Theory on the mechanism of DNA renaturation: Stochastic nucleation and zipping

Renaturation of complementary single strands of DNA is one of the important processes that requires better understanding in the view of molecular biology and biological physics. Here we develop a stochastic dynamical model on the DNA renaturation. According to our model there are at least three steps in the renaturation process viz. incorrect-contact formation, correct-contact formation and nucleation, and zipping. Most of the earlier two-state models combined nucleation with incorrect-contact formation step. In our model we suggest that it is considerably meaningful when we combine the nucleation with the zipping since nucleation is the initial step of zipping and the nucleated and zipping molecules are indistinguishable. Incorrect-contact formation step is a pure three-dimensional diffusion controlled collision process. Whereas nucleation involves several rounds of one-dimensional slithering dynamics of one single strand of DNA on the other complementary strand in the process of searching for the correct-contact and then initiate nucleation. Upon nucleation, the stochastic zipping follows to generate a fully renatured double stranded DNA. It seems that the square-root dependency of the overall renaturation rate constant on the length of reacting single strands originates mainly from the geometric constraints in the diffusion controlled incorrect-contact formation step. Further the inverse scaling of the renaturation rate on the viscosity of the reaction medium also originates from the incorrect-contact formation step. On the other hand the inverse scaling of the renaturation rate with the sequence complexity originates from the stochastic zipping which involves several rounds of crossing over the free-energy barrier at microscopic levels.Comment: 17 pages, 2 figure

Cooperative effects on DNA renaturation.

<p><b>A</b>. Probability density function associated with the one dimensional slithering length (<i>n</i> measured in bases) of cn-ssDNA in the process of searching for the correct-contact as given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.e042" target="_blank">Eq 13</a> for different values of the characteristic length ranging from 10 to 100 bases where <i>D</i>_<i>o</i> (base²s^-1) is the one dimensional diffusion coefficient associated with the slithering dynamics and <i>k</i>_<i>r</i> is the dissociation rate constant connected with cn-ssDNA. <b>B</b>. Zipping time (<i>τ</i>_<i>Z</i>, measured in seconds) in the presence of cooperative effects. Here sequence complexity (<i>c</i>) is same as that of the length (<i>L</i>) of c-ssDNA i.e. <i>c</i> = <i>L</i>. The number of correct-contacts <i>β</i> = <i>L</i>/<i>l</i>_<i>p</i> is a dimensionless quantity where <i>l</i>_<i>p</i> = 1 base and <i>L</i> is the length of the reacting c-ssDNA. Green solid line is calculation from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.e056" target="_blank">Eq 18</a> and blue solid line is calculation from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.e117" target="_blank">Eq C9</a> of <b>Appendix C</b>. Here we have set <i>K</i>_<i>Z</i> ~ 10⁻⁶ and <i>k</i>_<i>+</i> ~ 1 s^-1. Red solid line is the derivative of zipping time with respect to <i>β</i> as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.e060" target="_blank">Eq 21</a> which shows that the value of the derivative of overall zipping time with respect to <i>β</i> is < 10⁻² when <i>β</i> > 10². These plots suggest that when <i>K</i>_<i>Z</i> 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 <i>c</i> bases increases linearly with <i>c</i>. <b>C</b>. Variation of the overall renaturation rate <i>k</i>_<i>HR</i> with respect to length and complexity of c-ssDNA under relaxed conformational state. Here settings are M^-1s^-1, <i>Y</i>_<i>A</i> = 100 bases, <i>Y</i>_<i>E</i> = 1 bases and <i>n</i> = 10 bases. <b>D</b>. Variation of the overall renaturation rate <i>k</i>_<i>HR</i> as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.e038" target="_blank">Eq 12</a> with respect to the length of reacting c-ssDNA strands <i>L</i> and 1D slithering distance <i>n</i> under relaxed conformational state. Here settings are M^-1s^-1, <i>Y</i>_<i>A</i> = 100 bases, <i>Y</i>_<i>E</i> = 1 bases and <i>c</i> = 10 bases. In both <b>C</b> and <b>D</b>, <i>k</i>_<i>HR</i> shows a maximum at <i>L</i> = <i>L</i>_<i>opt</i>. Here Lopt can be obtained by solving <i>∂</i>_L<i>k</i>_<i>HR</i> = 0 for <i>L</i>. Explicitly one finds that . The dotted line in (<b>C)</b> is which breaks down beyond <i>L</i>_<i>opt</i>.</p

Mechanisms of DNA renaturation.

<p><b>A</b>. Collision between c-ssDNA strands leads to the formation of nonspecific contacts (cn-ssDNA) at a diffusion controlled bimolecular collision rate of <i>k</i>_<i>fQ</i> (Q = C for condensed conformation and Q = R for relaxed conformational state of c-ssDNA). <b>B</b>, <b>C</b>. Slithering and internal displacement mechanism are involved in the nucleation step of renaturation of c-ssDNA strands. Here slithering is a 1D diffusion dynamics (with unit base step size) of one of the cn-ssDNA strands on the other in the process of searching for the correct-contact to initiate nucleation and zipping. Slithering involves local dynamics of individual bases of one strand of cn-ssDNA over the other. Internal displacement mechanism involves inchworm type movement of one of the cn-ssDNA strand over the other. Here two different segments of the same cn-ssDNA strand involved in the inchworm type 1D diffusion dynamics where second nonspecific contact is formed between cn-ssDNA strands before the dissociation the former one with a dissociation rate <i>k</i>_<i>r</i>. In the illustration (<b>C</b>) three different locations of red colored strand of cn-ssDNA are marked as 1, 2 and 3. Initially position 2 of the probe c-ssDNA strand has a nonspecific contact with the template strand. In this condition the position 3 located on the freely moving overhang of probe strand makes contact with the template strand after which dissociation of the nonspecific contact at position 2 occurs. In this way the probe strand performs an inchworm type movement over the template strand. Occurrence of internal displacements in turn speeds up the 1D diffusion dynamics up to certain extent as in case of the intersegmental transfers via ring closure events associated with the site-specific DNA-protein interactions [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0153172#pone.0153172.ref031" target="_blank">31</a>]. Both slithering and internal displacement mechanism are thermally driven stochastic processes which independently contribute to the 1D diffusion coefficient <i>D</i>_<i>o</i>. Correct contact formation leads to nucleation with rate <i>k</i>_<i>N</i> beyond the critical nucleus size of <i>N</i> ~ 4–7 bases which in turn results in the zipping of cc-ssDNA strands into dsDNA with a rate of <i>k</i>_<i>Z</i>. <b>D</b>. Slithering seems to be analogous to the sliding mode of searching in the site-specific DNA-protein interactions whereas internal displacement is similar to that of the intersegmental transfer dynamics via ring closure events. Here two distal segments of the same DNA polymer come nearby in 3D space through thermally driven looping dynamics so that the nonspecifically bound protein molecule moves between them. As in DNA renaturation is the rate constant associated with the forward 3D diffusion mediated nonspecific binding of proteins with DNA and is the rate constant associated with the reverse dissociation step. Before reaching the CRMs (specific binding site) the protein molecules perform several rounds of 3D diffusion mediated association with DNA at random locations, 1D diffusion (which includes various modes of facilitating processes such as sliding, hopping and intersegmental transfers) along the DNA polymer and dissociations.</p

Earlier models on DNA renaturation kinetics.

<p>Renaturation of c-ssDNA strands was initially modelled as one-step bimolecular collision rate process as in <b>Scheme I</b> with an overall bimolecular association rate of <i>k</i>_<i>HQ</i>. <i>S</i> and <i>S’</i> are the concentrations of c-ssDNA and <i>H</i> is the concentration of completely renatured dsDNA. According to this model <b>Scheme I</b> the overall renaturation rate <i>k</i>_<i>HQ</i> should scale with the length of the reacting c-ssDNA strands in a linear manner. However experiments revealed a square-root dependency of the renaturation rate on the length of reacting c-ssDNA strands. To comply with the experimental observation a two-step mechanism was proposed as in <b>Scheme II</b> which comprised of nucleation and zipping. In this mechanism the nucleation rate (<i>k</i>_<i>N</i>) is inversely proportional to the square-root of the length of c-ssDNA strands. This scaling seems to emerge as a consequence of excluded volume effects of c-ssDNA polymer. Whereas the zipping rate (<i>k</i>_<i>Z</i>) is directly proportional to the length of c-ssDNA strands (<i>L</i>) and inversely proportional to the sequence complexity (<i>c</i>). Since the overall renaturation rate is directly proportional to both <i>k</i>_<i>N</i> and <i>k</i>_<i>Z</i> one observes a square-root dependency of the overall renaturation rate on the length of c-ssDNA strands. To generalize nucleation is modelled as a reversible process with a dissociation rate constant <i>k</i>_<i>r</i>. Here <i>Y</i>_<i>N</i> is the concentration of the nucleus. <b>Sequence complexity</b> of c-ssDNA is defined as the length of DNA with unique sequence pattern. For example consider sequences S1, S2 and S3 all with length of <i>L</i> = 30 bases. By definition the sequence complexity of S1 is <i>c</i> = 30 bases. Complexity of S2 is <i>c</i> = 10 bases since it has 3 repeats of ATGATCTACG with 10 bases length. In the same way, the complexity of S3 is <i>c</i> = 5 bases since it has 6 repeats of ATGAT with 5 bases length. The copy numbers <i>ρ</i> = <i>L</i>/<i>c</i> of S1, S2 and S3 are 1, 3 and 6 respectively. The zipping rate in two-step renaturation models as in <b>Scheme II</b> is directly proportional to this copy number <i>ρ</i>. This means that the overall renaturation rate is inversely proportional to the sequence complexity of the reacting c-ssDNA strands.</p

Basic steps of DNA renaturation phenomenon.

