound state, may get dissociated from the DNA surface (with rate -kon, opposite to association process) to the bulk solution. In addition, the equations for the adjacent sites to the target (L-th site) in a closed loop DNA assume the forms -.D (0) -0

= 𝑢 𝑃E 𝑡 + 𝑃B 𝑡

-.FGD (0) -0

(3)

+ 𝑘788 𝑃9 𝑡 − 2𝑢 + 𝑘788 𝑃3 𝑡

= 𝑢 𝑃B2E 𝑡 + 𝑃B 𝑡

(4)

+ 𝑘788 𝑃9 𝑡 − 2𝑢 + 𝑘788 𝑃B23 𝑡

The initial condition is 𝑃" (0) = 𝛿",9 . For n=L (target site), which indicates that if at t=0, the protein was at site L, the search process finishes immediately. ~

To solve eqs (1) - (4), we adopt the Laplace transformation 𝑃" 𝑠 =

O 2M0 𝑒 𝑃" (𝑡)𝑑𝑡 9

that transforms

the equations into simple algebraic forms as follows: ~

~

~

~

𝑠 + 2𝑢 + 𝑘788 𝑃" 𝑠 = 𝑢 𝑃"23 𝑠 + 𝑃"53 𝑠 ~

@A/

𝑠 + 𝑘7" 𝑃9 𝑠 =

B

~

+ 𝑘788 𝑃9 𝑠

~ B "C3 𝑃" (s)

(6)

~

~

(7)

𝑠 + 2𝑢 + 𝑘788 𝑃3 𝑠 = 𝑢 𝑃E 𝑠 + 1 + 𝑘788 𝑃9 𝑠 ~

(5)

~

~

𝑠 + 2𝑢 + 𝑘788 𝑃B23 𝑠 = 𝑢 𝑃B2E 𝑠 + 1 + 𝑘788 𝑃9 𝑠 ~

With the initial condition 𝑃B 𝑠 = 1. 8


(8)

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Equations (5) - (8) can be solved simultaneously by assuming a general solution of the form ~

𝑃" 𝑠 = 𝐴𝑦 " + 𝐵

This yields ~

𝑃" 𝑠 = 1 − 𝐵

S / 5S FG/ 35S F

+ 𝐵 for n > 0

(9)

~

For n = 0, the solution results 𝑃9 𝑠 = BM

@A/ (M5@ATT )UV M

(10)

M5@A/ 5@ATT 5@A/ @ATT UV M

where parameters y and B are given by 𝑦 𝑠 =

M5EW5@ATT 2

M5EW5@ATT

X

2YWX

EW

,𝐵 =

@ATT M5@ATT

~

(11)

𝑃9 𝑠

35S (32S F )

and the auxiliary function for closed loop DNA is given by 𝑆[ 𝑠 =

(12)

32S (35S F )

The solution is distinctly different from what Veksler et. al.33 has reported for the target search phenomena of a protein on linear/open chain DNA. The suitability of the solution (Eq 9) for a closed loop DNA structure can be confirmed from the fact that around the target site (L), proba˜

˜

˜

˜

˜

˜

bilities of the equally spaced sites are identical, such as 𝑃3 = 𝑃B23 , 𝑃E = 𝑃B2E , 𝑃] = 𝑃B2] and so on. It should also be noted that the explicit analytical expression for first-passage probability in the Laplace form is advantageous to delineate the search process in detail. For example, the mean firstpassage time33, 59 to locate the target site on closed DNA can be found from33 ~

𝑇9 = −

_.? M _M

(13)

|MC9

which readily yields to, 𝑇9 =

@ATT B5@A/ [B2UV (9)] @A/ @ATT UV (9)

9


(14)


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An identical approach can be taken while dealing with complex supercoiled DNA topology provided that the appropriate topological information is taken into account. To this end, we note that supercoil topology (Fig 1C-E) with the similar number of twists (Tw) differs from that of closed loop DNA in terms of writhe (Wr), reflected by the increase in the number of sites where the DNA helical axis crosses itself. At such crossover sites, two sequentially distant DNA sites come spatially close to each other and may promote the transfer of a protein molecule across the DNA segment. This can be viewed similarly to ‘intersegmental transfer’ mechanism35, where protein moves from one DNA site to another by binding transiently to both the sites. If the corresponding transition rate is kt, inter-communication of this nature between two sites m and mb (see Fig 1C) imposes a condition given by, -.c (0) -0

