To simulate and analyze a genetic regulatory network (GRN), we need to build an objects' array to store the complete information of each gene. It contains regulation relationships between genes, sequences of genes, sequences of promoters and so on. However, it's hard to find an appropriate database online containing all information we need in a simple file. RegulonDB has downloadable files about the regulation between transcription factors (TF) and genes. Files about genetic information, transcription unit information and promoter information can also be downloaded from the RegulonDB. All those files have been put into file “source data” in the root directory of our software. They contain all information the simulation needs and we use fetching module to achieve data extraction and integration. There are four steps: fetch regulation relationships, fetch gene information, fetch promoter information and integrate information above.

1 User Input

Some genes' regulation could be get from experiment. So, if users could get the unknow regulation between new gene and old ones, they could manually set the interactions which do not need model. Those regulations will be used in later simulation.

2 Simalarity Analysis

The Needleman-Wunsch algorithm was first published in1970 by Saul B. Needleman and Christian D. Wunsch. It performs a global alignment of two sequences and is mostly used in bioinformatics to align protein or nucleotide sequence. Our software applied this algorithm in the alignment of DNA and amino acid sequences.

The Needleman-Wunsch algorithm is one kind of dynamic programming and It was the first attempt in biological sequence comparison of dynamic programming.

Here is an example of Needleman-Wunsch algorithm. S(a,b) is the similarity of character a and character b. The scores of characters are shown in the similarity matrix. We assume this matrix was

And we uses linear gap penalty, denoted by d, here, we set the gap penalty as -5.Then the alignment:

A: AGACTAGTTAC
B: CGA - - - GACGT

would have the following score:

S(A,C)+S(G,C)+S(A,A)+(3)+S(G,G)+S(T,A)+S(T,C)+S(A,G)+S(C,T) = -3+7+10-(3x5)+7+(-4)+0+(-1)+0 = 1

To find the highest score of alignment, in this algorithm, a two dimensional matrix F with sequences and scores was allocated. The score in row i, column j is denoted by Fij. There is one column for each character in sequence A and one row for each character in sequence B. Therefore, if we align sequences with sizes of n and m, the amount of memory taken up here is O(n,m).

As the algorithm going on, Fij was calculated to be the optimal score by the principle as following:
Basis:

Fi0 = d*i
F0j = d*j

Recursion:

Fij = max(F(i-1,j-1) + S(Ai,Bj), F(i-1,j) + d, F(i,j-1) + d)

The pseudo-code of this algorithm would look like this:

After the matrix F was computed, Fnm would be the maximum score among all possible alignment.

If you want to see the optimal alignment, you can trace back from Fnm by comparing three possible sources mentioned in the above code (Match, Insert and Delete). If Match, then Aj and Bi are aligned, if Insert, Bi was aligned with a gap and if Delete, then Aj and a gap are aligned. Also, you may find there are not only one optimal alignment.

As for the example, we would get the following matrix by applying Needleman Wunsch algorithm:

And the optimal alignment would be:

- - AGACTAGTTAC
CGAGAC - - GT - - -

The rows and columns in the GRN matrix can be regarded as vectors containing the regulated or the regulating information. The behavior similarity of two units can be described by the dot product of two regulated vectors or two regulating vectors. Biologists usually think the more similar two sequences are, the more likely they have similar behaviors. Whether the ratio of genes with similar behaviors is positively correlated with gene similarity is essential to our project. So we obtained 1.6 million sets of data by pairwise alignment of all the 1748 units in the GRN of K-12. Each set of data consists of gene similarity and behavior similarity. The result is analyzed and plotted in the figure. The linear fit shows that the ratio is positively correlated with the similarity.

Although there are examples that a slight change in DNA sequence will significantly change the property of the gene, for example, sickle-cell disease, the influence is usually determined by the location and scale of the mutation. So the result is still convincing to some degree.

Normally, the similarity of two sequences will not be zero. Some computational experiments were carried out to study the random sequence similarities. We randomly chose a gene in the network and generated 1000 random sequences. The alignment result indicates that the random sequence similarities are Gauss distributed. The result suggests that some similarities are out of statistic significance.

We need the genes highly similar to the exogenous one to interact with it. The program will align the exogenous gene(query) with genes in the network(subject) and get the original similarities. In order to filter meaningless low values, a certain amount of random sequences are generated for each query-subject alignment. Normally, 100 is sufficient. Because the sequence length will influence alignment result, random sequences are fixed at the same length as the query one. Then align random sequences with the subject sequence. The statistic result of these random similarities is used as a threshold.

In the formula, μ is the average random similarity. σ is the standard deviation. x is used to control the filter determined by machine learning. If the original similarity is lower than the threshold, it is abandoned. It is usually means the original value is usually short of statistical significance.

An example about filtring and consistency is presented in “Example”.

If there is a three-unit network and they interact with each other as it is shown in the figure. The regulation is described by the GRN matrix.

If D is the exogenous unit, we can obtain three similarity data sets of D with the units in the original GRN:

Promoter sequence similarity

Gene sequence similarity

Amino acid sequence similarity.

The construction is equivalent to add a new column and a row into the original matrix.

When filling the column, D is compared with the regulators of the unit in each row. The regulations in the row are consider separately and marked as “positive group” and “negative group”. The average similarity of each group represents the distance between the exogenous unit and the group. D is supposed to have the larger one's regulatory direction(positive or negative). The regulatory intensity is the weight average regulation of the chose group. The weight here is the amino acid sequence similarity.