<p><b>A</b>. Three basic steps in the renaturation of complementary single strands of DNA (c-ssDNA) are viz. nonspecific-contact formation, nucleation and zipping. Two arbitrary locations on the probe c-ssDNA are marked as 1 and 2 (blue and green dots respectively). Nonspecific-contact formation (cn-ssDNA) is purely a three dimensional (3D) diffusion controlled collision rate process (I) where the rate constant associated with the formation of nonspecific-contact scales with the length of colliding c-ssDNA molecules in a square root manner and it scales with the solvent viscosity in an inverse manner. Nucleation involves a one dimensional (1D) slithering dynamics (II) of one strand on the other strand of cn-ssDNA in the process of searching for correct-contact (cc-ssDNA). Internal displacement dynamics through inchworm movements (III) of one complementary strand on the other can facilitate the 1D diffusion dynamics. Upon finding the correct-contact and forming the nucleus, zipping of cc-ssDNA step (IV) follows. <b>Scheme III.</b> According to this scheme both the nucleation and zipping are coupled stochastic dynamical processes. In this scheme there are three distinct steps in the process of renaturation viz. 3D diffusion mediated nonspecific contact formation with an on rate of <i>k</i>_<i>fQ</i> and off-rate of <i>k</i>_<i>r</i>, 1D and 3D diffusion mediated nucleation step with rate <i>k</i>_<i>N</i> and zipping which is a 1D diffusion process with a rate of <i>k</i>_<i>Z</i>. Before forming a successful nucleus with a critical size of <i>N</i> bases the colliding c-ssDNA strands undergo several rounds of nonspecific contact formation to form cn-ssDNA, 1D diffusion of one of the cn-ssDNA strands over the other and then dissociation. Upon formation of the nucleus (cc-ssDNA with <i>N</i> numbers of correct contacts) zipping process commences. Since nucleated cc-ssDNA is indistinguishable from zipping one, it is more appropriate to combine the nucleation with the zipping with an overall rate of <i>k</i>_<i>NZ</i> = 1/(1/<i>k</i>_<i>N</i> + 1/<i>k</i>_<i>Z</i>) rather than with the nonspecific-contact formation step as in <b>Scheme II</b>. Conformational state of the reacting c-ssDNA molecules seems to significantly affect the reaction mechanism and scaling relationships associated with the overall renaturation rate on the size of the system. <b>B</b>, <b>C</b>. We can model the c-ssDNA chains as clusters of nitrogen bases so that 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 <i>C</i>_<i>M</i> ~ <i>2πr</i>_<i>D</i><i>M</i> of a c-ssDNA molecule with a radius of <i>r</i>_<i>D</i> bases will be confined within the spherical solvent shell with surface area (<i>M</i> = <i>L</i> for template and <i>M</i> = <i>l</i> for probe c-ssDNA strands) where <i>r</i>_<i>M</i> is the radius of gyration of the respective c-ssDNA molecule. Under strongly condensed state of c-ssDNA one finds that <i>V</i>_<i>M</i> < <i>C</i>_<i>M</i> (<b>C</b>) and when the DNA polymer is in a relaxed state then one find that <i>V</i>_<i>M</i> > <i>C</i>_<i>M</i>. At a coarse grained level one can model the bases of c-ssDNA as a chain of spherical beads with radius <i>r</i>_<i>D</i>. Under relaxed conformational state all these nitrogen base beads are distributed on the surface of the spherical solvent shell that covers a c-ssDNA molecule (<b>B)</b>. Under condensed conformational state of c-ssDNA molecules significant fraction of nitrogen base beads will be inaccessible to the inflowing c-ssDNA molecules since they are buried inside the matrix of condensed c-ssDNA (<b>C</b>).</p

Various symbols and their definitions

<p>Various symbols and their definitions</p

Codon Usage Provide Insights into the Adaptation of Rice Genes under Stress Condition

Plants experience different stresses, i.e., abiotic, or biotic, and to combat them, plants re-program the expression of growth-, metabolism-, and resistance-related genes. These genes differ in their synonymous codon usage frequency and show codon usage bias. Here, we investigated the correlation among codon usage bias, gene expression, and underlying mechanisms in rice under abiotic and biotic stress conditions. The results indicated that genes with higher expression (up- or downregulated) levels had high GC content (≥60%), a low effective number of codon usage (≤40), and exhibited strong biases towards the codons with C/G at the third nucleotide position, irrespective of stress received. TTC, ATC, and CTC were the most preferred codons, while TAC, CAC, AAC, GAC, and TGC were moderately preferred under any stress (abiotic or biotic) condition. Additionally, downregulated genes are under mutational pressure (R2 ≥ 0.5) while upregulated genes are under natural selection pressure (R2 ≤ 0.5). Based on these results, we also identified the possible target codons that can be used to design an optimized set of genes with specific codons to develop climate-resilient varieties. Conclusively, under stress, rice has a bias towards codon usage which is correlated with GC content, gene expression level, and gene length