**Polyhedron Software Generator**|

**Catalytic Polyhedra Modeling**

**POLYHEDRON SOFTWARE GENERATOR**

[Temporary supressed. Updating new information about the software.]

**CATALYTIC POLYHEDRA MODELING**

The question we address here is: what kind of kinetic information the most basic properties of our cage could provide us?

**Intro**

Modeling multienzyme nanostructures is a very challenging problem. Many recent studies [1]-[3] using DNA scaffolds as a coassembly strategy have investigated the role played by interenzyme distances, their relative orientations and the DNA architecture to enhance catalytic reactions. The main idea behind all those studies relies on the optimization of intermediate transfer between active sites via: 1 - the increase of local substrate concentration, and 2 - a bounded diffusion mechanism (substrate channeling).

Because of the difficulty in dealing with the detailed geometric features of these scaffolded systems, which would require a more sophisticated approach, we choose here to explore a minimalistic model of our system.

**Minimal Deterministic Model**

So, the idea is to provide a kinetic enhancement indicative of performing “enclosed” enzymatic reactions in comparison with free reagents in solution. For now on, let’s consider a sequence of n generic reactions:

For our purposes here, we suppose that the reaction rates change linearly with the total enzymes concentrations. Then, if the reagents are free in solution (figure), we’ll have:

where the terms on the right represent the velocities of production and consumption respectively. The functions fi are dependent of the catalysis mechanism, for example, if they follow a Michaelis-Menten process, we would have:

But, what happens when we confine all the enzymes inside the nanocages (one complete pathway per cage)? In principle, we can say that only the intermediates inside the “boxes” A(in) could react, and those outside A(out) could not. In other words, the kinetics would be limited in one hand by a transfer process in-out the cages and, by the other hand, the super concentration of enzymes in a small cage volume would accelerate the catalytic cascade. In mathematical language, we get:

And, the corresponding kinetics equations stay:

where λ(i) and λ(-i) are constants related to the rates of Ai getting in and out of a cage, respectively. The last two terms on the right have the same form of the free enzymes kinetics, which is not fully correct, since inside the cages the reactions should not obey a first order law. In this sense, the scaffold is still a black box for this model, but, let’s suppose we did a good job and our cage design operates under optimal conditions, better than a sequence of first order reactions.

That said, we can imagine our system as equivalent to two separate compartiments: one without enzymes (out) and the other with a super concentration of enzymes (in) where the first order law still holds. If we know the total volume V and the volume of the enzymes compartment Vc ( = volume of one cage x number of cages), its easy to see that:

where [Ei] would be the concentration of a free enzymes solution. The factor α=V/Vc represents how much the enzymes are concentrated. Using the fact that:

we can rewrite the kinetic equation as:

Here, we identified the last term with the expression of a free enzyme reaction. Finally, we’re going to assume one more hipotesis: the transfer “reaction” ( A(out)↔ A(in) ) reachs the equilibrium very fast, since its just a diffusive process. This means that, at any moment,

Then, substituting this expression on the previous one, we get:

This formula compares the reaction rates of our nanoreactor (left side) and the same rection performed with free enzymes. In this format, its clear that a kinetic enhancement will be observed only if:

Since Vc/(V-Vc ) << 1 , the last inequality seems to hold to all substrates if there is no strong impediment to their free difusion through the cage. This offers an indicative of the potential capacity of enclosed multienzyme structures to speed up catalysis. However, no time scale at which this boost is significant is provided.

**Discussion and Perspectives**

Of course this idealized model is far from fully consider the implications of the detailed geometric organization of those scaffolded systems. Many subtle features in the assembly design (interenzyme distances, rotational orientation, substrate channeling, etc) can have large effects on the kinetic efficiency. However, once we have an optimized design that operates better than a first order kinetics (with respect to the enzymes concentrations) our model suggests a gain in production.

For more refined models, one could explore coarse-grained Brownian dynamics (BD) simulations to study, for example, bounded diffusion mechanisms (through hydration layers or eletrostatic interactions with enzyme or DNA surfaces, etc), concentration distribution, binding time distribution and so on. We think that a combined deterministic model with probabilities of substrate-enzyme association, provided by BD simulations, could shed light on new modeling strategies [4].

**References**

[1] C. C. Roberts and C. a. Chang, “Modeling of Enhanced Catalysis in Multienzyme Nanostructures: Effect of Molecular Scaffolds, Spatial Organization, and Concentration,” J. Chem. Theory Comput., vol. 11, pp. 286–292, 2015.

[2] J.-L. Lin, L. Palomec, and I. Wheeldon, “Design and Analysis of Enhanced Catalysis in Scaffolded Multienzyme Cascade Reactions,” ACS Catal., vol. 4, no. 2, pp. 505–511, 2014.

[3] J. Fu, M. Liu, Y. Liu, N. W. Woodbury, and H. Yan, “Interenzyme substrate diffusion for an enzyme cascade organized on spatially addressable DNA nanostructures,” J. Am. Chem. Soc., vol. 134, no. 12, pp. 5516–5519, 2012.

[4] O. Idan and H. Hess, “Origins of activity enhancement in enzyme cascades on scaffolds,” ACS Nano, vol. 7, no. 10, pp. 8658–8665, 2013.