Team:UT-Tokyo/Counter/Project/Modeling

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Modeling is an attempt to describe, in a precise way, an understanding of the elements of a system of interest, their states, and their interactions with other elements.

The purpose of our modeling team is to peel back the layer of appearance of the device to reveal it's underlying nature. We tried to improve the device, cooperating with the experiment team. To achieve our goal, we have developed three fundamental themes. These three themes divide the modeling part into three parts. At the beginning, we confirmed whether our circuit realizes a reaction:this for part 1. Next, we adjusted the parts and the conditions, for the device to reproduce a satisfactory value suitable for naming the device as a counter:this for part 2. Finally, we discussed what would be appropriate modeling, frequent issue to attack, in order to find the best strategy of modeling and wrote how we constructed our model: this for part 3.

In Part1(Deterministic Model,Stochastic Model), we approached the problem in two ways.

・Deterministic model:In this model,chemical reactions are described as differential equations and concentration of reaction products can be calculated by those of reactants. This model is intuitive, simple and hence popular to estimate the results of experiment.

・Stochastic model:The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). We used Gillespie Algorithm to solve CME.

In Part2(Result), changing measured values of gene copy numbers, strength of P_Const, sequence of taRNA and etc. in silico, we estimated in which combination of values the counter outputs a sufficient amount of data.

In Part3(Guide for Modeling), what is modeling, aims of modeling and different stochastic approaches and their interrelationShips

First of all, we constructed the deterministic model to estimate the behavior of the counter. In this model, biochemical reactions are described as differential equations and concentration of reaction product can be calculated by those of reactants. This model is intuitive, simple and hence popular to estimate the result of experiment. We were able to get some parameters for modeling of counter from previous works.[→ see parameter section]

We had simplified the construction of mathematical model before described time evolution in which concentrations of mRNAs and proteins change as differential equations. First, we regarded that the reaction between taRNA(transactivating RNA) and crRNA(cis-repressor RNA) in riboregulator is much faster than that of transcription or translation and equilibrium reaction. This diminution of parameters enable us to use the equilibrium constant as a parameter and prevent us from overfitting when we adapt this model to raw data.

equation (1):

So, the concentrations of mRNAs and the coupling of taRNA and crRNA are represented as stated above . Subscript means the gene encoded by the mRNA. We regarded that the affinities between crRNA and taRNA on different mRNAs were equal. The dissociation constant of equilibrium reaction was therefore shown as following.

equation (2):

Using dissociation constant, concentrations of reaction products such as [mcr_cr-σ] could be discribed as function of those of taRNA and mRNA of σ and GFP. We put X, A and B as the total quantity of taRNA, sigma and GFP, respectively.

equation (3):

equation (4):

equation (5):

Using these equations((3)-(5)) and equilibrium constant, concentrarions of free taRNA, free mRNAs and taRNA-mRNA complexes are described as following. These are all of simplifications.

Finally, we built up differential equations about concentrations of reaction products including mRNA of sigma which has no riboregulator. (It makes positive feedback loop.) This model assumed relationship between promoter and the amount of transcriptional product increasing per unit time. The amount is in proportion to the number of promoter if the promoter expressed constitutively and can be approximated by Hill equation if the inducer controled its promoter. It was also assumed that the degradation of proteins and RNAs followed first order kinetics. Some of used parameters were cited from references.[1]~[6]

We aimed to determine parameters about sigma through experiment and used provisional parameter deter- mined in reference to other promotor.

The transctiptional activity of mRNA encoding sigma factor initiated by positive feedback loop and that of mRNA encoding anti-sigma induced by IPTG are described as following.

In our project, IPTG induction was aimed at enough production of anti-sigma to reset the counter and the sensitivity of P_lac(lactose promoter) was not our main interest. Therefore, we used simple equation,(15) to describe how lactose promoter behave. P_lac depend on the concentration of IPTG but we regarded it as a fixed number in this modeling.

Taking into account that translation coincide with transcription in prokaryotes, the linearity between the concentration of transcriptional product and the change rate of the amount of translational product was assumed. this linearity does not depend on the kind of translational product. We also assumed that anti-sigma combine with sigma and form inert matter irreversibly, and the reaction velocity of that is proportional to product of these.

Using above-mentioned differential equations, we simulated behavior of the counter by Euler's method.

