Team:Michigan/Model

Team Michigan: Attributions

Mathematical Modelling

Overview

In order to better understand the behavior of our temperature-controlled lysis system, we created a deterministic model representing the genetic circuit as a system of ordinary differential equations. This model was used to explore the parameter space of the system at steady state and to identify key biological parameters that might be used to alter circuit behavior. This working model that can be fit to experimental data to determine the mode of circuit failure or to fine-tune circuit behavior to meet desired specifications.

Diagram of temperature-dependent lysis circuit

The species involved in the synthetic genetic circuit include endolysin (which is capable of degrading the peptidoglycan layer in the presence of holin), holin (which perforates the cell membrane), anti-holin (which reversibly binds and inactivates holin), and TlpA36 (which is capable of repressing its respective promoter at temperatures of 36C or lower). The expression of anti-holin is driven by a promoter under the control of TlpA36. The expression of TlpA36 is driven by both a constitutive promoter directly upstream of the tlpA36 gene and by the TlpA36-regulated promoter.

ODE model

This genetic circuit can be modelled using a system of deterministic non-linear differential equations. Transcription and translation of each gene is combined into a single process. Each protein is assigned a degradation rate, and repression by Tlpa36 is modelled using a hill function with cooperativity of 1. Endolysin is assumed to be non-limiting and to only be capable of inducing toxicity in the presence of holin; therefore, it is excluded from the ODE model. Degradation of holin and endolysin is assumed to be independent of whether they are bound or unbound.

Steady state assumption and Kon, Koff >> transcription/degradation

All derivatives are set to zero in order to obtain a system of algebraic equations which describes the system at steady state. [Tlpa36] is then put in terms of constant reaction parameters.

We then reframe the system in terms of totals for H and A (where the total includes both the species alone and in a complex). We also assume that Koff and Kon are much faster than other reactions in the system (the reaction quotient for H and A binding is very close to the dissociation constant for H and A).

Assume tight binding between holin and anti-holin

We then simplify our model by assuming that the dissociation constant for H and A binding is much lower than the concentrations of H and A which are relevant for cell lysis, which gives a new equation for [HA] at steady state which is simply dependent on the difference between the total amounts of H and A.

Since [H] is ultimately the species determining lysis, we use the relationship

to create a system the following system of equations which represents the behavior of the model under the aforementioned assumptions.

Conclusion and discussion

Although the equation which determines the steady state concentration of [TlpA36] remains unwieldy due to self-regulation (a feature carried over from the natural system), we now have a fairly simple model which allows us to make important observations about the effects of various parameters on cell lysis. Because cell lysis is expected to occur at some threshold value of [H], and temperature affects the value of [H] by altering the degradation rate for TlpA36, the key design goal of the system is that the value of [H] must be below the lysis threshold when the degradation rate for TlpA36 is high but above the threshold when degradation of TlpA36 is low. With this goal in mind, we can see that testing constitutive promoters of various strength to drive holin expression may be sufficient to obtain [H] values that are above and below the lysis threshold for low and high temperatures, respectively. However, if no lysis is observed even under the strongest constitutive promoter used for holin expression, this likely indicates that [H] is still below the lysis threshold because holin expression is not high enough relative to anti-holin expression. This may have been the case in our experimental realization of this design, which failed to show large decreases in growth even when a strong promoter was used to drive holin expression. With this in mind, we can return to the model to identify a few other readily tunable parameters. Most notably, increasing the degradation rate of [A] through use of a protein tag would directly decrease the value of [Atot], possibly causing [H] to pass the lysis threshold at low temperatures.