= 𝑢 𝑃d23 𝑡 + 𝑃d53 𝑡

+ 𝑘0 𝑃de 𝑡 + 𝑘788 𝑃9 𝑡 − 2𝑢 + 𝑘788 + 𝑘0 𝑃d 𝑡

(15)

The expression is similar to Eq. 1 with the additional second term that accounts for short jumps with rate kt between sites m and mb that are spatially close due to the supercoiled topology of the DNA molecule. The corresponding Laplace transformation results into ~

~

~

𝑠 + 2𝑢 + 𝑘788 + 𝑘0 𝑃d 𝑠 = 𝑢 𝑃d23 𝑠 + 𝑃d53 𝑠

~

~

+ 𝑘0 𝑃de 𝑠 + 𝑘788 𝑃9 𝑠

(16)

Therefore, the average search time for a protein on a supercoiled DNA with one crossover site (Fig 1C) can be estimated by solving Eqs (5) - (8) and Eq 16 simultaneously as described for closed loop DNA. For more complex supercoil topology, additional equations corresponding to each crossover site can be incorporated readily. The explicit form of T0 remains the same as in Eq. 14 with differences in auxiliary function Si(s). For supercoiled DNA with one crossover site 𝑆[ (𝑠) is given by,

10


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𝑆[ 𝑠 =

M5EW5@ATT 5@g S F 2S c 5@g S ce 2S FhcGce 5W 35S X S F 5S Xc 2ES c (S F 5S X )

S S c 23

where 𝜃 = S S Xc23

𝑦d 1 + 𝑦 1 − 𝑦B 𝜃 (1 − 𝑦)

M5EW5@ATT 5@g S XF 2S Xc 5@g S c S ce 2S XFGce 5W 35S X S XF 5S ic 2ES Xc (S XF 5S X )

(17)

Simulation Methods To study the target search process of a protein on a closed loop and supercoiled DNA topologies using Monte Carlo (MC) simulation, we use the method primarily described by Das et. al.60. The DNA molecule of L base pairs is designed by a self-avoiding random walk on a 3D lattice of size 100 x 100 x 100 with periodic boundary conditions. For closed loop DNA, two ends of the DNA chains are fused together, whereas, for supercoiled DNA topology with n number of juxtaposition sites, additional conditions are imposed that ensure if the protein reaches the desired sites (crossover sites), it is allowed to perform short jumps with MC acceptance criterion. For simplicity, we assume that the lattice sites are equally spaced. Any site in the lattice that is not occupied by the DNA chain is considered as the bulk solution. Initially, the protein is placed randomly in the solution phase from where it can associate to any of the lattice sites on the DNA with total association rate kon (kon/L for each DNA site). If the protein is non-specifically (other than target placed at Lth site) bound to DNA, it may either slide along the DNA with rate u or can perform 3D diffusion to move to bulk solution phase with rate koff. In the case of supercoiled DNA, if the protein reaches to crossover site m, it can jump to site mb with rate kt. The simulation strategy for closed loop DNA is as follows: At each MC step, we generate a random number, r (0 ≤ 𝑟 ≤ 1) and multiply it with

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the sum of the rates S (= 𝑢 + 𝑘788 + 𝑘7" ). If 𝑟𝑆 ≤ 𝑘7" , the searching protein binds to a random vacant DNA site. For 𝑘7" < 𝑟𝑆 ≤ (𝑘7" + 𝑘788 ), the non-specifically bound protein diffuses into the solution, whereas for (𝑘7" + 𝑘788 ) < 𝑟𝑆 ≤ (𝑢 + 𝑘7" + 𝑘788 ), the protein slides to adjacent DNA sites. For supercoiled DNA, the intersegmental jumps from the crossover site happen when (𝑢 + 𝑘7" + 𝑘788 ) < 𝑟𝑆 ≤ (𝑢 + 𝑘7" + 𝑘788 + 𝑘0 ), where 𝑆 = 𝑢 + 𝑘788 + 𝑘7" + 𝑘0 . After each MC step the

time is advanced by 1 𝑆. Once the protein reaches its target site either by 3D diffusion or 1D sliding the simulation finishes immediately and total search time is noted. To this end, one should note that for moderate values of L, the presence of crossover sites limits the conformational dynamics of the DNA molecule and is, therefore not taken into consideration during the simulation.

