Introduction

Aim of this repository and target users

This repository is made in order to provide python programs that make simulations of the number of molecules produced by a gene. Of course, it is a very wide world because we can have different reactions and different models that describe such process in biology. In this repository we explore three main genetic circuits described in section Models and provide also programs to analyse simulation data (that can not just be generated and left there!). We have wrapped this repository into a unique name called PyExTra because of the acronym with gene expression and translation processes that are simulated with his python programs. Users which can be interested in PyExTra are beginners with gene expression simulations and analysis. You can just use it or explore its code to have some help for your research work. PyExTra, indeed, was born with the aim to investigate about NF-kB model in colon cancer cells for my master degree thesis in Physics (it is Thesis.tex and Thesis.pdf available in Thesis folder and the thesis presentation MDThesisPresentation_ManuelaCarriero.pptx).

Dependencies

Python verion: 3.9.x
Python modules:

• numpy, scipy, pandas, stats.models, matplotlib, seaborn;
  
• scikit-learn, keras, tensorflow;
  
• ssqueezepy, pywt, scaleogram.

OS: Windows10, Linux (Ubuntu).

Structure of the project

1. First of all, make the simulation using the Stochastic Simulation Algorithm (SSA):
   ssa_simulation.py SSA simulation of the first protein synthesis model;
   ssa_simulation_autorepressor.py SSA simulation of the Autorepressor model;
   ssa_simulation_2_ind_genes.py SSA simulation of two independent genes;
   ssa_simulation_toggleswitch.py SSA simulation of the Toggle-Switch model.
   ssa_simulation_nfkb.py SSA simulation of the first NF-kB model.
   
2. Then using:
   plots.py you can plot the number of molecules as function of time and the distribution of states;
   acf.py you can plot the autocorrelation functions refered to the temporal series. statistics.py is a python script to calculate and plot the quantiles autocorrelation.
   You can also use fft.py and wavelet.py: they are python scripts for the Fast Fourier Transform and the Continuous Wavelet Transform analysis.
   
3. Furthermore, acfk_DNNregressor.py is a script that given the autocorrelation values, it estimates the model parameters using a deep neural network built with keras. The autocorrelation values are generated by either firstmodel_ssa_acfs_ks.py or autorepressor_ssa_acfs_ks.py or toggleswitch_ssa_acfs_ks.py.
   
4. Other types of algorithms to simulate biological stochastic processes have been implemented:
   tauleap_simulation.py: it makes simulations using the Tau-leap algorithm;
   hybrid_simulation.py: it makes the simulation using an algorithm that combines SSA algorithm with the Tau-leap (NOTE: it is new ! The algorithm has been developed by me and my Supervisor during the thesis work).
5. ODEvsStoch.py is a script to compare deterministic approach (Ordinary Differential Equation) vs stochastic simulation results.

Testing

As you can notice from the Dependencies section, we have made use of many built-in functions. However, this work is mainly based on the simulations and the autocorrelation function, hence tests are performed on these two main topics, on which this research is based, in order to ensure that the basics work properly.

Models

First protein synthesis model

A simple model for central dogma of biology is represented by the following biological circuit:

The number of messenger RNAs m and proteins n produced is a stochastic process and the Chemical Master Equation describes the time evolution of the probability of a stochastic system to be in a particular configuration.

Autorepressor model

In this model there is a gene and the gene product p represses its own transcription.

Toggle-switch model

The genetic toggle-switch is a system of two mutually repressing genes.

First NF-kB model

Algorithms

In this documentation, the stochastic simulation algorithms are explained by considering the first protein synthesis model. If you want to have a look at all the transitions and the reaction probability rates considered for the other models, see the presentation MDThesisPresentation_ManuelaCarriero.pptx.