Parameter

We explain how we determined the parameters of the deterministic model. PoPS (promoter per second) of P_Const is 0.03\cite{promoter}, so its promoter activity is 0.03/(6.0× 10²³× 1.0× 10^-15)[M/sec] = 0.051[nM/sec]. The switch point and hill coefficients of P_BAD is written in \cite{pBAD1}. PoPS of P_BAD is 5/60\cite{pBAD1} , so its RPU (relative promoter unit) is (5/60)/(0.03) = 2.78. We set the RPU of P_lac as 2 when induced. We don't consider the leak expression from P_lac.

The average half life of mRNA is 2-5 min\cite{Uri}, so we set the degradation rate of mRNA as 0.010[/sec]. The half life of GFP is infinite\cite{GFP}, so we set the degradation of GFP as 0.0[sec]. The degradation rate of sigma factor[2](reference) is fast. So we set as 0.0001[/sec]. The degradation rate of anti-sigma is unknown, so we set as 6.0× 10^-6, the average degradation rate of protin(reference). The equilibrium constant of the equations (1)(2) is 80.0[nM]\cite{taRNA}. The reaction rate of the association of sigma and anti-sigma is unknown. We assumed this reaction is fast so we set as 10.0[/M sec]. The number of plasmids copied is 100~300\cite{plasmid1}\cite{plasmid2} , so we set as 200. The number of ribosomes on a mRNA is about 30(reference) and the time for a ribosome to translate is about 2 minute(reference), so we set the translational rate as 30/120 = 0.25[/sec].

The summary of the parameters of this model is given in Table 1.

Result

The unit of vertical axis is [nM], and that of the horizontal axis is [sec]. We can see that only after the second induction GFP was expressed.

By conducting sensitivity analysis, we can know what parameters have the most influential to the system.

horizontal axis : the pulse length of the arabinose induction

vertical axis : fluorescence after the first induction/fluorescence after the second induction

Formulation of the Model

If there are a lot of molecules, modeling usually uses ordinary differential equations, but some in vivo reactions involve only a few molecules. For example, transcription involves the cell's genomic DNA which is one copy or plasmids which are about 200 copies \cite{plasmid1}\cite{plasmid2} in a cell of Escherichia coli. The average size of a cell of E. coli is about 1.0× 10^-15[L]\cite{volume}, so the concentration of DNA is about 1.7[nM] and the concentration of plasmids is about 200 times of it. This is obviously weak. Reactions like this are well affected by fluctuations due to the reactants's limited copy numbers. So, we need to take this fluctuations into our modeling which is derived from stochastic methods. We also introduce delay effect.

First we explain about the Gillespie algorithm which is often used in stochastic simulations. In the Gillespie algorithm, we treated not the concentration of molecules but the number of them. Reactions are also viewed as discrete, essentially instantaneous physical events. What we have to determine when using the Gillespie algorithm is (A) when the next reaction is going to occur and (B) which type of the reaction it will be. Looking more closely at the Gillespie algorithm by the next set of reaction formulas:

Let n₁, n₂, and n₃ denote the respective copy number of the components X₁, X₂, and X₃. Notice that they are all integer. First we have to determine how easily each reactions could happen. It depends on the number of components copied. In stochastic simulations, we often determine the parameter called stochastic rate constant, which is often written as "c. We assume that each possible combinations of reactant molecules have the same probability c per unit time to react. In other words, c× dt gives the probability that a particular combination of reactant molecules will react in a short time interval [t,t+dt). We call the stochastic rate constant of a reaction j, c_j. Considering the all combinations of reactant molecules, the probability that the reaction 0 occur in [t,t+dt) is c× n₁× n₂. We now define the propensity function as the function of which product with dt gives the probability that a particular reaction will occur in the next infinitesimal time dt, which is often written as "a. Later on, the propensity function of a reaction j is a_j. Following the equation:

Notice that c_j is invariant parameter, but a_j changes as the state changes. In the same way, a₁ = c₁× n₃.

First we answer the question (A) when is the next reaction going to occur? Now, to simplify the situation we assume the situation that only the reaction 0 occurs. Set the time as 0, and define P(t) as the probability that the reaction 0 doesn't occur in [0,t). Then from the definition of a,we obtain the equation; P(t+dt) = P(t)(1-a× dt). (Because the probability that the reaction 0 doesn't occur in [0,t+dt) is the product of the probability that the reaction 0 doesn't occur in [0,t) with the probability that the reaction 0 doesn't occur in [t,t+dt).) Using P(t+dt) = P(t) + dP(t)/dt, we get :

Because the probability that the reaction0 doesn't occur in a 0 second interval is zero; P(0)=1. Solving the above ordinary differential eqaution we get :

If r₁ is a uniform number from [0,1], the time of the next reaction should be determined by solving P(t) = r₁. Using (2), we get t = -a₀/log r₁.