Results and Discussion We note that the Eqs (12) and (17) corresponding to closed loop and supercoiled DNA with one crossover site are entirely different. The mismatch stems from the fact that protein can perform additional jumps at crossover sites on supercoiled DNA with rate kt. With each jump, protein is likely to bypass scanning the fragment of DNA sequences in between the two sites involved in forming the crossover position. A similar intersegmental transfer mechanism of proteins on DNA was previously reported and is speculated to be one of the major determinants in speeding up the search kinetics20, 61. To critically assess the effectiveness of such intersegmental jumps in facilitating the target search process of DBPs, we fix site m at L/2 and vary the position of the conjugate site (mb). Depending on the position of mb, protein needs to translocate the rest of the distance ld = |L- mb | before it encounters the target site. Position dependent target search kinetics

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In Fig. 2, we present the ratio of target search times (α) on supercoiled DNA (T0 (1cs)) with one crossover site and closed (T0 (closed)) DNA as a function of ld for DNA of length 200 bp. Required rate constants are adopted from experimental measurements of lac repressor proteins28, 41. Lower α values indicate shorter the target search time on supercoiled DNA with one crossover site compared to that on closed loop DNA. For ld values greater than ~20 bp, α gradually reaches to ~ 1, suggesting comparable target search time in both the DNA topologies inspite of the intersegmental jump dynamics of the searching protein on supercoiled DNA. The following argument can be presented to explain the phenomenon. With the parameters chosen here, the sliding length62 (the

Figure 2. Position dependent acceleration in the target search process. The ratio of the relative search times of the supercoiled DNA with one crossover site and closed DNA is presented as a function of ld=|L-mb|, which is defined as the distance between the target site L and the crossover site mb (see Fig 1C). ld is varied by changing the position of mb. Following parameters are adopted: L=200 bp, m=L/2, koff = 500 s-1, u = kon = 105 s-1 and kt = 106 s-1.

average distance on DNA covered by a protein using 1D random walk during a single sliding event) is 𝜆 = 𝑢 𝑘 ≈ 14𝑏𝑝. Therefore, for 𝑙- > 20𝑏𝑝, the protein cannot scan the DNA segment 788 (𝑙- ) by performing only one sliding event, instead it requires multiple sliding events or combination

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of 1D and slower 3D diffusion events. Associated search time is longer and is comparable to that on closed loop DNA. In this context, it is interesting to note that the introduction of 1D diffusion was argued to facilitate the target search process. The resultant association rate was thought to be higher than the 3D diffusion-limited reactions because of the strong electrostatic attractions between the searching protein and the DNA as well as due to the reduction of dimensionality of the target search space63. The recent studies however, suggest that the sliding dynamics in presence of the DNA sequence-specific interactions can actually retards the overall 1D diffusivity of the protein64-68. Our results, suggests that the efficacy of intersegmental transfer to convey the protein to the target site is strongly dependent on position of landing site (mb) and is maximum (~ 3 times to the search time on closed DNA topology) when mb juxtaposes with target site L. The target site, in that case, is maximum approachable by the searching protein as it directly lands on the target site as soon it performs an intersegmental jump from a juxtaposition site. Role of DNA topology in target search kinetics

