Models

Overview

We regard mathematical modeling as priority of our software, FLAME. In order to achieve a balance between efficiency and accuracy in simulation, we grasp several core elements of genetic circuits, and stick to them throughout the process. When delving into different circuits, we categorize some of them into different logic groups based on their functions, and improve the equations for further use. Here are the features of our models and algorithms:

* For simplicity, we mainly consider the effects of promoters and RBSs on expression, and utilize ordinary differential equations (ODE) and Hill equations in our models.

* Most of the biochemical processes are made up of activation and/or repression, so we use two equations, each representing activation and repression respectively, as the fundamental of our models.

* Instead of sticking to only general principles, we realize the importance of building models for different circuits to improve the accuracy.

* We do not engage ourselves in finding values of all the parameters to exactly simulate a process; instead, some of parameters have little impact on simulation but grossly complicate data-processing, so we left them out. By utilizing only the parameters that is of greater importance, we improve the efficiency and avoid vain efforts.

* Leakage rate is considered to better reflect complex systems and experimental results^[14].

Parameters

Parameter	Description	Range	Unit	Remark
[P]	Concentration of proteins	≥ 0	Cell-1
[R]	Concentration of repressors	≥ 0	Cell-1	Most repressors are proteins
[A]	Concentration of activator	≥ 0	Cell-1	Most activators are proteins
α	Maximum rate of protein synthesis	≥ 0	RiPS	For simplicity, α is equal to RiPS.
β	Degradation rate of protein	1×10-8~1	1
γ	Leakage rate	0~1	Cell-1
K	Reaction constant	≥ 0	1	equivalent to concentration at which the protein is half-activated or half-repressed

Deterministic models

Our software aims to simulate the complex biological system and introduce interdisciplinary method to solve problems. We learned from the models adopted by our team last year, and developed new models to further illustrate the relations between parts and circuits.

We look into these basic processes and find that biological process is actually the result of activation or repression between proteins; in other words, we can use the activation and repression to further represent most biological processes. So we look up the related reference to find out formulas for the activation and repression. We utilize the basic Hill equation as following, and improve the equation.

  Basic Hill equation:     

where θ represents the fraction of the ligand-binding sites on receptor proteins occupied by their specific ligands; [L] is the free (unbound) ligand concentration; K_A is the ligand concentration in which half of the ligand-biding sites are occupied. This is also the microscopic dissociation constant; n is the Hill coefficient;

Since biological processes are the results of activation and repression between proteins, they also display similar results in the equations form and we therefore deduce their ordinary differential equations^[1][2][3][4]:

  Repression:      

  Activation:      

where [P] is the concentration of the target protein; [R] is the concentration of the repressor protein; [A] is the concentration of activator protein; α is the maximum rate of protein synthesis; β is the degradation rate of the protein; γ is the leakage rate; K is the reaction constant equivalent to the concentration at which the protein is half-activated or half-repressed; n is the Hill coefficient

These models capture the main features of biological processes and therefore achieve relative accuracy. We further adapt these equations into different biological structures such as toggle switch and oscillator in order to allow the researchers to design circuits at the structural level and therefore gain an overall and comprehensive insight.

1. Toggle Switch^[5][6]

The behavior of the toggle switch and the conditions for bistability can be understood using the following dimensionless model for the network:

where repressor1 and repressor2 mean two outputs of the toggle switch. α₁, α₂ is the effective synthesis rate of two repressor proteins, and β, γ, reflects the extent to which two proteins repress each other (See Figure 1).

Figure 1. A toggle switch, The presence of Repressor1 can repress the transcription of Repressor2, but such repression can be abolished by Inducer1; while Repressor2 can repress the transcription of Repressor1 and be abolished by Inducer2. (Adapted from Gardner et al., 2000)

Therefore, the pattern of toggle switch is: when there is an input of Inducer1 with no Inducer2, there is no output; when there is input of Inducer2 with no Inducer1, there would be an output.

Following the standard mentioned above, we deduce the following form:

2. Oscillator^[7][8][9]

Using Hill equation one can represent the system as

α_i is the conversion rate when there is no activator; β_i is the degradation rate; γ_i is the leakage rate; n is the Hill coefficient.

We transform these equations into the same form: since such process also reflects the relation of activation and repression between proteins, we can still use function F and G mentioned above to illustrate such reaction.

Therefore, we deduce:

3. Inverter^[10]

We study a structure from its function. We know that inverters serve to alter the concentration level from a high one to a low one, or vice versa. In the integrated circuit design, we can use a NOT GATE to realize the switch between high and low electric potential. Therefore, In our software, we design an inverter structure resembling the NOT GATE to mimic the real situation.

4. AND GATE as a Dual System^[11][12][13]

AND GATE are different from cases above. An AND GATE receives two inputs and returns only one output as a result. Therefore, we cannot utilize the F and G functions as mentioned above. However, it is reported that AND GATE is composed of activation and repression between proteins; therefore, they are equivalent in biological terms. We multiply these factors to indicate their interrelationship and get function H:

Dual-Activation system:

Activation-repression coexist system:

Dual-repression system:

Algorithm

1. Chose different structures, and then we identify the relationship behind that structure.

2. Invoke the corresponding functions such as F(the activation) or G(the repression).

3. Use the SPICY package built in Python to solve ordinary differentiated equations and returned the values.

Reference

[1] Edelstein-Keshet, L. Mathematical Models in Biology. McGraw-Hill, New York (1988).

[2] Kaplan, D. & Glass, L. Understanding Nonlinear Dynamics. Springer, New York (1995).

[3] Yagil, G. & Yagil, E. On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys J 11, 11-27 (1971).

[4] Rubinow, S. I. Introduction to Mathematical Biology. Wiley, New York. (1975).

[5] Gardner, T.S., Cantor, C.R. & Collins, J.J. Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339-342 (2000).

[6] Hasty, J., McMillen, D. & Collins, J.J. Engineered gene circuits. Nature 420, 224-230 (2002).

[7] Purcell, O., Savery, N.J., Grierson, C.S. & di Bernardo, M. A comparative analysis of synthetic genetic oscillators. Journal of The Royal Society Interface 7, 1503-1524 (2010).

[8] O Brien, E.L., Van Itallie, E. & Bennett, M.R. Modeling synthetic gene oscillators. Math Biosci 236, 1-15 (2012).

[9] Letters to nature. Nature 76, 321 (1995).

[10] Yokobayashi, Y., Weiss, R. & Arnold, F.H. Directed Evolution of a Genetic Circuit. P Natl Acad Sci Usa 99, 16587-16591 (2002).

[11] Wang, B., Barahona, M. & Buck, M. A modular cell-based biosensor using engineered genetic logic circuits to detect and integrate multiple environmental signals. Biosens Bioelectron 40, 368-376 (2013).

[12] Anderson, J.C. Environmental signal integration by a modular AND gate--supplementary., 1 (2007).

[13] Duarte-Galvan, C. et al. FPGA-Based Smart Sensor for Drought Stress Detection in Tomato Plants Using Novel Physiological Variables and Discrete Wavelet Transform. Sensors (Basel) 14, 18650-18669 (2014).

[14] https://2013.igem.org/Team:SYSU-Software/model