Stochastic Simulation Algorithm (SSA)

Gillespie algorithm (i.e. stochastic simulation algorithm) samples the probability distribution described by the master equation. The basic idea is that events are rare, discrete, separate events, i.e., each event is an arrivial of a Markov process and the algorithm is based on the following steps:

1. Choose some initial states (in the first protein synthesis model case, initial state of gene that is active or inactive, initial number of mRNA molecules m₀ and initial number of proteins p₀);
2. A state change will happen. In this model, the event that can happen is one of the following:

   • gene activation: the state of the gene switches from inactive to active and, in such case, there is no increase or decrease in the number of RNA or protein molecules;

   • gene inactivation: the state of the gene switches from active to inactive and, in such case, there is no increase or decrease in the number of RNA or protein molecules;

   • RNA synthesis: the number of RNA molecules increases by 1;

   • RNA degradation: the number of RNA molecules decreases by 1;

   • protein synthesis: the number of protein molecules increases by 1;

   • protein degradation: the number of protein molecules decreases by 1.

   We can also call these events as transitions and each one happens with a certain transition rate. Higher rate of transition means that the event happens faster and so this event is more likely to occur. In this modeling, when the gene is activated, RNA molecules are produced with rate k₁, and degraded with rate k₂. When the gene is inactive, RNA molecules can only be degraded (Φ is the degraded molecule), but not produced. Proteins are produced with rate k₃ and degraded with rate k₄ and this happens only if the number of RNA molecules is greater than 0. Hence, the rate of protein production depends on the number of mRNAs but not vice-versa. Finally, activation and deactivation happen respectively with rate k_a and k_i. Thus, the second step of the algorithm is to define all the possible transitions that can happen and to calculate the transition probabilities. All the reactions are summarized in the table below:

   transition	transition rate	m	p
   Gene(I) → Gene(A)	ka	0	0
   Gene(A) → Gene(I)	ki	0	0
   Gene(A) → Gene(A) + RNA	k1	+1	0
   RNA → Φ	k2	-1	0
   RNA → RNA + Protein	k3	0	+1
   Protein → Φ	k4	0	-1

3. Calculate the time of residency, that is how much time the system is in that specific state. Since the distance between consecutive events in Markov processes is Exponentially distributed, the time of residency has an exponential distribution with a characteristic time equal to the inverse of the sum of the total rates.
4. Choose what state change, i.e. transition, will happen. For this purpose we use random.choices method that returns a list with the randomly selected element (i.e. a randomly selected transition) from the specified list of transitions with the possibility to choose the probability of each element selection.
5. Increment time by time step you calculate in step 3.
6. Update the state according to the state change chosen in step 4.
7. If the total time spent is less than a pre-determined stopping time, go to step 2. Else stop.

You can look at the flowchart that explains in a more "intuitive" way the SSA algorithm:

Tau-leap algorithm

1. Choose some initial states and reaction time step $\tau$ (tau);
2. Identification of all possible reaction events;
3. Calculation of reaction probabilities;
4. The number of reactions in the reaction time step is given by a Poisson distribution with mean equal to reactions probabilities times $\tau$;
5. Increment time by $\tau$;
6. Update the state according to the state change decided in step 4;
7. If the total time spent is less than a pre-determined stopping time, go to step 2. Else stop.

You can look at the flowchart that explains in a more "intuitive" way the Tauleap algorithm:

SSA/Tau-leap algorithm

SSA/Tau-leap algorithm, as such called "hybrid", is presented directly through the flowchart:

1. The SSA updates more numerous molecular species such as genes, the tau-leap less numerous molecular species such as RNAs and proteins;
2. The tau-leap algorithm uses the gene state residency time as time step in the algorithm;
3. There are two important checks: the number of updated molecules by the tauleap algorithm has to be greater or equal than 0 and the difference between reaction probabilities has to lower or equal to a threashold (to monitor accuracy of results);
4. If step 3 is not satisfied, it means that we need to make smaller steps, to we divide it by one half; we check if the new smaller tau is greater of equal than 3 times the characteristic time of the system state given by the SSA considering all the reactions because, if it is smaller, it is worth to make the SSA algorithm; If yes, we go on applying the tau-leap algorithm to the new residency time untill we reach a time that is close to the initial gene state residency time value (the code is math.isclose(time_count, gil_time, rel_tol=0.5)).