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Figure 3. (A) Target search time as a function of DNA length for different DNA topology. For open and closed DNA, the target is positioned at the site L and for supercoiled DNA the juxtaposition sites are placed symmetrically at L/2, L/2 - L/3, L/4 - 3L/8 - L/2 for supercoiled DNA with one, two and three juxtaposition sites respectively. Lines represent analytical calculations whereas results of Monte Carlo Simulations are represented by circles. (B) Target search time as a function of different DNA topology of lengths 5000 bp and 104 bp with juxtaposition sites up to 15. Parameters used are: koff = 102 s-1, u = kon = 105 s-1 and kt = 106 s-1. Results for supercoiled DNA with ten juxtaposition sites and fifteen juxtaposition sites are obtained through MC simulations only. An important question is how the numbers of juxtaposition sites affect the target search kinetics? To enquire this issue, we assume supercoiled DNA featuring two and three target sites juxtaposed with arbitrary DNA sites as shown in Fig 1D and 1E. Although juxtaposition involving all the target sites may not be the case always in living organisms, the dynamics of site juxtaposition can potentially lead to such situation16. We estimate the average target search time T0(s) for all the DNA topologies presented in Fig 1, and present the results in Fig 3A as a function of DNA length. Juxtaposition sites are assumed to be distributed symmetrically along the DNA length. Our analytical results, supported by extensive Monte Carlo simulations reveal that in addition to DNA topology, the number of juxtaposition sites also determine the efficiency of target search process. For example, compared to a 200 bp long open chain DNA conformation, protein finds the target site on a closed DNA ~2.5 times faster. This stems from the fact that in open chain DNA, the target site can be approached from one side, while in closed loop DNA or in circular DNA the target site

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is accessible from both the sides, resulting in lesser search time to find the target. This result is in agreement with previous theoretical and experimental observations, where facilitated diffusion has been reported in the presence of loop DNA54, 69. Similarly, if the DNA has supercoiled topology with one juxtaposition site placed at the middle of the DNA, we find ~8 times acceleration in the target site identification compared to that of open chain DNA. While both the examples delineate the impact of DNA topology, the importance of increasing number of juxtaposition sites can be understood from the fact that ~11 and ~14 times quicker target search are observed for supercoiled DNA with two and three juxtaposition sites respectively in comparison to open chain DNA with one target site. The faster search kinetics is due to two factors: 1) with increasing number of juxtaposition sites, protein performs intersegmental jumps often and thereby can bypass scanning a significant segment of the DNA. 2) The presence of higher number of juxtaposition sites also provides additional routes to approach the target sites along with the existing 1D diffusion paths from adjacent sites. It is, however, interesting to note that the rate of acceleration indicates a nonlinear dependence on DNA topology. For a more direct measurement, we estimate T0(s) for different DNA topology with varying lengths and present in Fig 3B. Our results indicate that the average target search time decreases exponentially as the complexity in DNA topology increases from open chain to closed loop to supercoiled DNA with increasing number of juxtaposition sites. Even if the number of target site remains fixed, the DNA topology alone can promote ~23% increment (Fig S1, Supporting Information) in search kinetics on plectonemic supercoiled (Fig 1E) topology compared to open chain DNA. The acceleration rate in target search process is also dependent on DNA length and slows down with increasing DNA length. For example, target search on 22000 bp supercoiled DNA with three juxtaposition sites is only ~8 times faster compared to that in open chain DNA conformation. However, if the DNA length is reduced to 200 bp for the

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same DNA topologies, the increment is as high as ~14.5 times. This indicates that for faster target site identification, it is important to have an optimal number of juxtaposition sites distributed over the entire length of DNA molecule. Effect of spatial distribution of juxtaposition sites

Figure 4. Normalized search times as a function of relative position of the sole juxtaposition site on a supercoiled DNA of length (A) 100 bp, (B) 1000 bp, (C) 10000 bp under different transition rates kt. The parameters used for these calculations are: koff = 102 s-1 and u= kon = 105 s-1. Solid lines represent theoretical results and the circles indicate Monte Carlo Simulation results.

It is not only the number of juxtaposition sites that primarily determines the target search kinetics, but the spatial distribution of such sites is also important. For example, Fig. 4 shows how the relative search times vary if the position of juxtaposition site on a supercoiled DNA changes under different transition rate kt. When the transition rate is small (kt ~ 1-10 s-1), it is less important where the juxtaposition site is located. The rationale behind this is that under this condition, protein rarely performs intersegmental jumps and therefore, the search time is nearly independent of the location of juxtaposition site. In contrast, when the transition rate is high enough (kt ranges from 103 s-1 to 106 s-1) to promote frequent intersegmental jumps, because of symmetry arguments the target search process is the quickest when juxtaposition point m is positioned at the middle of a 100 bp

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DNA chain (Fig 4A). The results presented in Fig 4B (L=1000) and 4C (L=10000) however, illustrates an interesting phenomenon. For L>>1, which is practically relevant to most of the cellular DNA, we do not find a pronounced minimum in the search time for the specific symmetric location of the juxtaposition site. Except when m=L or m=1, a case that corresponds to closed DNA topology, for all other locations of juxtaposition site we observe optimal and similar target search times. Does this mean the protein adopts different target search technique depending on the length of the DNA? Dynamic Phase Diagram of target search kinetics

To this end, we find the theory is extremely advantageous as it allows analyzing the target search dynamics for all ranges of relevant parameters precisely. We probe the DNA length dependent search dynamics by estimating the average search times as a function of the scanning length 𝜆. The result is presented in Fig. 5A, which shows three distinct dynamic phases that depend on the relative values of the length (L) of DNA and the average scanning length 𝜆 = 𝑢 𝑘 . For example, 788 the regime corresponding to 𝜆 ≥ 𝐿 represents a situation where due to high affinity between the protein and DNA molecules, the former exclusively slides or hops (1D diffusion) along the DNA. This can result in an effectively longer scanning length (𝜆) compared to the length of the DNA (L). Under this condition, Eq. 12 takes the form (see Supporting Information for details),

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𝑆[ (0) ≃ 𝐿 + 𝐿(1 − 𝐿E )

tGX 3E

+ 𝑂(𝜆2] )

(18)

using which in Eq 14, one finds that the average search time (𝑇9 ) of a protein on a closed loop DNA is proportional to 𝐿E . If the target is positioned at L-th site, the protein takes

Figure 5. Dynamic phase diagram for the protein search with different DNA topology. (A) Search times as a function of the scanning length are shown for open DNA, closed DNA and supercoiled DNA with one, two and three juxtaposition points. The shaded region corresponds to the experimentally observed scanning lengths. (B) - (D) represent the DNA length dependency of the target search times for different DNA topologies under the three regimes of dynamic search phases. These regimes are (B) when the scanning length λ is greater than the length of the DNA molecule, i.e 𝜆 ≥ 𝐿, (C) 1 ≤ 𝜆 ≤ 𝐿 and (D) 𝜆 < 1. The parameters used for both the calculations are: L=104 bp, u= kon = 105 s-1 and kt = 106 s-1. For an open chain, closed loop and supercoiled DNA with one juxtaposition site, the target sites are positioned at L-th site, whereas for supercoiled DNA with multiple juxtaposition points, targets are distributed symmetrically over the entire DNA length. Scanning length λ changes with the variation of koff. Solid lines correspond to theoretical calculations and the circles are from Monte Carlo Simulations.

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𝑇9 ≃

@A/ 5@ATT BX

Page 20 of 31

(19)

@A/ @ATT 3EtX

time to reach the target site through pure 1D motion. Previously, Veksler et.al34 has shown that the same target search time of a protein on linear/open DNA is 𝑇9 ≃

@A/ 5@ATT BX @A/ @ATT ]tX

. Comparing the re-

sults, one can easily identify that the search process is approximately four times faster on a closed loop DNA compared to that on an open DNA chain. Following the same procedure for supercoiled DNA topologies with one juxtaposition site, we find (see Supporting Information as well), 𝑇9 ≃

@A/ 5@ATT @g BX 53vBW @A/ @ATT Yw(@g 5ix)tX

(20)

F

where the target site is assumed to juxtapose with site 𝐿 2. The same for the supercoiled DNA with two and three juxtaposition sites are given by (see Supporting Information for details), 𝑇9 ≃

@A/ 5@ATT E@g BX 5yzBW

(21)

@A/ @ATT 3YY(@g 5D{x)tX F

with juxtaposition sites placed at L/2 and L/3 sites and 𝑇9 ≃

@A/ 5@ATT ]|@g BX 53wvEBW @A/ @ATT ]9|E(@g 5}~x)tX

(22)

F

with juxtaposition sites placed at L/4, 3L/8 and L/2 sites respectively. In Fig 5B, we present the length dependencies of target search time for mentioned DNA topologies. Considering 𝑘788 = 10𝑠 23 , the scanning length is estimated to be 100 bp. The shaded region of the plot shows that for 𝜆 ≥ 𝐿 , the average search time 𝑇9 varies with 𝐿E . In addition, it should also be noted t hat Eq. 20,