Now we suppose there is N types of reactions. Let a₁,a₂,…,a_N denote the respective propensity function of reaction 1,2,…,N. From previous method;

Let dt be so small that we can ignore the term of higher than two orders of dt. The equation(3) becomes:

Solving (4) (a = Σ_{j=1}^{N} a_j

Setting τ as the time of the next reaction, we get:

Second we answer the question (B) what types of the reaction will it be? We determined the time of the next reaction, so what we have left to do is to determine what kind of reaction occurs. Some people may feel queer, but in the Gillespie algorithm, first the time of next reaction will be determined, and second the kind of reaction will be determined. It is natural to determine that the probability that the reaction j occurs is a_j/a. If r₂ is a uniform number from [0,1], j is the only number that meets below in equations:

In the case a₀ ≧ a× r₂, the reaction that occurred is reaction 0.

Now we can run the Gillespie algorithm by following the next steps.(t_MAX is the finish time of the simulation.)
1.Initialize the system at t = 0 with initial numbers of molecules for each spices, n₀,… ,n_s
2.For each j = 0,1,…,r, calculate a_j(n) based on the current state n using (21)
3.Calculate the exit rate a(n) = Σ_{j=0}^{r} a</sub>j</sub>(n).
4.Compute a sample tau of the time until the next time using (27)
5.Update the time t = t + tau
6.Compute a sample j of the reaction index using (28)
7.Update the state n according to the reaction j.
8.If t < t_MAX, return to Step 2

Stochastic rate constant can be determined by the parameters we used in the deterministic model (if we modeled the reaction in the deterministic model) . If there are a lot of reactant molecules, stochastic simulations have to show similar results as those of determinisitic simulations. For this reason, stochastic rate constant, c, can be calculated from the chemical reaction rate constant, k. See \cite{gillespie1} if you want to know the deriving process. Here we just write the result.

For a unimolecular reaction, c numerically equals to k, whereas for a bimolecular reaction, c equals to k/N_AV if the species of the reactant molecules are different, or 2k/N_AV if they are the same.V is the volume of the system and N_A is the Avogadro's constant.

However, these results should not be taken to imply that the mathematical forms of the propensity functions are just heuristic extrapolations. The propensity functions are grounded in molecular physics, and the formulas of deterministic chemical kinetics are approximate consequences of the formulas of stochastic chemical kinetics, not the other way around.

The Gillespie algorithm is so clear and useful that it is often used. However, this algorithm is not suitable for describing transcriptions and translations because they are very slow and complex reactions involving many kinds of reactant molecules. If we treat transcription from plasmids as one reaction, assuming the copy number of plasmids as 200, then the propensity function a equals to the stochastic rate constant multiplied by 200 (200× c). So it will take about one of a two hundred times of an average transcription time to finish one transcription. Of course, in the time scale of average transcription time it is not a big problem, but this may not be good for simulating, like in our project, the system that uses the time for transcriptions and translations cannot be shortened. We introduce time-delay into the Gillespie algorithm based on \cite{delay1}~\cite{delay3}. The mathematical correctness of this algorithm is proved in \cite{delay3}. Time-delay means treating reactions as following:

Furthermore, transcriptions and translations are too complex to list up all of the reactions step by step. Therefore it is better to treat them as time-delay than reaction formulas.

Now we begin to model our project, sigma Re-counter. In our model, there are only three reactions: transcription, translation, and an association and disassociation of crRNA and taRNA. We introduce time-delay into only transcription and translation. Then, we explain how we treat these three reactions in general.

First we explain transcription's model\cite{stochastic}. When the RNA polymerase binds to the promoter region, first they take the RNAP・promoter close complex. At this state, the complex can dissociate. But with a certain probability, the close complex turn to the open complex which doesn't dissociate. After the RNA polymerase and the promoter region take the open complex, a transcription starts. Then the reaction formula of transcription can be described as following's reactions:

combining reaction3' and reaction3, we get:

Second, we refer to the translational model [8]. Similary to the transcrptional model we model as following;

combining reaction2' and reaction2, we get:

Last, the model of association and disassociation of crRNA and taRNA is a reversible reaction. So we model as following:

We can conclude that reaction formulas of our model are as follows: