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Computation and amplification processes based on Networks of Chemical Reactions are at the heart of our understanding of the regulation and error correction of life systems. The recent advances in DNA nanotechnology, with the creation of the modular structures origamis and the development of dynamical networks using the toe hold mediated strand displacement, open fertile areas to construct Hierarchical Cascades of Chemical Reactions with an increasing complexity inspired from systems in biology. DNA strands have the great advantage to design autonomous and homogeneous Networks of Chemical Reactions leaving aside companion chemical reactions as it occurs in biological systems. In the present paper, we use the Fokker Planck equation to extract predictions that address a wider class of systems beyond the case of diluted solutions. We introduce the concept of toehold strength and output strength that leads to an exponential square dependence of the toehold strength divided by the output strength on the escape rate and the probability for the output strand to leave the gate. We highlight the influence of the boundary conditions that may have an important consequence in confined environment when modular structures like origamis are employed.

Over the last decades, DNA became a material to conceive and fabricate news structures and functions. Thanks to N.C. Seeman approach [

In the seminal work done by B. Yurke et al. [

Other approaches using DNA to design nano devices have also been developed. G sequences that form tetraplex are used with fuel strands to transform tetraplex to duplex and vice versa building an alcalin responsive nano mechanical devices [

The works that motivated the present study were the numerous experimental studies and quantitative analysis made recently [

What we want to emphasize is that those chemical reactions exhibit a symmetry breaking, without which no flow can occur, and make a biased random walk to happen. DNA nano devices are dynamical systems working out of equilibrium. On the other hand, to transduce the chemical energy into a mechanical work, the downstream flow involves dissipation [

We start with an evaluation of a few physical properties when the nucleotides are bind together to form the canonical helical structure. The approach is based on a simple mechanistic approach. The Nucleotide is described as a bead in spite of the fact that it is in itself a complex unit. We also do not consider the chemical specificity and use an average bead free binding energy of about 2.5k_{B}T [_{B}T, the damping coefficient using the Stokes relationship Damp = 3phf where h is the viscosity of the water at room temperature. The values of the associate spring constant, damping coefficient and characteristic times are:

where M is the average molecular weigth of a nucleotide (300 g/mole), ^{−12} - 10^{−13} s, and a much slower inter well equilibrium time when the barrier height separating the wells is larger than k_{B}T. Keeping this picture we evaluate the hopping rate of a basic unit. The Kramers’ expression [

with

The calculated rate (Equation (1)) gives a hopping time over the barrier height of a few ten nanoseconds. A much shorter time, typically 3 orders of magnitude faster than the hopping time deduced from the experimental data. Note also that the measurement of the self-diffusion of DNA gives time constants around 10^{−5} s [

We now evaluate the diffusion limited rate constant of DNA hybridization in dilute solutions. To calculate the diffusion constant D of a short oligonucleotide, say 10 to 20 bp long, we use the persistent length of a single strand f = 1 nm and the Einstein relationship, then the average time for the particle to diffuse over its length is computed:

The hybridization rate is the product of the Diffusion constant and a characteristic length of the species. We use again the persistent length, and with the Avogadro number N_{a} we get:

while the observed second order chemical rate constant if often measured to be:

^{1}Using a random walk description gives a total time for N steps corresponding to N nucleotides:

We may argue that the hybridisation rate might depend on the length of the strands. However, even with a slow stepping time of 10 ms, the probability for a complete reaction weakly depends on the total strand migration time^{1}. Here again, there is a 2 orders of magnitude difference between the experimental results and the calculated value. The nucleation rate borrows the same kind of feature than the one used, the Kramers result, to calculate the bead hopping rate over a potential barrier. Everything happens as if an effective barrier height of 7k_{B}T must be added:

Note that this height of the activation barrier is close to the one found, U = 8k_{B}T, to compute the diffusion constant of a protein, the Kinesin, walking along a microtubule track [^{−6} M^{−1}∙s^{−1}. This number of nucleotides was shown to be the lowest toehold length required to reach an asymptotic plateau for an equivalent bimolecular process describing the migration with a strand displacement [

It is worth noting that the above numerical applications assume that the thermal energy is fully dedicated to the translational motion. However, only a few degrees of freedom might be able to pump the thermal energy available to activate a back or forward step along the track. As a consequence, a tiny amount of the thermal energy available may contribute to the nucleotide translational diffusion, which will lead to an overestimation of the barrier height.