Running PyExTra

Installation

In order to use PyExTra you can just clone this repository in the folder that you desire. So first open your terminal, move in the directory you want to put PyExTra and then type:
git clone <git_repo_url> <directory_name_where_you_clone_PyExTra>
For example, if you want to clone this repository in a folder named "PyExTra", you can type this command:
git clone https://github.com/ManuelaCarriero/PyExTra PyExTra

Usage

PyExTra is mainly based on command lines to let you use the available Python programs in a easier and friendly way.

Command line syntax

PyExTra command lines are described following the Structure of the project order.

1. Starting from the basics, if you want to run the SSA simulation of one of the studied models, the command line synthax is the following:
   python <simulation_model.py> <configuration.txt> -run
   If you want to run more than one simulation, beware of specifying the number of simulations that you desire in the configuration.txt file and change the command line simply in this way:
   python <simulation_model.py> <configuration.txt> -run_multiplesimulations
   After running one of these, you will have a file called gillespiesimulation_results.csv in your working directory or, if you choose to run more than one simulation, the program gives a list of files named gillespieresults_seed<number_of_the_random_seed> (i.e. run 4 simulations, you will obtain 4 files named gillespieresults_seed1, gillespieresults_seed2, gillespieresults_seed3 and gillespieresults_seed4).
   You can ask for the help python <simulation_model.py> -h in order to know all the possible commands you can ask to the program with the description of what they do.

2. Stay in the same working directory where the simulation results are. You can run this command line in order to plot the number of molecules over time of observation:
   python plots.py <configuration.txt> -time_plot
   This is in case of first model simulation results. For the other models the syntax is similar:
   python plots.py <configuration.txt> -<model>_time_plot
   You need to specify the type of model you are considering.
   For the precise list of commands you can ask for the help even in this case: python plots.py -h
   Importantly, you can plot the distribution of states (i.e. the stationary distribution) using:
   python plots.py <configuration.txt> -distribution

3. In order to plot the autocorrelation values as function of sampling time:
   python acf.py <configuration.txt> -plot_acf_RNA
   This is the case you want to plot the autocorrelation of only the RNA number of molecules in case of SSA simulation results.
   Also in this case, you can ask for the help: python acf.py -h
   

4. Follow the command line syntax already described in 1. and 2. for hybrid_simulation.py and tauleap_simulation.py.

The others are scripts that you can use for the analysis of results.

Example of usage and results

Let us consider the first protein synthesis model and play with its parameters configuration.
Open file configuration.txt. You can start from a basic configuration where rate constants: ka = 1, ki = 0.5, k1 = 1, k2 = 0.1, k3 = 1, k4 = 0.1, k5 = 0, that is a gene that tends to be more active than inactive.
Run the SSA simulation python ssa_simulation.py configuration.txt -run
Then you would like to observe the time course behavior: python plots.py configuration.txt -time_plot

And mostly the stationary distribution: python plots.py configuration.txt -distribution after modifying the time limit value from 1000 to 10000 in configuration.txt in order to obtain the Poisson distribution:

If you change the type of regulation by making the gene more inactive than active (for instance, ka = 0.1 and ki = 1), you should see a distribution of states whose states with higher residency time are those with lower number of molecules (in particular, a peak at zero molecules). Thereby, the Poisson distribution will change its shape assuming a longer tail on the right.

Try yourself and, if you want, let me know ! PyExTra let you simulate gene expression in a way that you can manipulate your biological system.

References

https://github.com/UniboDIFABiophysics/programmingCourseDIFA

In particular:

Lesson_AF_03_continuous_time_random_walks.html