21 and 22 suggest dependency of 𝑇9 on the sliding rate 𝑢 and inter-segmental transition rate 𝑘0 as well. For a fixed sliding rate u=100000 s-1, Fig S2 signifies that increasing 𝑘0 accelerates the target site recognition process. However, the modes of dynamics are often coupled, such as enhancement of sliding during 1D diffusion lowers the hopping propensity38. Similarly, if the protein engages

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in intersegmental transition at the cost of the lowering of sliding time, Fig S3 indicates that the target search time decreases initially with increasing 𝑘0 but beyond a certain 𝑘0 value, the search process gets delayed (𝑇9 increases). This is because of the slower sliding rate that points out towards the fact that while a faster intersegmental transfer is essential to bypass scanning of a DNA segment and thereby lower the search time, a comparable sliding rate is also required to ensure recognition of the juxtaposition sites accurately from where the protein can jump to the target site. Failing to which due to slow sliding rate retards the target search process. Therefore, it is not just naive intersegmental transfer, rather a weighted combination of intersegmental jumps and 1D diffusion that optimises the target search process. Another dynamic search space represents the regime 1 ≤ 𝜆 ≤ 𝐿 , where the scanning length is smaller than the DNA length but larger than the target site. This means that the protein binds to the DNA, scans it, unbinds and diffuses into the solution in a repeated fashion before it reaches the target site. Therefore, this regime combines both 1D and 3D search pathways. If the length of DNA is too long, the number of such repeat cycles would be very large, leading to approximately identical target search times irrespective of the position of juxtaposition site (see Fig 4B and C). The average search time for a protein in this regime is also dependent on the DNA topology as can be seen in Fig 5C. Considering 𝑘788 = 1000𝑠 23 , the estimated scanning length is only ten bp. The shaded region, therefore, denotes the 1 ≤ 𝜆 ≤ 𝐿 regime, where average target search time varies linearly with L. The same was concluded previously that for an open chain DNA with target site placed at L-th site, the auxiliary function, 𝑆[ (0) becomes33, 𝑆[ (0) ≃

35 35YtX

(23)

E

which leads to an average target search time

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𝑇9 ≃

@A/ 5@ATT B

Page 22 of 31

(24)

@A/ @ATT t

according to Eq 14. For closed loop DNA topology in this regime the auxiliary function varies slightly and is given by, 𝑆[ (0) ≃ 1 + 4𝜆E

(25)

according to Eq 12. Corresponding average target search time is 𝑇9 ≃

@A/ 5@ATT B

(26)

@A/ @ATT Et

If the DNA exists in supercoiled topology, the average target search times are given by, 𝑇9 ≃

@A/ 5@ATT B @A/ @ATT Et

1−

@g

(27)

@ATT 5E@g

for one juxtaposition site placed at L/2, 𝑇9 ≃

@A/ 5@ATT B

(@g 5@ATT )X

(28)

@A/ @ATT Et @ATT (35@g )5@gX

for two juxtaposition sites located at L/2 and L/3 sites. For supercoiled DNA with three juxtaposition sites placed at L/4, 3L/8 and L/2 sites respectively, the analytical expression for average search time features complex terms. The underlying biophysics, however, remains the same and the average target search time exhibits identical length dependency as can be seen in Fig 5C. To this end, we emphasize that this search regime closely resembles the in vivo target search process of proteins that features a combination of both 1D and 3D diffusions. Although the theoretical limit of 𝑘788 has been extended up to ~1011 in order to examine both the pure 1D and 3D diffusion regimes, we note that in reality, the variation in sliding length is less scattered. Considering the experimentally estimated parameters for lac repressor, we find that the 𝜆 varies from one to 102 bp as shown in Fig 5A. The corresponding target search time for this scanning length predicted from our theory 22


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agrees well with the recent study carried out by Bauer et. al.47 for a target site size, 𝑁 ≤ 100 and the target DNA sequence playing crucial role in transcription factor (TF) binding. The third search regime corresponds to λ < 1, where the protein molecule can not even scan the neighbouring sites on a DNA because of high dissociation rate (𝑘788 = 10v 𝑠 23 ). The protein effectively finds the target site directly from the solution using 3D diffusion only. Expectedly, under that situation, the topology of DNA molecule is irrelevant and has no role to play in determining the target search dynamics. This is reflected in Fig 5D showing identical search times (proportional to the length of the DNA molecule, L) for all sorts of DNA topology.

