Mathematical modelling plays a central role in Synthetic Biology, due to its ability to serve as a crucial link between the concept and realization of a biological circuit: what we propose in this page is a mathematical modelling approach to the entire project, which has proven extremely useful before and after the "wet lab" activities.

Thus, immediately at the beginning, when there was little knowledge, a mathematical model based on a system of differential equations was derived and implemented using a set of reasonable values of model parameters, to validate the feasibility of the project. Once this became clear, starting from the characterization of each simple subpart created in the wet lab, some of the parameters of the mathematical model were estimated thanks to several ad-hoc experiments we performed within the iGEM project (others were derived from literature) and they were used to predict the final behaviour of the whole engineered closed-loop circuit. This approach is consistent with the typical one adopted for the analysis and synthesis of a biological circuit, as exemplified by Pasotti L et al. 2011.

After a brief overview on the importance of the mathematical modelling approach, we deeply analyze the system of equations, underlining the role and function of the parameters involved.
Experimental procedures for parameter estimation are discussed and, finally, a different type of circuit is presented. Simulations were performed, using ODEs with MATLAB and used to explain the difference between a closed-loop control system model and an open one.

Mathematical modelling reveals fundamental in the challenge of understanding and engineering complex biological systems. Indeed, these are characterized by a high degree of interconnection among the single constituent parts, requiring a comprehensive analysis of their behavior through mathematical formalisms and computational tools

• Simulation: mathematical models allow to analyse complex system dynamics and to reveal the relationships between the involved variables, starting from the knowledge of the single subparts behavior and from simple hypotheses of their interconnection. (Endler L et al. 2009)
• Knowledge elicitation: mathematical models summarize into a small set of parameters the results of several experiments (parameter identification), allowing a robust comparison among different experimental conditions and providing an efficient way to synthesize knowledge about biological processes. Then, through the simulation process, they make possible the re-usability of the knowledge coming from different experiments, engineering complex systems from the composition of its constituent subparts under appropriate experimental/environmental conditions (Braun D et al. 2005; Canton B et al 2008).

Equations (1) and (2) have identical structure, differing only in the parameters involved. They represent the synthesis, degradation and diluition of both the enzymes in the circuit, LuxI and AiiA, respectively in the first and second equation: in each of them both transcription and translation processes have been condensed. The mathematical formulation is analogous to the one used by Pasotti L et al. 2011, Suppl. Inf., even if we do not take LuxR-HSL complex formation into account, as explained below.

These equations are composed of 2 parts:

1. The first term describes, through Hill's equation, the synthesis rate of the protein of interest (either LuxI or AiiA) depending on the concentration of the inducer (anhydrotetracicline -aTc- or HSL respectively), responsible for the activation of the regulatory element composed of promoter and RBS. In the parameter table (see below), α refers to the maximum activation of the promoter, while δ stands for its leakage activity (this means that the promoter is slightly active even if there is no induction). In particular, in equation (1), the almost entire inhibition of pTet promoter is given by the constitutive production of TetR by our MGZ1 strain. In equation (2), pLux is almost inactive in the absence of the complex LuxR-HSL.

   Furthermore, in both equations k stands for the dissociation constant of the promoter from the inducer (respectively aTc and HSL in eq. 1 and 2), while η is the cooperativity constant.

   
   
2. The second term in equations (1) and (2) is in turn composed of 2 parts. The former one (γ*LuxI or γ*AiiA respectively) describes, with an exponential decay, the degradation rate per cell of the protein. The latter (μ*(Nmax-N)/Nmax)*LuxI or μ*(Nmax-N)/Nmax)*AiiA, respectively) takes into account the dilution factor against cell growth which is related to the cell replication process.

Here the kinetics of HSL is modeled, through enzymatic reactions either related to the production or the degradation of HSL. This equation is composed of 3 parts:

1. The first term represents the production of HSL due to LuxI expression. We modeled this process with a saturation curve in which V_max is the HSL maximum transcription rate, while k_M,LuxI is LuxI dependent half-saturation constant.
   