The slow down is even much more pronounced when DNA migration through Holiday junction motion is considered with a stepping time ranging between the ms up to hundred of ms [^{2+} and Na^{+} [

The use of the Kramer’ approach is mostly of Heuristic interest. What we want to address is the question upon the basic ingredients we need to understand the physical origin of the strand displacement that is the transduction of a chemical energy, the folding of the toehold, to a mechanical work in confined environment and diluted solution.

The toy model is based on the works done to describe a ratchet device or a biased random walk [_{off} rates for the good and wrong molecules.

We use a simple piecewise potential that allows extracting simple analytical expressions from the Fokker- Planck equation [

The elementary unit is built with 4 or 5 nucleotides as it was discussed in the previous paragraph. Therefore, the case of a toehold made of one nucleotide will be given by a fraction of this unit either 0.25 or 0.20. Note also that, from first principle, the motion of a nucleotide will have a perfect brownien motion with an equal probability

to move either to the left or to the right (

The toehold strength is described with a varying slope that can either be smaller or greater than that of the slope of the barrier height controlling the output strand escape rate. The biased forward motion of the invader and the displacement of the output strand are described as the motion of a bead in an asymmetric potential well (_{0}, meaning that once the bead has reached the top of the barrier height it diffuses away being not allowed to go back into the potential. This absorbing condition represents the behavior of the displaced strand. Depending on the way the device is used, the boundary condition at x = -x_{1} can be different (see Annex 2). In solution the most appropriate is also to use an absorption condition. The potential is sketched as follow (

We write the potential in unit of k_{B}T:

We inject a particle at x = 0 at the time t = 0, and look for the probability for the particle to be still in the well at time t. Basically, this will be given by a residence time, which is the reciprocal of the rate at which the particle escapes from the bottom up to the location x_{0}. The time evolution of the probability follows the Fokker Planck equation:

The current is given by:

Applying the Laplace transform, Equation (4) rewrites:

Equation (6) gives a characteristic equation from which it is convenient to use the notations:

In the left side, the diffusion varies as Dn^{2}, thus the drift velocity scales as n^{2}. Such a variation could be of interest to evaluate the kinetics competition between different configurations and has to be compare to the fluctuation time of a fluctuating barrier height of the potential [

With p > 0, q > 0, the physical solutions of Equation (6) are noted as a function of the coefficients A(s), B(s), C(s) and D(s) and have the form:

The coefficients A(s), B(s), C(s) and D(s) are obtained by ensuring the continuity of the probability and the current at x = 0 and the boundary conditions at x = x_{0} and x = -x_{1} (see Annex 1):

With Q(t) the probability that the particle is still in the potential well at time t, the escape rate k is given by:

Then using the Laplace transform of

Integration of Equation (10) and using Equation (9) gives:

The coordinate x_{1}, the location at which the bead escapes from the left, has a linear dependence with n, thus the toehold strength gives an n^{2} exponential dependence. The coordinate x_{0}, the location at which the bead diffuses away when it reaches the right side end, has a linear dependence with the length of the displaced strand. Let’s note this length m, where m is a number of units. It is not obvious to give a precise number of nucleotides for a given value of U_{0} or n. For instance U_{0} = 12.5k_{B}T might correspond to about 4 GC nucleotides, while U_{0} = 10k_{B}T might correspond to 5 - 6 AT nucleotides. Therefore, it is worth using the concept of toehold strength, for n, and displaced strand strength for m, rather than a DNA length as used in experimental data.