There are two conditions when fill the new row:
1. There are units having the same promoter as the exogenous unit.
2. There is no units having the same promoter as the exogenous unit.

In condition 1, the units sharing the same promoter with the new member are picked out, and the following steps are the same as the construction of the column. The difference is the similarity used here is the gene sequence similarity. As explained in the regulation model part, the promoter is the main regulatory region, but the following sequence is also considered. Now the promoter is the same, so what we focus on are the gene sequences.

In condition 2, the process is almost the same as constructing the new column. Promoter similarity is used because it is the main region.

Figure 8. Construct New GRN

3 Clustering

Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in other groups (clusters). It is a main task of exploratory data mining, and a common technique for statistical data analysis, used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics.

For get a better regulation, we use online database DAVID to cluster all the genes in our whole GRN. Avoid of supersoftless, we hope to create an online communication with DAVID. After getting the cluster of our genes, we multiply the genes simalarity with a factor if they are in the same cluster.

Though the source code of this part has already done, we lack the experiment information to set a propriate factor. All source code were pushed up to our github.

Network analysis includes finding stable condition of network, adding new gene, finding new stable condition and changes from original condition to new condition. We use densities of materials to describe network condition. If all material densities are time-invariant, we can say the network condition is stable.

Regulation relationship in genetic network includes positive regulation, negative regulation, positive-or-negative regulation and no regulation. We store regulation relationship in matrix R. Rji means the unit in line j and row i. For the material of original network, Rji=1 means material i enhance material j, Rji=-1 means material i repress material j, Rji=0 means material i has no influence on material j, Rji=2 means material i enhance or repress material j. For the new material, Rji ranges from -1 to 1. Rji<0 means the possibility of positive regulation is Rji; Rji>0 means the possibility of negative regulation is –Rji; Rji=0 means there is no regulation from i to j. We use Hill equations to describe intensity of regulation. Equations are like following:



The left side of the equation is the derivative x(density) on t(time).”qi”,”pi”,”ri”,”mi”,”ni” are parameters, which determine the intensity of regulation."ri" is degradation rate. Mji is exponent. M is a matrix whose dimensions are equivalent to R's. Mji is 0 or ranges from 0.5 to 1.2 or ranges from -1.2 to 0.5. For the material of original network, if Rji=1,Mji ranges from 0.5 to 1.2;if Rji=-1, Mji ranges from -1.2 to -0.5; if Rji=2;Mji ranges from -1.2 to 0.5 or 0.5 to 1. These Mjis' absolute values are given randomly by program. If Rji=0, Mji=0.
For the new material,

Stable condition is the condition in which densities are time-invariant. We store material densities in a vector and solve the differential equations with Euler's formula, which is like below



We know the network will be stable at last, so every material density has a limitation.

Record the original stable condition, set new material density to 0 and this is the new initial density vector. Solve new equations and record density vectors before the new condition is stable and store these data in a text file.

To evaluate the new network, we introduce the grading system.



"xi" and "Xi" are densities of material i, which is not the new material."ny" is the number of materials. The more new densities are close to the original, the less the influence the cell endues. In general, cells close to the original cell are more likely to survive than those who are far different from the original cell. That is the thought of the grading system.

We did a lot of running and found that the “AbsValue” ranges from 0 to 370, so "ScoreA" ranges from 0 to 4.9.We get the integer part and store it in an array, which has five sections. Generate 100 or 200 matrix M from matrix R and run the original and new network for each M, so we can get 100 or 200 of "ScoreA"s. The section which has maximum "ScoreA"s is the eventual score.

Before reverse analysis, we use the same idea about constructing a new GRN. So we create a virtual gene which replace the gene what users want to get. In calculation, it means that we add a row and a column to the matrix of GRN.

Before prediction, the expression of specific genes which the experimenter needs should be input into our software as well as the improvement or depression. The number of target gene is SIX at most.

It is a must that figuring out the strongest and weakest expression strength before inputting the extreme cases into the target expression. The way to find out the strongest and weakest expression is modeling the GRN's steady state by a large amount of random regulation from -1 and 1. We ran it for 1000 times to get the range of gene expression. On the other hand, the expression of genes unpicked by the users should be stable as much as possible. The initial strength of expression is calculated by modeling the original GRN with Hill's equation.

For getting the best regulation, our software uses PSO algorithm based on 30 particles to simulate the GRN's changing. First of all, the interactions of regulator and regulated-by have been put into those particles in random so that each particle will have the whole set of regulation. Secondly, the variance between target expressions and stable expression of new GRN have been regarded as the optimize requirements in PSO algorithm. As a result, the minimal variance of 30 particles is the global optimum and the minimal variance of the procession in one particle is the local optimum. Then, taking a step towards global and local optimum as well as considering the inertia and perturbation avoids falling into the sub-optimal condition.

At last, when the variance of expression reaches an acceptable range, our software picks out and saves the best global optimum particle following by the movement of those particles stop.

We constantly revises the factors in PSO algorithm by machine learning method for accurate simulation with a fast PSO particle-motion equation. At the same time, our software also filter the result of regulatory value which is more intuitive.

To improve the efficiency of choosing a suitable gene after getting a series of regulatory value, our software picks out some obvious regulation. The value of regulation is between -1 to 1 in which -1 means negative effect and 1 means positive effect. As a result, what our software has done is filtering out the absolute value which is lower than 0.9. Because the difference of regulatory intensity lower than 0.1 has very little effect to the stable expression, the final result of regulation is indicated by Boolean value.

The format of regulatory prediction in “Result”:
Gene_name->Gene_name regulation(+/-)