# Mathematical Modeling

Increasingly mathematical modeling is becoming a crucial part of synthetic biology. Mathematical modelling and simulations implementing those models play a pivotal role between the concept and realization of a biological circuit [9]. Specifically, mathematical modeling can serve two distinct tasks: offering a detailed understanding of the dynamics of a biological circuit and/or the design of a novel circuit. Figure M1 shows one possible way to outline the modelling procedure. Generally, we begin with a mathematical model that captures the dynamics of the biological circuit in question. For instance, this biological circuit could be a gene regulatory network [6]; we could model such a system by a graph structure capturing the pattern of connections between the genes in the network. The topology of connection is only of limited used with regards to a complete understanding such a system; thus, we could impose a dynamical system over this graphical model. Now, the nature of this dynamical model would depend on the task at a hand; depending on our need we might be looking at a deterministic model or a stochastic model. Once we have a model, it is imperative to decide upon the values for the associated parameters. For an explanatory task, we would estimate them from experimental data; and for circuit design these values would be driven by biological constraints and/or a sweeping of the parameter space to identify values for which the system exhibits the desired behavior.Realizing the significance of the assistance modelling offers in circuit design, in this section we provide details for the various in-silico experiments conducted by our team. The basic circuit that team iGem IIT Delhi has worked with is the quorum sensing version of the Amplified Negative Feedback Oscillator [8]. This particular configuration has been implemented in [4] as has been discussed earlier. We start with the original model of the Danino oscillator [4]; discuss it's implications and suggest modifications to this model so as to incorporate the idea of reconfigurability.

Danino Oscillator: Synchronized quorum of
Gene oscillatory Networks

For the sake of completeness, the model for Danino is replicated here as well from equations 2.1 - 2.4. This is a deterministic model for intracellular concentrations of LuxI(I ), AiiA(A), internal AHL(H

_{i}) and external AHL(H

_{e})

In equations 2.1 and 2.2

describes the delayed production of corresponding proteins, it depends on the past concentration of the internal AHL, H

_{τ}(t) = H

_{i}(t-τ). Description for the various parameters in these equations can be found in [4]. The two parameters of immediate concern to us are the prefactor

*[1-(d/d*and the flow rate

_{0})^{4}]*μ*. The prefactor

*[1- (d/d*describes slowing down of protein synthesis at high cell density

_{0})^{4}]*d*due to lower nutrient supply and high waste concentration; thus, to ensure proper functioning we maintained a constant cell density of 0.6 (od 600). A sufficient flow rate is necessary to ensure diffusion of AHL to ensure oscillations in the system. If we look at equation 2.1 qualitatively, it tells us that AiiA is activated by the past concentration of internal AHL in the cell; thus, capturing the causal nature of AiiA’s activation by the AHL produced by LuxI. Similarly, equation 2.2 shows this process for LuxI, capturing that fact LuxI is a self activator here. For internal AHL in 2.3, we can see that it is being activated by LuxI as well as it is positively affected by the difference with external AHL concentration. In equation 2.4 the last term is quite interesting. This is the term that captures the synchronized nature of the oscillators; the movement of AHL across the cell membranes allows the cells to communicate with their neighbors regarding the state of oscillations.For HOTFM, we need to modify the Danino model to account for self-repression on AiiA. To achieve this, we add two more equations to the model and modify equaation 2.1. Thus, the modified and new equations are as follows:

To study the quantitative solutions of the Danino model

^{1}, we look at two broad scenarios: 1. First, we analyze the bulk scenario, where we ignore the diffusion term (D

_{1}= 0) 2. Second, we include the diffusion term and observe the cells as synchronization takes place.

Bulk Phenomena: No Diffusion

The aim of this experiment is to establish whether an individual cell is oscillating or not. Essentially, we have the null hupothesis that there are no oscillations and by solving the Danino model we show the alternative hypothesis to be true. The result for this in-silico experiment is shown in Figure M1. From this figure we can qualitativly explain the dynamics of the oscillator. Initially, both AiiA and LuxI accumulate latently. As the AiiA concentration shoots beyond a level, it suppresses the AHL concentration, thus further halting the production of both AiiA and LuxI. And as AiiA degrades enzymatically below a threshold, production of AHL resumes and again we have oscillations.

Synchronization Experiments

For this set of experiments, we assume a nonzero value for the diffussion coefficient and observe oscillation synchronizaton under different initial conditions. We perform these experiments both for Danino oscillator and HOTFM and compare the results. We take a linear array of 100 cells; and assign a set of 4 equations to each cell for the Danino Oscillator and a set of 6 equations for HOTFM. All heatmaps and results are for cells from index 10 to 90 to neglect the boundary cells.

Parameters and the role of diffusion

If we had hypothetical oscillators with identical parameters, then, in principle, starting with identical initial conditions would make synchronization redundant. However, for real oscillators there is variability in parameters across cells and this necessitates synchronization even with identical initial conditions. Figures M2 and M3 show the spatiotemporal heatmaps for the concentration of AiiA for Danino and HOTFM, respectively.

If we add noise to the parameters and start the cells with slightly different intial conditions, simulations show that in the absence of diffusion cells are completely off phase with each other as shown in Figures M4(Danino) and M5(HOTFM). Further, the time evolution for each cell is shown in Figures M6(Danino) and M7(HOTFM). Here, we see that at each time point the phases for all the cells are not visibly not in synchronization, thus yielding a very low amplitude for the mean value.Further, video V1 shows the temporal evolution of AiiA concentration for cells numbered 10, 30, 50, 70 and the mean concentration as well, thus supporting the hypothesis that for zero diffusion there is no synchronization. Equation V2 is yet another

Where comparison between Danino and HOTFM is concerned, Figure M8 shows the oscillations for cell number 10 for both Danino and HOTFM when diffusion is zero. Here, it is quite evident that addition of self repression on AiiA has reduced the amplitude and thus the time period of oscillations, as hypothesized; Danino has 21 peaks over a time frame of 1500 minutes, while HOTFM shows 26 peaks.

Having established the importance of diffussion for synchronization, it is imperative to conduct simulations to compare the diffusion behavior between Danino and HOTFM. For this purpose, we have conducted three different sets of experiments:

Diffusion with initial conditions nonzero across all cells

Here, we assume that initially all the cells have nonzero concentration of LuxI. This is done to compare the effects of diffusion with that of no diffusion. We perform two sub-experiments in this category with two different values of diffusion.

- Diffusion = 200 μm2/s The results for this experiment are shown in the following Fi gures: Spatio-temporal HeatMaps M9(Danino) M10(HOTFM) Temporal variation of AiiA concentration across all cells and mean value M11(Danino) M12(HOTFM)
- Diffusion = 800 μm2/s The results for this experiment are shown in the following Figures: Spatio-temporal HeatMaps M13(Danino) M14(HOTFM) Temporal variation of AiiA concentration across all cells and mean value M15(Danino) M16(HOTFM) Comparison of means for Danino and HOTFM M17 Video for temporal amplitude variation for cells = 10, 30, 50, 70 and the mean amplitude from cells 10 to 90 V1

The inferences from all these results can be summarized as follows:

- As we increase the value for diffusion synchronization is supported for larger period of time both for Danino and HOTFM.
- Further, from figures M11, M12, M15 and M16 it is clear than HOTFM shows better synchronization and that too for a longer period of time.
- Also, we get better synchronization with larger diffusion.
- The frequency of oscillation is always greater for HOTFM compared to Danino.
- From M17 it is clear that Danino has a larger amplitude than HOTFM on average.

Further, video V3 shows the temporal evolution of AiiA concentration for cells numbered 10, 30, 50, 70 and the mean concentration as well; here it is visibly evident that AiiA concentration across cells and the mean oscillate in phase. Equation V4 is yet another implication of this fact; it shows the time evolution of amplitude histograms. Ideally, for synchronized similar oscillators amplitude histogram should oscillate at the system frequency with a very narrow spread about its mean. Though the histogram is not quite narrow at times, but we can see a similar trend.

Figure M18 is the spatio-temporal heatmap for mixing between Danino and HOTFM. We have an 1D array of 200 cells; cells numbered 1 to 100 are the Danino oscillators, while those numbered 101 to 200 are HOTFM systems. Initially, only a couple of cells have non zero LuxI concentrations for both the systems. As time evolves, oscillations spread across the system; and once these reach the boundary between the systems, there is mixing of the two systems in the spatial sense.

×