Random fluctuations and noise are ubiquitous within nature and will generally occur on some length scale. Consider long linear DNA strands suspended within a solution at some temperature $T$ and salt concentration Or in-vivo, compactified within a living cell, within a biological environment with average ambient temperature T. Thermal fluctuations will cause the DNA ribbon to fluctuate weakly about its average geometry. In addition, the thermal fluctuations will also result in the spontaneous opening and closing of the bonds or bridges spanning the edges of the DNA ribbon. The DNA ribbon is therefore not a static structure but fluctuates randomly, essentially becoming a stochastic or random geometry.} The aim therefore, will be to describe the fluctuating DNA ribbon mathematically, in terms of random geometrical and topological terms. First, some statistical mechanical issues are considered. The following proposition follows the arguments in Huang. The following assumptions are made:

A DNA suspension within a liquid at temperature and some given salt concentration, or in=vivo within a cell. It can be considered within the canonical ensemble as a subsystem within a heat bath at fixed temperature. The DNA strand of length $\ell$ and sugar-phosphate backbone edges $h,\bar{h})$ has N bonds or weak hydrogen bridges across the ribbon binding the A-T and G-C nucleic acids. At temperature T the ribbon weakly fluctuates and the bridges open and close. A open bond can be assigned an energy $E$ and an open bond has energy $\mathcal{E}=0$. A link can be open only if all the other links along the ribbon are already open. The possible states are labeled by the number of open bridges or links $\alpha=0,1,2,...,N$ so that the energy with $\alpha$ open links is $E_{\alpha}=\alpha{E}$ The at low temperatures $E/T\gg 1$, and so there are few open links in that $\langle\alpha\rangle\sim \exp(-E/T)$. At high temperatures $E/T\ll 1$, and so most of the links are open in that $\langle\alpha\rangle\sim \alpha$
The partition function is

$Z_{n}=\sum_{\alpha=0}^{n}\exp[-\alpha{E}/T]\equiv\frac{1-\exp[-\beta(n+1)E}{1-exp[-\beta E]}$

where $\beta=1/T$ is the inverse temperature. The expected number of open links at temperature $T$ is then given by the thermodynamic relation

$\langle\alpha\rangle=-\frac{1}{E}\frac{\partial \ln[Z_{n}]}{\partial\beta}$ so that $\langle\alpha\rangle = \frac{\sum_{\alpha=0}^{N}\alpha n\exp[-nE/T]}{\sum_{\alpha=0}^{n}\alpha \exp[-nE/T]}=\frac{\exp[-E/T]}{1 - \exp[-E/T]}-\frac{(n+1)\exp[-(n+1)E/T}{1-\exp[-(n+1)E/T]}$

Then at low temperatures $E/T\gg 1$, and so there are few open links in that $\langle\alpha\rangle\sim \exp(-{E}/T)$. At high temperatures $E/T\ll 1$, and so most of the links are open in that $\langle\alpha\rangle\sim \alpha$

This suggests that DNA will exist and function only within certain temperature ranges. The molecule dissociates at high temperatures. (Which also implies that DNA-based life cannot arise and evolve in high-temperature conditions in the universe.) Superhelicity also plays a role in that the free energy $\delta G$ associated with supercoiling is of the quadratic form $\Delta G \sim K(Lk-Lk_{0})^{2}$, where K is a proporitonality constant. This quadratic relation has been experimentally shown (refs.) DNA molecules that differ only in $\mathbf{LK}$ are  topoisomers. The concentration $C[Lk,T]$ of a given topoisomer with linking number $Lk$ at temperature T, is the Boltzmann distribution

$C[\mathcal{LK},T]=\frac{1}{Z}\exp[[-\Delta G/RT]\sim \frac{1}{Z}\exp[-K(\mathcal{Lk}-\mathcal{L}_{r})^{2}/R$ which is a normal or Gaussian distribution centred on $\mathbf{LR}$, the relaxed linking number, and with standard deviation $\sim \sqrt{RT/2K}$. Proteins which regulate the supercoiling and maintain this distribution are known as topoisomerases. For high temperatures $C[\mathcal{LK},T]\rightarrow 0$.