The residence time of the bead is assimilated to the migration time of the strand and will be expressed with the reduced coordinate

For a toehold strength below n = 1, the residence time is mostly driven by the left escape rate at the x_{1} location. Therefore, for n < 1, the probability for the invader to leave the device is greater than the probability for the output strand to escape. For the symmetric case, which is obtained for the couple of values (n = 1, m = 1), there is an equal probability for the bead to escape either from the right (output strand) or from the left (input strand). After the maximum, the residence time decreases as the toehold strength increases reaching an asymptotic value, which is only given by the right escape rate at the x_{0} location. For n large enough, the escape rate no longer depends or weakly depends on the m value. This is a known result and in experiments a toehold length of 7 or 8 nucleotides (n of about 2) is often chosen.

The asymmetry of the potential shape is governed by the couple of parameters (n, m), that appears in the exponential argument as nx_{1}/x_{0} (or else n^{2}/m)^{2}. As expected, when the output strength increases the whole curve is shifted toward the high value of n. For m = 3, the symmetric value (n = 1, m = 1) is reached for n nearly equal to 2, meaning that for n < 2 the bead mostly escapes from the left, while with m = 1 and n = 2 we expect that the bead mostly escapes from the right. In _{0} = 10 and 12.5k_{B}T. The main difference is a decrease of _{0} decreases and a slight shift of the maximum toward higher values of n.

The residence time is a function of both the left and right escape rates and does not give an unambiguous information on the output strand probability to leave the device. The probability for the bead to move forward, e.g. to the right is given by the current:

In

bility for the output strand to diffuse away from the device. With m = 1 a plateau is reached at a value of n slightly higher than 1, the symmetrical value where the probabilities for the bead to escape on either sides are equal. For U_{0} = 10k_{B}T the probability varies over 5 orders of magnitude, while for U_{0} = 12.5k_{B}T the probability varies over 6 orders of magnitude. Once the invader strand has touched the device, the probability for the output strand to escape from the right side provides a weighting value to evaluate the effective migration bimolecular rate k_{on} showing that k_{on} can vary from 10^{6} to 1 M^{−1}∙s^{−1} as a function of the toehold strength. The overall shape of the computed variation is in good agreement with the experimental data (_{on} where the product of the bimolecular reaction is the output strand. Once the target has been hit it does not mean the reaction occurs, thus P = 1 corresponds to 10^{6} M^{−1}∙s^{−1} and P = 10^{−6} to 1 M^{−1}∙s^{−1}. Also it is worth recalling that n = 1 corresponds to 4 - 5 base pairs. Again, the effect of the parameter m is to shift the whole curve toward higher values of n. The model predicts that to reach the plateau with m = 3 requires to have a toehold strength about twice greater than the one required for m = 1.

An alternative to improve the computation with DNA strands is to use the DNA origamis as platforms for DNA devices. Beyond the confinement that enhances the concentration at the local scale in turn increases the speed of the chemical reaction [_{on} × clocal in s^{−1}, with c_{local} the concentration of DNA strands at the local scale), the close proximity of the gate with the input and fuel strands may significantly modify the reaction scheme in turn the design of the device. As a matter of fact, the close proximity of the interacting strands makes truly difficult to sequester the active part of the gate output strand. Even a drastic reduction of the toehold strength cannot avoid unwanted leaks of the devices [_{1} (scheme

As shown on scheme 6, the potential structure is the same as the one used previously, the unique difference is the reflecting boundary condition at x = -x_{1} which writes:

This boundary condition seems to be a bit artificial as it forces the current to always be directed from the left to the right. However, we observed that with a seesaw gate on a origami, the fuel strand was able to trigger a gate output without the need of an input strand, something which is unlikely to happen in solution [

The residence time is (see Annex 2):

In

curves compare the behavior of the devices in diluted and confined situation once the invader has touched the gate without taking into account a local concentration effect. When the invader cannot escape, the invader can still do a work with a very weak toehold strength, case n < 1, while it cannot in diluted solution. Moreover, when the displaced strand strength is increased, case m = 3, the tethered invader remains much more efficient than the one in solution that only has to displace a shorter strand (m = 1).

