Computational and mathematical models are useful and effective tools for understanding biology. Molecular biology experiments in vitro are very expensive and time consuming, building models of biological processes as a preliminary step helps to circumvent some of the drawbacks of performing hypothesis-testing in the wet lab. Using models could provide a faster and cheaper shortcut for biologists to gain a better understanding of the newly engineered system. This is why we feel that computational modeling is important and useful. Particularly with the extent of synthetic biology, many of the biological systems that are being researched could not be found in nature because they are genetically engineered, so their behaviors are unknown and need to be characterized.

Analysis and design based on models:

• A model provides a prediction of how the system will behave
• Models help sort out what is going on
• Models don’t have to be exact to be useful: they just need to help explain (and predict)

For one green world, an Intelligent Microbial Heat Regulating Engine (I’M HeRE), as shown in figure 2, was designed to satisfy our needs described in the project. The engine consists of 2 systems namely the heat-resistant system and the quorum-control system.

Fig.1 : The gene regulatory network of the intelligent microbial heat regulating heat engine. (a) shows the regulatory relationships of these genes. (b) is the detailed diagram of the GRN of the quorum control system.

When the number of cells increased to a certain extent, the gene regulatory network was started to control the number of live cells.

• A is the chemical species of interest, and the number of molecules of chemical species A at time t is simply denoted as A(t)
• The symbol φ denotes chemical species which are of no further interest in what follows.
• k is the rate constant of the reaction and is defined so that k*dt gives the probability that a randomly chosen molecule of chemical species A reacts (is degraded) during the time interval [t, t + dt)
• the probability that r < A(t)k∆t is equal to A(t)k∆t, since r is a random number uniformly distributed in the interval (0, 1)
• A(t)*kdt gives the probability that exactly one reaction occurs during the infinitesimal time interval [t, t+dt)

The goal is to compute the number of molecules A(t) for times t > 0.

• (a1) Generate a random number r uniformly distributed in the interval (0, 1).
• (b1) If r < A(t)k ∆t, then put A(t + ∆t) = A(t) − 1; otherwise, A(t + ∆t) = A(t).
• Then continue with step (a1) for time t + ∆t.

With k = 0.1, protein (0) = 20 and ∆t = 0.005.

Fig.2 Stochastic simulation of protein degradation for k = 0.1, protein (0) = 20 and ∆t = 0.005. Number of molecules of proteins as a function of time for two realizations

Fig.3 Stochastic simulation of protein degradation for k = 0.1, protein (0) = 20 and ∆t = 0.005. Number of molecules of proteins as a function of time for ten realizations. The blue dash line shows the mean value of the stochastic results.

• Stochastic simulation of production and degradation

With Protein (0) = 0, k1 =0.1 and k2= 1

Fig.4 two realizations

Fig. 5 ten realizations

Fig.6 stationary distribution of the molecules, it’s analyzed to be poison disribution

• Stochastic simulation of dimer formation

With A(0)=0, B(0)=0, k1=10-3,k2=10-2, k3=1.2 and k4=1.

Fig.7 number of A molecules

Fig.8 number of B molecules

The GRN is too complicated to conducted biological experiments. Thus dynamic simulation of the GRN by mathematical modelling is particularly important. Before the wet experiments were conducted, mathematical models were built to predict the behavior of our heat regulating engine. Such bottom-up design of synthetic gene circuit will not only speed up the biology research, but provide us a valuable insight into the system as well. We want to know:

• In a particular state which genes were expressed?
• What’s the influence of the gene product on cell physiological activities?
• To provide a guidance on the design of experiments by further analysis of the dynamic behavior of gene regulatory networks.

Models of Synthetic Gene Networks

• Adopt Jacob’s and Monod’s postulate: biological phenotypes, however complex, can be explained in terms of cascades of bio-molecular interactions.
• All interactions represented by reactions and dictated by thermodynamics and kinetics.
• Model all interactions at the molecular level: Transcription, Translation, Regulation.

Bacteria communicate with each other using chemical signaling molecules. The process of producing, releasing, detecting, and responding to the signaling molecules in bacteria is referred to as quorum sensing. Quorum sensing enables bacteria to obtain information from their environment and alter their behavior in a population size that is scaled in response to changes the number of bacteria present in their community. Thus, this method of intercellular communication allows populations of bacteria to work together, in contrast to the traditional unicellular view of prokaryotes.

In general, quorum-sensing bacteria produce and release signaling molecules into their environments which are called auto-inducers. The concentrations of auto-inducers increase in the bacteria environment as the bacteria population density grows. The bacteria detect the accumulation of a minimal threshold concentration of the auto-inducer and regulate their gene expression in a population-wide scale in response. As a result, using these signaling systems, all the bacteria in a community alter their behavior at the same time in response to an external signal, and therefore function similar to a multicellular organism.

Many different quorum-sensing systems have been discovered. Some of them are extremely specific for a particular species and some of them are common in all bacteria. In this work, we sought to establish a synthetic auto-inducer (AHL) signaling system.

It is assumed that AHL was produced and degraded at a certain rate to maintain a certain concentration in a single cell. That is to say the number of AHL molecules within a single cell is constant. Therefore, there is a linear relationship between the AHL molecule concentration and cell concentration. The regulating engine was started when the number of AHL molecules reaches a certain threshold, or when the number of cells reaches a certain threshold. (indexed from Wikipedia)

Fig.9 the GRN of the quorum sensing device. The right table shows the chemical reactions of the GRN.

Fig.10 stochastic simulation of the GRN in a single cell

As we could see from the simulation result in fig.11. AHL was produced and degraded at a certain rate to maintain a certain concentration in a single cell. A AHL*LuxR complex was formed once LuxR was produced. Thus the concentration of LuxR was low.

The oscillator is a synthetic biology construct composed of three genes which mutually repress each other in sequence according to a ring structure.

Fig.11 detailed GRN of the oscillator

Fig.12 ODE simulation of the oscillator

Fig. 14 the red curve shows the result with the lethal effect considered, the blue curve shows the normal logistic growth result.

The initial inoculation amount of this model is 1 unit. The proposed intrinsic rate is 0.05. And we set the maximum number of cells that the environment can support is 100 units. As can be seen from the simulation , from 0 to 200 units of time, the number of cells are in a growth state in both cases, no lethal effect is greater than the growth rate of cells in the presence simulation lethal effect of simulations; after 200 units of time, two kinds of simulations have tended to steady state, the growth of non-lethal effects of simulated results of the final number of the environment can accommodate the largest state, while the presence of the number of lethal effect of significantly less than the maximum number of steady-state simulation results is about 90 units.

Fig.15 time evolution of heat accumulated in the fermenter

We could get the information that the heat in the fermenter will accumulate over time without a cooling system from the above figure. Once the cooling system was turned on, a steady state could be achieved as shown in the figure below.