Difference between revisions of "Template:Team:Utrecht/MainBody"

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By comparing the two figures, we can now see that the probability of cleavage for the slow cleaver is much smaller than that of the fast cleaver in the timespan that the transient complex persists. Of course, the concentration of the substrate-mediated complex decreases over time, so the total cleavage decreases when it cleaves later. To investigate how much the transient and substrate-mediated complex contribute to signal development for both Protease Chain variants, we define:

By comparing the two figures, we can now see that the probability of cleavage for the slow cleaver is much smaller than that of the fast cleaver in the timespan that the transient complex persists. Of course, the concentration of the substrate-mediated complex decreases over time, so the total cleavage decreases when it cleaves later. To investigate how much the transient and substrate-mediated complex contribute to signal development for both Protease Chain variants, we define:

Wherein S’ is the increase in signal, given by the probability of cleavage (p_c) for the remaining uncleaved complex. The remaining uncleaved complex is given by the remaining complex fraction (C) and how likely it is that it has not already been cleaved (one minus the integral of p_c from 0 until that timepoint).

Wherein S’ is the increase in signal, given by the probability of cleavage (p_c) for the remaining uncleaved complex. The remaining uncleaved complex is given by the remaining complex fraction (C) and how likely it is that it has not already been cleaved (one minus the integral of p_c from 0 until that timepoint).

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From these relations, it is already clear that there can only be a stable signal potential, i.e. a stable fraction of the target chains remains uncleaved, when the production of target chain outweighs the decay and cleavage of it, whilst the acquisition of cleaved target chain is balanced by its decay, such that the following conditions apply:

From these relations, it is already clear that there can only be a stable signal potential, i.e. a stable fraction of the target chains remains uncleaved, when the production of target chain outweighs the decay and cleavage of it, whilst the acquisition of cleaved target chain is balanced by its decay, such that the following conditions apply:

Which together forms:

Which together forms:

However, the concentrations of the intermediary complexes is dependent on the concentration and hence production of protease chain P. We know that, at equilibrium concentrations, the following must hold:

However, the concentrations of the intermediary complexes is dependent on the concentration and hence production of protease chain P. We know that, at equilibrium concentrations, the following must hold:

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By combining these equations, we can get an expression for T depending on the production, decay and complex concentrations:

By combining these equations, we can get an expression for T depending on the production, decay and complex concentrations:

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By assuming that the DNA binding dynamics of the system occur at much faster timescales than protein production and decay, we can assume that the substrate binding of the protease and target chains is at steady state, yielding the following expressions:

By assuming that the DNA binding dynamics of the system occur at much faster timescales than protein production and decay, we can assume that the substrate binding of the protease and target chains is at steady state, yielding the following expressions:

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We could then substitute these three concentrations for their expressions in the expression of the target chain concentration. Making further quasi steady state assumptions on the formation of the pre-cleavage and post-cleavage complexes reduces the expression by two more dependencies. This was done in mathematica notebook, found <a target=_BLANK href="https://static.igem.org/mediawiki/2017/2/21/UuModelingQSSAWorkouts.txt" class="url_external">here</a>.

We could then substitute these three concentrations for their expressions in the expression of the target chain concentration. Making further quasi steady state assumptions on the formation of the pre-cleavage and post-cleavage complexes reduces the expression by two more dependencies. This was done in mathematica notebook, found <a target=_BLANK href="https://static.igem.org/mediawiki/2017/2/21/UuModelingQSSAWorkouts.txt" class="url_external">here</a>.

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