Murugan R

We develop a theory that explains how the thermally driven conformational fluctuations in the DNA binding domains (DBDs) of the DNA binding proteins (DBPs) are effectively coupled to the one-dimensional searching dynamics of DBPs for their cognate sites on DNA. We show that the rate γopt, associated with the flipping of conformational states of DBDs beyond which the maximum search efficiency of DBPs is achieved, varies with the one-dimensional sliding length L as γopt ∝ L-2 and with the number of roadblock protein molecules present on the same DNA m as γopt ∝ m2. The required free energy barrier ERTO associated with this flipping transition seems to be varying with L as ERTO ∝ In L2. When the barrier height associated with the conformational flipping of DBDs is comparable with that of the thermal free energy, then the possible value of L under in vivo conditions seems to be L ≤ 70 bps. © 2010 by the Biophysical Society.

Analytical solution to the Michaelis–Menten (MM) rate equations for single-substrate enzyme catalysed reaction is not known. Here we introduce an effective scaling scheme and identify the critical parameters which can completely characterize the entire dynamics of single substrate MM enzymes. Using this scaling framework, we reformulate the differential rate equations of MM enzymes over velocity-substrate, velocity-product, substrate-product and velocity-substrate-product spaces and obtain various approximations for both pre- and post-steady state dynamical regimes. Using this framework, under certain limiting conditions we successfully compute the timescales corresponding to steady state, pre- and post-steady states and also compute the approximate steady state values of velocity, substrate and product. We further define the dynamical efficiency of MM enzymes as the ratio between the reaction path length in the velocity-substrate-product space and the average reaction time required to convert the entire substrate into product. Here dynamical efficiency characterizes the phase-space dynamics and it would tell us how fast an enzyme can clear a harmful substrate from the environment. We finally perform a detailed error level analysis over various pre- and post-steady state approximations along with the already existing quasi steady state approximations and progress curve models and discuss the positive and negative points corresponding to various steady state and progress curve models.

We develop a generalized theoretical framework on the binding of transcription factor proteins (TFs) with specific sites on DNA that takes into account the interplay of various factors regarding overall electrostatic potential at the DNAprotein interface, occurrence of kinetic traps along the DNA sequence, presence of other roadblock protein molecules along DNA and crowded environment, conformational fluctuations in the DNA binding domains (DBDs) of TFs, and the conformational state of the DNA. Starting from a Smolochowski type theoretical framework on site-specific binding of TFs we logically build our model by adding the effects of these factors one by one. Our generalized two-step model suggests that the electrostatic attractive forces present inbetween the positively charged DBDs of TFs and the negatively charged phosphate backbone of DNA, along with the counteracting shielding effects of solvent ions, is the core factor that creates a fluidic type environment at the DNAprotein interface. This in turn facilitates various one-dimensional diffusion (1Dd) processes such as sliding, hopping and intersegmental transfers. These facilitating processes as well as flipping dynamics of conformational states of DBDs of TFs between stationary and mobile states can enhance the 1Dd coeffcient on a par with three-dimensional diffusion (3Dd). The random coil conformation of DNA also plays critical roles in enhancing the site-specific association rate. The extent of enhancement over the 3Dd controlled rate seems to be directly proportional to the maximum possible 1Dd length. We show that the overall site-specific binding rate scales with the length of DNA in an asymptotic way. For relaxed DNA, the specific binding rate will be independent of the length of DNA as length increases towards infinity. For condensed DNA as in in vivo conditions, the specific binding rate depends on the length of DNA in a turnover way with a maximum. This maximum rate seems to scale with the maximum possible 1Dd length of TFs in a square root manner. Results suggest that 1Dd processes contribute much less to the enhancement of specific binding rate under in vivo conditions for condensed DNA. There exists a critical length of binding stretch of TFs beyond which the probability associated with the random occurrence of similar specific binding sites will be close to zero. TFs in natural systems from prokaryotes to eukaryotes seem to handle sequencemediated kinetic traps via increasing the length of their recognition stretch or combinatorial binding. TFs overcome the hurdles of roadblocks via switching effciently between sliding, hopping and intersegmental transfer modes. The site-specific binding rate as well as the maximum possible 1Dd length seem to be directly proportional to the square root of the probability (pR) of finding a nonspecific binding site to be free from dynamic roadblocks. Here pR seems to be a function of the number of nsbs available per DNA binding protein (φ) inside the living cell. It seems that pR > 0.8 when φ > 10 which is true for the Escherichia coli cell system.

We show that nucleosomes exert a maximal amount of hindrance to the one-dimensional diffusion of transcription factors (TFs) when they are present between TFs and their cognate sites on DNA. The effective one-dimensional diffusion coefficient of TFs (χTF) decreases with a rise in the free-energy barrier (μNU) of the sliding of nucleosomes as χTF∝exp(−μNU). The average time (ηL) required by TFs to slide over L sites on DNA increases with μNU as ηL∝exp(μNU). When TFs move close to nucleosomes, then they exhibit typical subdiffusion. Nucleosomes can enhance the search dynamics of TFs when TFs are present between nucleosomes and TF binding sites. These results suggest that nucleosome-depleted regions around the cognate sites of TFs are mandatory for efficient site-specific binding of TFs. Remarkably, the genome-wide in vivo positioning pattern of TFs shows a maximum at their specific binding sites where the occupancy of nucleosomes shows a minimum. This could be a consequence of an increasing level of breathing dynamics of nucleosome cores and decreasing levels of fluctuations in the DNA binding domains of TFs as they move across TF binding sites. The dynamics of TFs becomes slow as they approach their cognate sites so that TFs form a tight site-specific complex, whereas the dynamics of nucleosomes becomes rapid so that they quickly pass through the cognate sites of TFs. Several in vivo data sets on the genome-wide positioning pattern of nucleosomes and TFs agree well with our arguments. The retarding effects of nucleosomes can be minimized when the degree of condensation of DNA is such that it can permit a jump size associated with the dynamics of TFs beyond ∼160–180 bp.

Eukaryotic genes are typically split into exons that need to be spliced together to form the mature mRNA. The splicing process depends on the dynamics and interactions among transcription by the RNA polymerase II complex (RNAPII) and the spliceosomal complex consisting of multiple small nuclear ribonucleo proteins (snRNPs). Here we propose a biophysically plausible initial theory of splicing that aims to explain the effects of the stochastic dynamics of snRNPs on the splicing patterns of eukaryotic genes. We consider two different ways to model the dynamics of snRNPs: pure three-dimensional diffusion and a combination of three- and one-dimensional diffusion along the emerging pre-mRNA. Our theoretical analysis shows that there exists an optimum position of the splice sites on the growing pre-mRNA at which the time required for snRNPs to find the 5′ donor site is minimized. The minimization of the overall search time is achieved mainly via the increase in non-specific interactions between the snRNPs and the growing pre-mRNA. The theory further predicts that there exists an optimum transcript length that maximizes the probabilities for exons to interact with the snRNPs. We evaluate these theoretical predictions by considering human and mouse exon microarray data as well as RNAseq data from multiple different tissues. We observe that there is a broad optimum position of splice sites on the growing pre-mRNA and an optimum transcript length, which are roughly consistent with the theoretical predictions. The theoretical and experimental analyses suggest that there is a strong interaction between the dynamics of RNAPII and the stochastic nature of snRNP search for 5′ donor splicing sites. © 2012 Murugan, Kreiman.

In this paper, we develop a theory on the mechanism of distal action of the transcription factors, which are bound at their respective cis-regulatory enhancer modules on the promoter-RNA polymerase II (PR) complexes to initiate the transcription event in eukaryotes. We consider both the looping and tracking modes of their distal communication and calculate the mean first passage time that is required for the distal interactions of the complex of enhancer and transcription factor with the PR via both these modes. We further investigate how this mean first passage time is dependent on the length of the DNA segment (L, base-pairs) that connects the cis-regulatory binding site and the respective promoter. When the radius of curvature of this connecting segment of DNA is R that was induced upon binding of the transcription factor at the cis-acting element and RNAPII at the promoter in cis-positions, our calculations indicate that the looping mode of distal action will dominate when L is such that L > 2πR and the tracking mode of distal action will be favored when L < 2πR. The time required for the distal action will be minimum when L = 2πR where the typical value of R for the binding of histones will beR ∼ 16 bps and L ∼ 102 bps. It seems that the free energy associated with the binding of the transcription factor with its cis-acting element and the distance of this cis-acting element from the corresponding promoter of the gene of interest is negatively correlated. Our results suggest that the looping and tracking modes of distal action are concurrently operating on the transcription activation and the physics that determines the timescales associated with the looping/tracking in the mechanism of action of these transcription factors on the initiation of the transcription event must put a selection pressure on the distribution of the distances of cis-regulatory modules from their respective promoters of the genes. The computational analysis of the upstream sequences of promoters of various genes in the human and mouse genomes for the presence of putative cisregulatory elements for a set of known transcription factors using the position weight matrices available with the JASPAR database indicates the presence of cis-acting elements with maximum probability at a distance of ∼102 bps from the promoters which substantiates our theoretical predictions. © 2010 IOP Publishing Ltd.

We demonstrate that a protein molecule of interest undergoing the one-dimensional Brownian dynamics along DNA can exhibit a directional dependent net transport either toward or away from its target site depending on the distribution of the initial positions of the other classes of protein molecules present on the same DNA. Directionality arises as a consequence of the confinement of the search space and dynamic reflections by other protein molecules present on the same DNA chain. Energy cost for such directionality comes from the free energy spent on setting the initial positions of the other protein molecules. In the mechanism of action of cis-acting elements on the initiation of transcription, such free-energy inputs are derived from the site-specific binding affinities of the inflowing transcriptional factors toward their cis-acting elements. If the initial distribution of other protein molecules is a random one, then the protein molecule of interest exhibits a net transport away from its target site. This directionality originates from unequal natures of enhancing and retarding effects of the randomly distributed other classes of protein molecules. The protein molecule of interest overcomes the retarding effects of the other classes of protein molecules in a dynamical manner by increasing the number of dissociation-association events when it is far away from its target site and then by switching back to the sliding dynamics due to increase in the enhancing effects as it moves closer to its target site. © 2009 The American Physical Society.

We derive a functional relationship between the mean first passage time associated with the concurrent binding of multiple transcription factors (TFs) at their respective combinatorial cis-regulatory module sites (CRMs) and the number n of TFs involved in the regulation of the initiation of transcription of a gene of interest. Our results suggest that the overall search time τs that is required by the n TFs to locate their CRMs which are all located on the same DNA chain scales with n as τs∝n α where α ∼ (2/5). When the jump size k that is associated with the dynamics of all the n TFs along DNA is higher than that of the critical jump size kc that scales with the size of DNA N as kc ∼ N2/3, we observe a similar power law scaling relationship and also the exponent α. When k < kc, α shows a strong dependence on both n and k. Apparently there is a critical number of combinatorial TFs nc ∼ 20 that is required to efficiently regulate the initiation of transcription of a given gene below which (2/5) < α < 1 and beyond which α > 1. These results seem to be independent of the initial distances between the TFs and their corresponding CRMs and also suggest that the maximum number of TFs involved in a given combinatorial regulation of the initiation of transcription of a gene of interest seems to be restricted by the degree of condensation of the genomic DNA. The optimum number mopt of roadblock protein molecules per genome at which the search time associated with these n TFs to locate their binding sites is a minimum seems to scale as mopt ∝ Ln α/2 where L is the sliding length of TFs whose maximum value seems to be such that L ≤ 104 bps for the E. coli bacterial genome. © 2010 IOP Publishing Ltd.

Feedforward loops (FFLs) consist of three genes which code for three different transcription factors A, B and C where B regulates C and A regulates both B and C. We develop a detailed model to describe the dynamical behavior of various types of coherent and incoherent FFLs in the transcription factor networks. We consider the deterministic and stochastic dynamics of both promoter-states and synthesis and degradation of mRNAs of various genes associated with FFL motifs. Detailed analysis shows that the response times of FFLs strongly dependent on the ratios (wh = γpc/γph where h = a, b, c corresponding to genes A, B and C) between the lifetimes of mRNAs (1/γmh) of genes A, B and C and the protein of C (1/γpc). Under strong binding conditions we can categorize all the possible types of FFLs into groups I, II and III based on the dependence of the response times of FFLs on wh. Group I that includes C1 and I1 type FFLs seem to be less sensitive to the changes in wh. The coherent C1 type seems to be more robust against changes in other system parameters. We argue that this could be one of the reasons for the abundant nature of C1 type coherent FFLs. © 2012 Rajamanickam Murugan.

The speed of site-specific binding of transcription factor (TFs) proteins with genomic DNA seems to be strongly retarded by the randomly occurring sequence traps. Traps are those DNA sequences sharing significant similarity with the original specific binding sites (SBSs). It is an intriguing question how the naturally occurring TFs and their SBSs are designed to manage the retarding effects of such randomly occurring traps. We develop a simple random walk model on the site-specific binding of TFs with genomic DNA in the presence of sequence traps. Our dynamical model predicts that (a) the retarding effects of traps will be minimum when the traps are arranged around the SBS such that there is a negative correlation between the binding strength of TFs with traps and the distance of traps from the SBS and (b) the retarding effects of sequence traps can be appeased by the condensed conformational state of DNA. Our computational analysis results on the distribution of sequence traps around the putative binding sites of various TFs in mouse and human genome clearly agree well the theoretical predictions. We propose that the distribution of traps can be used as an additional metric to efficiently identify the SBSs of TFs on genomic DNA.