   
2. The second term represents the degradation of HSL due to the AiiA expression. Similarly to LuxI, k_cat represents the maximum degradation per unit of HSL concentration, while k_M,AiiA is the concentration at which AiiA dependent HSL degradation rate is (k_cat*HSL)/2. The formalism is similar to that found in the Supplementary Information of Danino T et al 2010.
   
   
3. The third term (γ_HSL*HSL) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.

This is the typical cells growth equation, depending on the rate μ and the maximum number N_max of cells per well reachable (Pasotti L et al. 2009).

The aim of the model is to predict the behavior of the final closed loop circuit starting from the characterization of single BioBrick parts through a set of well-designed ad hoc experiments. This section presents the experiments performed. As explained before in NOTE, considering a set of 4 RBSs for each subpart expands the range of dynamics and helps us to better understand the interactions between state variables.

These are the first parts tested, with the target of learning more about pTet and pLux promoters. In particular, as previously explained in NOTE, for each promoter, we tested four different combinations of promoter-RBS, providing us a set of fundamental building blocks for the subsequent assebly of the closed-loop circuit.
As shown in the figure below, we considered a range of inductions and we monitored, in time, absorbance (O.D. stands for "optical density") and fluorescence; the two vertical segments for each graph highlight the exponential phase of bacterial growth. S_cell (namely, synthesis rate per cell) can be derived as a function of inducer concentration, thereby providing the desired input-output relation (inducer concentration versus promoter+RBS activity), which was modelled as a Hill curve: However, also Relative Promoter Unit (RPU, Kelly JR et al. 2009) has been calculated as a ratio of S_cell of promoter of interest and the S_cell of BBa_J23101 (reference to HP₃). As shown in the figure, α, as already mentioned, represents the protein maximum synthesis rate, which is reached, in accordance with Hill equation, when the inducer concentration tends to infinite, and, more practically, when the inducer concentration is sufficiently higher than the dissociation constant. Meanwhile the product α*δ stands for the leakage activity (at no induction), liable for protein production (LuxI and AiiA respectively) even in the absence of inducer. The paramenter η is the Hill's cooperativity constant and it affects the ripidity of transition from the lower and upper boundary of the curve relating S_cell to the inducer concentration. Lastly, k stands for the semi-saturation constant and, in case of η=1, it indicates the concentration of substrate at which half the synthesis rate is achieved.

T9002 introduction

LuxI and AiiA tests have been always performed exploiting the well-characterized BioBrick BBa_T9002, by which it's possible to quantify exactly the concentration of HSL. This is a biosensor which receives HSL concentration as input and returns GFP intensity (more precisely Scell) as output. (Canton et al. 2008). According to this, it is necessary to understand the input-output relationship: so, a T9002 "calibration" curve is plotted for each test performed.

AiiA & LuxI

Then, as a second step, we monitor in separate experiments HSL synthesis and degradation caused by the activities of the enzymes. In other words, our idea is to control the degradation of HSL versus time. ATc activates pTet and, later, a certain concentration of HSL is introduced. Then, at fixed times, O.D.₆₀₀ and HSL concentration are monitored using Tecan and T9002 biosensor. For example for AiiA dependent HSL degradation, we have:

Therefore, considering for a determined promoter-RBSx couple, several induction of aTc and, for each of them, several samples of HSL concentration during time, parameters V_max, k_M,LuxI, k_cat and k_M,AiiA can be estimated, through numerous iterations of an algorithm implemented in MATLAB.

N

Degradation rates

Simulations

Sensitivity Analysis of the steady state of enzyme expression in exponential phase

In this paragraph we investigate the theoretical behaviour of our circuit in the cell culture exponential growth phase. According to this, we first derive, under feasible hypoteses, the steady state condition for the enzymes and HSL concentration in that phase. Then we perform a sensitivity analysis relating the output of our system (HSL) to input (atc) and system parameters.

Steady state of enzyme expression

Two major hypoteses allow to formulate the steady state during the exponential growth phase. The former involves the number of cells N (in the order of 3*10^3), which is far lower than N_max (2*10^9). The latter pertains to γ*HSL parameter, whch can be neglected compared to the other two terms of the third equation. Based on this assumptions, equation (4) of the system becomes dN/dt=μN. Moreover, from equation (3), after having removed the third term, we can simplify the N parameter, since it is common to the remaining two terms. On a biological point of view, this implies that AiiA, LuxI and HSL undergo only minor changes through time, thereby allowing to derive their steady state expressions:

The first equation is independent from the second and third ones, enabling us to directly determine LuxI steady state expression during the exponential growth phase. On the contrary, second and third equations depend each other in defining the value of AiiA and HSL respectively, because the former is a function of HSL, while the latter is a function of AiiA. So we could resolve a system of two equations, first by expliciting one of the two variables with respect to the other, and then substituting its expression in order to determine the other variable possible values. This would bring a complex mathematical formulation, which is not helpful in understanding the influence of the various model parameters on the output HSL. On the other hand, AiiA and HSL values can also be graphically determined from the intersection of the curves derived from these two equations, if we explicit HSL as a function of AiiA (or, alternatively, AiiA as a function of HSL). It is easy to discover that these two curves represent rectangular hyperbolae (the first one only under a simple approximation, explained below) whose tails intersect each other at a particular point, corresponding to the searched values for AiiA and HSL.

centered at O(-d/c;a/c), with the vertical asymptote x=-d/c and the horizontal asymptote y=a/c

From equation 2, we have:

We can introduce the simplification to remove the ηplux exponent to the entire expression in the right hand side of the equation; even if this leads to a slight change in the curve behaviour, it allows to more clearly understand the relation between HSL and AiiA. Indeed, this approximation results in a rectangular hyperbola:

Parameter	Expression
a
b
c
d

The table below provides the corresponding asymptotes

Horizontal asymptote	Vertical asymptote

As pertains to equation 3, its steady state approximation during the exponential growth is more immediately identifiable as a rectangular hyperbola. In particular, the table below clarifies it has the ordinate axis as its vertical asymptote.

a	b	c	d
			0

Ver. asym.	Hor. asym.
0

Sensitivity analysis

Now, it is interesting to conduct some qualitative and quantitative considerations about our system sensitivity to its parameters and aTc input signal.

From the above figures, it is also clear that HSL steady state value is not very sensitive to α_pLux, at least when this parameter presents values greater than unity, because this brings the two curves to intersect in their low slope regions.

Referring to the second RH, the only adjustable asymptote is the horizontal one, that we can move upward or downward by altering VLuxI, which indirectly depends on aTc.

A further deepening in the exponential phase analysis involves determining the characteristics of the input-output relation. Our closed loop system can be realized with two alternative purposes in mind:

1. realizing a circuit able to adapt HSL output depending on aTc input concentration. This requires a good sensitivity between input and output.
2. designing a robust HSL concentration controller, which is immune to the input noise and offers a constant and defined amount of HSL. In this case HSL level should be appropriately tuned during the design stage, by choosing the correct strength of promoter-RBSx complexes.

Now,again considering the system of equations, it is easy to observe that HSL dependence on aTc input passes through two Hill equations. The former describes aTc driven LuxI synthesis, while the latter models LuxI dependent HSL synthesis rate. Therefore, in order to achieve a high aTc sensitivity, it is advisable to tune aTc and LuxI levels so that they place outside the saturation regions of their Hill curves. In this regard, it is possible to determine a closed form expression relating HSL to aTc, if we hypothesize that both aTc and LuxI are far lower than their respective half-saturation constant (k_ptet and Km), thus simplifying the Hills with a first order relation:

References