Taking into account thermodynamic, mechanical and biophysical properties of fluctuating DNA molecules within a thermal bath is complex. However, the aim of this section is to focus on a tractable stochastic mathematical description of a fluctuating DNA ribbon, in terms of a ‘stochastic geometry and topology’ which arises from contact with a thermal bath, or more precisely, a thermal noise bath.
\item This introduces analogies with Brownian motion and turbulence, which are studied and described on length scales as stochastic processes and where the deeper underlying statistical mechanics can be ignored. Indeed, the stochastic fluctuations of the DNA double-helix structure when immersed in a thermal bath whether in-vitro or in vivo, is a form of Brownian motion, arising from bombardment of the double helix by a ‘warm sea’ of smaller surrounding molecules.

For proteins, which consist of mobile helices, strands and sheets, Brownian bombardment from water molecules will also play an important role in the folding of the protein. Simple Brownian motion of a particle in a thermal bath at temperature T is described by a linear SDE of the for

$d\widehat{X}(t)=dX(t)dt + \rho d\widehat{\mathcal{W}}(t)$

where $X(t)$ is the position of the Brownian particle. The random field $\widehat{\mathcal{W}}(t)$ is a thermal white noise which described the effect of the thermal Brownian bombardment of the particle by the smaller water molecules. Generally

$\mathbb{E}\langle\widehat{\mathcal{W}}(t)\rangle=0$ and by the

equipartition theory $\mathbb{E}\langle \widehat{\mathcal{W}}(t)\otimes\widehat{\mathcal{W}}(s)\rangle \sim kT \delta(t-s)$, where $\mathbb{E}\langle...\rangle$ is the statistical expectation, k is Boltzmann constant. The solution is the well-defined stochastic Ito integral

$X(t)=\int_{0}^{t}X(u)du + \rho\int_{0}^{t}d\widehat {\mathcal{W}}(u)= \int_{0}^{t}X(u)du + \rho\int_{0}^{t}\widehat{\psi}(u)du$

Suppose by analogy, we consider the tangent vectors $X_{i}(x(s))$ and $X_{j}(y(s))$ along the edges of the DNA ribbon. If the DNA ribbon undergoes a Brownian bombardment from smaller molecules, these vectors will be randomly perturbed as the ribbon fluctuates randomly about its average geometry/topology.

$d\widehat{X}_{i}(x(s))=dX_{i}(x(s))+d\widehat{\mathcal{F}}_{i}(x(s))$

$d\widehat{X}_{j}(y(s))=dX_{j}(y(s))+d\widehat{\mathcal{F}}_{j}(y(s)$

where $\widehat{\mathcal{F}}_{i}(x(s))$ is a Gaussian random vector field defined at all points in a domain $\mathbb{D}$ containing the ribbon. GRVFs are discussed in the Appendix. The linking number will not be affected by thermal fluctuations of the DNA ribbon but the linking number $\mathcal{LK}=\mathcal{TW}+\mathcal{WR}$ will be randomly partitioned into twist and writhe which will randomly change. The total remains constant however. This can be made more precise as follows:

Let $R(h,\bar{h},\alpha,\beta,\epsilon,p:Lk,Sh\subset\mathbb{D}$ be a DNA ribbon with linking number $Lk$ and superhelicity $Sh$ and let
$X_{i}(x)$ and $X_{j}(y)$ be tangent vectors at $x^{i}\in h$ and $y^{j}\in \bar{h}$. The Wilson lines along the helices or ribbon edges are

$W[h]=exp\bigg(i\int_{h}X_{i}(x)dx^{i}\bigg)=\exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2}Lk(h,\bar{h}))$

$W[\bar{h}]=exp\bigg(i\int_{\bar{h}}X_{i}(x)dx^{i}\bigg)=exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h}))$

If the domain $\mathbb{D}$ is a thermal bath at some temperature $T$ then the tangent vectors can be interpreted as random Gaussian vector fields $\widehat{X}_{i}(x)$ and $\widehat{X}_{j}(y)$ such that

$\widehat{X}_{i}(x)= X_{i}(x) + \widehat{\mathcal{F}}_{i}(x) \widehat{X}_{j}(y)= X_{j}(y) + \widehat{\mathcal{F}}_{j}(y)$

where the GRVF $\widehat{\mathcal{F}}_{i}(x)$ obeys the statistics
$\mathbf{E}(\widehat{\mathcal{F}}_{i}(x))=0,~~~~\mathbf{E}(\widehat{\mathcal{F}}_{i}(x)\otimes \widehat{\mathcal{F}}_{j}(y))=C_{ij}(x,y;\zeta)$ and where $C_{ij}(x,y;\zeta)$ is a 2-point function or covariance regulated at $x=y$ and $\zeta$ is a correlation length such that $C_{ij}(x-y)\rightarrow 0$ for $|x-y|\gg \zeta$.
(Appendix A.)

