Team:USP UNIFESP-Brazil/Model
AlgAranha Team USP_UNIFESP-Brazil
In the development process of synthetic biology projects not everything can be understood easily. Sometimes, in order to save money, reactants and lab time it is necessary to appeal to other realm of tool sets. This is when modeling comes in, with mathematics as it's hammer and every project as it's nail. Jokes apart, modeling is a useful tool to understand possible limitations in complex systems that could lead to project failure, or even give some insights into underlying processes that would be otherwise hidden. During the initial development of Algaranha several issues that could be tackled within a mathematical framework were found, but one in particular caught our attention: the great difficulty to achieve large polymer chains in the insert. While catching up with the bibliography we found that most of silk chains that could be generated had at average 10-20mers, and the record length was a bit less than 100mer. Several possible reasons were described, such as the large quantity of Cytosine and Guanine in the gene composition, rendering it difficult to synthesize; the impossibility to insert large genes into the plasmid, and the impossibility to insert this large plasmid inside its vector organism.<\p>
As an approach to overcome these difficulties we decided to build our plasmid via repeated addition of smaller sub units in hope of achieving a larger chain. The system consists in a plasmid with a specific sticky end in both sides that pairs only with a specific coding sub unit, which we will call them A and C. These sub units have a specific sticky end as well, which binds only with an intermediate sub unit B. This B sub unit can bind either with the end terminals A and C or with another B sub unit. So we then have the possibility of constructing large chains by adding B units to the plasmid. The system described above can be understood as a set of coupled reactions, represented in the scheme bellow. As we can see, there are a few ramifications the reactions could take, where theoretically it could go on to infinite chain size. This scheme can be put on series of coupled reactions as we can see below:
In the first step we have the reactions for A and C binding in the plasmid (Pl):
At the second step we have the B sub unit binding to an Pl + A or a Pl + C:
And we can keep adding n B's to the chain until it finally ads a C and closes the loop:
These reactions can be further understood by a set of rate equations given by:
But since the number o equations and constants become untreatable it is necessary to take a few simplifications so we can continue. First of all we will consider only the first two steps, where the largest chain possible will be PlAB2C, so we can keep track of what is going on. This way there are only a few equations left to be integrated. Since we don't have any of the rate constants we will assume three scenarios: homogeneous consumption constants (k1 = k2 = ... = kn) and homogeneous dissociation constants (γ1 = γ2 = ... = γn); progressively decreasing consumption constants (k1 > k2 > ... > kn) and homogeneous dissociation constants (γ1 = γ2 = ... = γn); homogeneous consumption constants (k1 = k2 = ... = kn) and progressively increasing dissociation constants (γ1 < γ2 < ... < γn). This way it is possible to have some insights on what processes may be ruling our system.
Now it is possible to integrate these equations numerically and find the equilibrium conditions for each sub product of the reactions. Our interest is to find the ratios between [PlABC] and [PlAB2C] when it is in equilibrium, i.e.:
Unfortunately the equations demonstrated to be considerably unstable, probably due to its coupling and the presence of the reactants A,B and C in almost all of the equations. But it is possible to have some insights on the behavior of theses equations given the scenarios described above. In the first scenario (homogeneous constants) it is expected that the final concentration of PlABC and PlAB2C should be equal or at least approximately equal. The final concentrations of A,B,C and Pl should be dependent on the ratio of the dissociation constant γ and the consumption rate k, if γ = 0 all of the reactants should be consumed and all of the final products would be in the end reactions. But if γ ≠ 0 the amount of reactants would be also different from zero. In the second scenario (progressively decreasing k) it is expected that [PlABC] > [PlAB2C], since its reaction ratio would be larger and given that k>> γ the amounts of the end sub units C and A would be close to 0, since it would be harder for them to break than to form a closed plasmid. Finally, in the third scenario (progressively increasing γ) the behavior should be similar, given that the larger molecules of PlABn and BnCPl would be rather unstable, dissociating before a C or A sub unit could bind and close the loop.