Figure 6. Acceleration in the search time as a function of scanning length for three different dynamic search regimes. The parameters used for these calculations are: L=104 bp, u= kon = 105 s-1 and kt = 106 s-1. Scanning length λ changes with the variation of koff The overall role of DNA topology on the target search time can also be qualified by measuring the acceleration an in locating the target site on a given DNA topology with respect to open chain conformation. 𝑎" =

•A‚ƒ/„…†V/ •gA‚A‡Aˆ‰ (")

, 𝑛 = 0,1,2,3

23



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Here n = 0, 1, 2, 3 correspond to closed chain DNA and supercoiled DNA topology with one, two and three juxtaposition points respectively, distributed symmetrically along the DNA length. The results are presented in Fig 6. One can see that with increasing number of juxtaposition sites, the protein searches the target site ~ 4 - 27 times faster compared to linear DNA in the random walk regime (𝜆 ≥ 𝐿). For intermediate (1 ≤ 𝜆 ≤ 𝐿) region, where protein searches the target site using a combination of 1D sliding and 3D diffusion, as they do in in vivo the target search kinetics can be accelerated up to ~ 25 times on supercoiled DNA. Topology dependent acceleration in the search processes Having seen the role of DNA topology, number and position of juxtaposition sites in controlling the target search dynamics of DBPs, we finally probe if the average target search time is influenced by the relative positions of the consecutive juxtaposition sites along the DNA length. For this, we vary the relative positions 𝑙 of juxtaposition sites along the length of the supercoiled DNA and monitor the acceleration in target search process compared to that on an open chain DNA. The results for 200 bp supercoiled DNA with two and three juxtaposition sites are presented in Fig 7 featuring a common trend that the acceleration in target search process increases to reach a maximum followed by a steady fall. While probing the reason, we find that the relative positions of juxtaposition sites regulate the topology of supercoiled DNA. For example, in supercoiled DNA with two juxtaposition sites, if the sites are too close to each other or they overlap with each other,

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Figure 7. Acceleration in the search time as a function of relative positions of juxtaposition sites on a supercoiled DNA topology. For supercoiled DNA with two juxtaposition sites, we place juxtaposition sites initially at L/2+l and L/2-l respectively, where l=0. The relative positions of the sites are varied by gradually increasing l from zero to L/2. For supercoiled DNA with three juxtaposition sites, we vary the relative positions of juxtaposition sites according to two different schemes. In scheme 1, the juxtaposition sites are initially positioned at L/3-l, 2L/3-l/2 and 2L/3+l, whereas, according to scheme 2, these sites are placed at L/3-l/2, L/3+l and 2L/3+l/2. The relative positions are obtained by altering l from zero to L/3 and zero to 2L/3 for scheme 1 and 2 respectively. The parameters used for these calculations are: L=200 bp, koff = 102 s-1, u= kon = 105 s-1 and kt = 106 s-1. the resultant DNA essentially leads to a supercoiled topology with one juxtaposition site. This can be seen in the plot associated with two juxtaposition sites (green line in Fig 7), where for l=0, i.e., when the distance between two juxtaposition sites is non existing, and both the sites overlap with each other, corresponding acceleration in target search a2 is ~ 8 times compared to the search process observed on an open chain DNA33. This matches with the acceleration on supercoiled DNA with one juxtaposition site (a1), suggesting a transformation in the DNA topology due to the positioning of the juxtaposition sites. With the gradual separation between the two juxtaposition sites, acceleration increases and reaches the maximum (a2 ~ 14.9) for juxtaposition sites positioned symmetrically at L/3 and 2L/3. A further change in the position lowers the acceleration sharply because

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of the deviation from symmetry. When both the juxtaposition sites are moved by l=L/2, they coincide with the target site L, resulting into an abolition of the existence of juxtaposition sites. The supercoiled DNA topology with two juxtaposition sites, therefore, turns into a simple closed loop DNA and the associated acceleration a2 is ~ 2.4, a characteristic for a closed looped DNA. The same can be observed for supercoiled DNA with three or more juxtaposition sites. Here, we adopt two different schemes to monitor the change in acceleration due to shifting in the relative positions of three juxtaposition sites. In scheme 1, we start by positioning the juxtaposition sites at L/3-l, 2L/3-l/2 and 2L/3+l, whereas, according to scheme 2, these sites are L/3-l/2, L/3+l and 2L/3+l/2 respectively. Clearly, for both the schemes, if l=0, two out of three juxtaposition sites coincide with each other to lead to a supercoiled DNA topology with two juxtaposition sites. Resulting acceleration a3 is ~ 14.9 that matches the maximum value obtained for supercoiled DNA with two juxtaposition sites (green line). The positions of the juxtaposition sites in scheme 1 and 2 can be shifted maximally by l=L/3 and l=2L/3 respectively. For these, the three juxtaposition sites in scheme 1 coincide with each other such that together they can juxtapose with target site L. This leads to a supercoiled DNA topology with one juxtaposition site. However, according to scheme 2, a maximal shift of all the juxtaposition sites leads to a coincidence of all the juxtaposition sites with the target site L itself. The resultant DNA conformation assumes a closed loop topology only. Another important observation is that for scheme 1, the maximum acceleration is observed when the three juxtaposition sites are positioned at 5L/22, 5L/8 and 10L/13 (blue line) respectively. The same in scheme 2 is found if the sites are located at L/4, L/2 and 3L/4 (red line) respectively, indicating the role of symmetrical distribution of juxtaposition sites in accelerating the target search process. Altogether, these observations suggest that the degree of acceleration in target

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search process is not only a function of DNA topology and the number of juxtaposition sites, rather the relative positions of juxtaposition sites are equally important.

Conclusions In this study, we present a theoretical framework that provides a straightforward way to explain the role of DNA topology during the search of DBPs for the specific binding site on DNA. This approach, to the best of our knowledge, is first of its kind that explicitly takes into account the topological information of DNA molecule to evaluate its impact on search mechanism of DBPs. Applying our theory to closed loop and supercoiled DNA topologies with increasing juxtaposition sites, we find that the DNA topology is a dominant factor that affects the target search process of DBPs in two prong manners: 1) increase the frequency of performing intersegmental jumps on DNA if its topology promotes juxtaposition of distal DNA sites, 2) enhance the approachability of the target site for the protein by providing alternative routes through the juxtaposition sites. We emphasize the fact that for current calculations, we have ignored the mobility of juxtaposition sites, as the corresponding kinetics is reported extremely slow in nature16 compared to the target search process of proteins. Apart from that, our analytical results, which are also validated by extensive Monte Carlo simulations, propose that the number, position and the relative distribution of the juxtaposition sites are crucial and their variations can drastically affect the search dynamics. Thus, when a protein diffuses on a DNA, all these factors contribute in harmony to govern how and to what extent the topology of the DNA molecule may facilitate the target search process of the protein. While these results provide a testable hypothesis that can be verified by advanced experimental techniques, it is nonetheless, desirable to compare with the existing experimental observations. To this end, we note that an experiment based on EcoRV on 3466 bp supercoiled plasmid DNA has reported approximately four times hike in performance at bringing the EcoRV to its 27



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recognition site compared to relaxed plasmid topology20. For a similar sized DNA, our theory predicts 4.25 times increment in target search kinetics in supercoiled DNA with three juxtaposition sites as compared to closed loop DNA. The close agreement in the result indicates the generality of the method that can be adopted for investigating the role of various DNA topologies in the target search phenomenon of DBPs.

Acknowledgement We are grateful to Joseph M. Whitmeyer, University of North Carolina and Anatoly B. Kolomeisky, Rice University for insightful discussions and DST India (DST/INSPIRE/04/2013/000100, DST-SERB ECR/2016/000188, DST PURSE) and JNU (UPoE, Project ID 259) for providing financial assistances. A. M acknowledges the financial support from CSIR India in the form of Junior Research Fellow. Supporting Information. Additional details of the Theoretical Methods using auxiliary functions for different supercoiled DNA topologies; scaling behaviour for the search time versus DNA length; figures emphasizing the impacts of DNA topology on target search kinetics of DNA binding proteins.

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Understanding the Role of DNA Topology in Target Search Dynamics

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