The stochastic Wilson lines along the helices are then
$W[h]=exp\bigg(i\int_{h}\widehat{X}_{i}(x)dx^{i}\bigg)=exp\bigg(i\int_{h}(\widehat{X}_{i}(x) +\widehat{\mathcal{F}}_{i}(x))dx^{i}\bigg)=exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h}))exp\bigg(i\int_{h}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)$

$W[\bar{h}])=exp\bigg(i\int_{h}\widehat{X}_{i}(x)dx^{i}\bigg)=exp\bigg(i\int_{h}(\widehat{X}_{i}(x) +\widehat{\mathcal{F}}_{i}(x))dx^{i}\bigg)=exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h}))exp\bigg(i\int_{\bar{H}}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)$

The stochastic expectations of the Wilson lines are
$\mathbf{W}[h]=\mathbf{E}(W[h])=\mathbf{E}\bigg(exp\bigg(i\int_{h}\widehat{X}_{i}(x)dx^{i}\bigg)\bigg)=exp\bigg(i\int_{h}(\widehat{X}_{i}(x) +\widehat{\mathcal{F}}_{i}(x))dx^{i}\bigg) = exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h}))\mathbf{E}\bigg(exp\bigg(i\int_{h}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)\bigg)$

\$\mathbf{W}[\bar{h}]=\mathbf{E}(W[H])=\mathbf{E}\bigg(exp\bigg(i\int_{h}\widehat{X}_{i}(x)dx^{i}\bigg)\bigg)= exp\bigg(i\int_{h}(\widehat{X}_{i}(x) +\widehat{\mathcal{F}}_{i}(x))dx^{i}\bigg)=exp(\tfrac{1}{2}i(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h}))\mathbf{E}\bigg(exp\bigg(i\int_{\bar{h}}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)\bigg)$

$\mathbf{W}[h]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h})\mathbf{\mathcal{W}}(h)$

$\mathbf{W}[\bar{h}]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h})\mathbf{\mathcal{W}}(\bar{h}$

The stochastic integral

$\mathbf{E}(exp(i\int_{h}\widehat{\mathcal{F}}_{i}(x)dx^{i})$

can be evaluated by a cluster integral decomposition, exploiting the Gaussian property of the random field $\widehat{\mathcal{F}}_{i}(x)$. This is briefly stated and proved in the following theorem. Let $\widehat{\mathcal{F}}_{i}(x)$ be a random Gaussian vector field defined for all $x^{i}\in\mathbb{D}$ with respect to a probability space $(\Omega, F,\mathbf{P})$ and having a 2-point function $C_{ij}(x,y;\zeta)$. Then the expectations of the stochastic Wilson loops along h and $\bar{h}$ are then

$\mathbf{E}\bigg(exp\bigg(i\int_{h}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)\bigg)=exp\bigg(-\int_{h}\int_{\bar{h}}C_{ij}(x,y;\zeta)dx^{i}dy^{j}\bigg)$latex  \mathbf{\mathcal{W}}[\bar{h}]=\mathbf{E}\bigg(exp\bigg(i\int_{\bar{h}}\widehat{\mathcal{F}}_{i}(x)dx^{i}\bigg)\bigg)= exp\bigg(-\int_{h}\int_{\bar{h}}C_{ij}(x,y;\zeta)dx^{i}dy^{j}\bigg)\$

$\mathbf{W}[h]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h})exp\bigg(-\int_{h}\int_{h}C_{ij}(x,y;\zeta)dx^{i}dy^{j}\bigg)$

$\mathbf{W}[\bar{h}]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h})\exp\bigg(-\int_{h}\int_{\bar{h}}C_{ij}(x,y;\zeta)dx^{i}dy^{j}\bigg)$

$\mathbf{W}[h]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2}Lk(h,\bar{h})\exp (-Q_{ij})$

$\mathbf{W}[\bar{h}]=\mathbf{E}(\widehat{W}[h])=exp(\tfrac{1}{2}(\alpha^{2}+p^{2})^{1/2} Lk(h,\bar{h})\exp(-Q_{ij})$

where

$Q_{ij}(x,y;\zeta)=\int_{h}\int_{\bar{h}} C_{ij}(x,y;\zeta)dx^{i}dy^{j}$. If $Q_{ij}(x,y;\zeta)\rightarrow 0$ for $|x-y|\gg \zeta$ then $exp(-Q_{ij})\rightarrow 1$ as