The above calculation gives an exponential square dependence as a function of the toehold strength n, where n corresponds to a number of units that are not restricted to the scale of a nucleotide. This way to do, while being set in a qualitative way, is an attempt to understand the biased motion of nucleotides in DNA nano devices. There is no reason to expect a nucleotide located in the middle of the device to move preferably in either direction. The model can be seen as a modified detailed balance approach in which is added a forcing given by a toehold force scaling as

Using the Fokker Planck Equation, the shape of the escape rate, involving both sides, and the probability for the output strand to leave the gate as a function of the toehold and output strength are derived for asymptotic long time. At this stage, it is worth noting that changing the energy landscape does not change the main trends given below (see Annex 3):

The output strength leads to an exponential

・ The invader escapes faster than the output strand for n < 1 and m = 1.

・ The output strand escapes faster than the invader for n > 1 with m = 1.

・ The residence time does not exhibit a monotonous variation as a function of the toe hold strength. The above model predicts that there is a domain (n, m) where the device is slowed, which can be of importance with respect to competitive kinetics between several different gates.

・ When m increases, there is an increasing domain of toe hold strength where the invader escapes faster than the output strand. The maximum of the residence time decreases and shifts toward higher values of n. Similarly, the probability for the output strand to leave the gate decreases with a shift toward higher values of n.

The second interest of the use of the Fokker Planck equation is to address the question upon the effect of the boundary conditions. This allows us to investigate a wider class of devices, either in solution or else arranged in a confined environment such as on an Origami nano platform or inside a vesicle. To emphasize the influence of the boundary conditions, we applied the absorbing and the reflecting condition on the side of the invader. Due to the obvious difference between these two boundary conditions, the difference is striking. When the reflecting condition is used the invader always succeeds, whatever the toehold strength, keeping in mind that it may take quite a long time as shown with the shape of the residence time (

From the behavior of the device as a function of the couple of parameter (n, m), we extract two main predictions that may be of interest to design DNA nano devices.

In confined environment, in addition to a reduction of ever possible cross talks, we can redesign the device by reducing the size of the domains involved. For example, it might not be necessary to use fuel strands to ensure a catalytic behavior of the invader. On the opposite, as shown with the reflecting condition, we may increase the size of the template and of the output strand without reducing the efficiency of the device.

In solution it would be of interest to verify the effect of the ratio n/m, which to our knowledge had never been systematically investigated with the simplest toehold mediated strand displacement device.

Devices based on DNA strands allow building autonomous and homogeneous systems (see for instance Ref. [

This work is supported by the French National Agency and the project PIA VIBBnano ANR-10-NANO-04-01 and the COST Action TD1003 “BioInspired Nanotechnologies: from concept to application”.

Absorbing condition at both end of the device:

The continuity of the probability and current at x = 0 gives:

and the absorbing condition at both end:

From which we obtain:

Residence time and escape rate in an asymmetric piecewise potential well is given by:

Case of a DNA device on origamis: Reflecting condition at the location x = -x_{1}

The potential structure is the same as the one used in Annex 1 with the same type of solutions. The unique difference is the reflecting boundary condition at x = -x_{1} which writes:

The solution are of the form:

And the residence time:

Other energy landscapes can be chosen that may appear more suitable for DNA device: for instance with left and right slopes proportional to n + m and m respectively, the asymmetry is maintained and the general trend is preserved. In this case the current and reduced residence times are:

- Absorbing conditions at both end:

where the reduced time is scaled at the monomer unit length, with x_{0} = mu.

- Absorbing at the right side and reflecting at the left side. The reduced time is given by: