Constraint Based Flux Analysis is a mathematical way of representing biological system information and allows for easy manipulation of this data. It is based on a stoichiometric matrix of reactions which correspond to individual enzymatic or transport reactions which have been characterized inside of the organism. In other words, it computationally represents each reaction using a linear array of numbers.

The flux can be defined as the amount of substrate moving to product for each individual reaction. The model assumes that the system is at steady state therefore, the overall flux is zero since each product becomes a substrate of another reaction. All substrates entering the system will have the same amount leaving the system. The products leaving the system can be removed by changing the boundary condition of the compound (that is making it unavailable to the system) or by using it to produce growth of the organism. A master growth equation determines which products are required for the cell to grow and this represented in units of DW/unit time. Systems Biology Markup Language (SBML) and the Cobra Toolbox (both produced from System’s Biology Research Group) allows for flux analysis to be performed in the MATLAB program.

We can perform single gene knockouts in the E. coli metabolic gene model and determine the resulting growth rate. Knockouts that result in a growth rate of approximately zero are put in a list for the researcher to study later. Deletion of any one of these genes will result in no growth. These genes are clearly essential.

FindEssentialGenes.m

The Cobra Toolbox makes this single gene deletion analysis for every gene very easy. We found 187 of the model's genes to be essential with this code.

After enriching the in silico media with various substances, the code was run again, and a list of 112 genes resulted. These results were analyzed, and another iteration of media enriching occurred, resulting in a list of 98 genes. Clearly, enriching the media relieves some genes of their function, perhaps by allowing the bypassing of a reaction catalyzed by a protein coded for by that gene. Comparing this list of 98 genes to our previously complied list of researched literature essential genes, we found many differences which we found to be very helpful. Many of these missing genes were added to our master list of essential genes.

Obviously, since the code only analyses single gene deletions, the E. coli cell would not survive with just these 98 genes (these 98 genes are not the minimal genome). Multiple gene deletions were not taken into account. For instance, consider 2 genes that at first appear inessential. When one is knocked out, the cell lives. When the other is knocked out, the cell lives. But lets say that once both in the pair of genes is deleted, the cell dies. The single gene deletion analysis would not detect this.

Another code was written to find out how many pair of genes are essentially entangled in this way. We found that there are 81 of these pairs in the model (assuming glucose minimal media). To validate this model's claim of paired essentiality, we performed single and double gene knockouts in the lab for these essential pairs. See the Model Validation section for details.

It was our goal to find the model's minimal genome. It is clear that single gene knockout analysis would not suffice. Since performing every possible combination of multiple gene knockouts in the model would take most computers many years to accomplish, a different strategy was taken.

FindMinimalGenomes.m

After running this code for the glucose minimal media model, we found minimal genomes varying in size that averaged around 300 genes (which is around 24% of the original size) with a 73% decrease from its original growth rate. After making many growth media changes (see the MediaChange code near the bottom of the page), the minimal genomes that we found had an average size of around 190 (~15% of the model's original genome size). That's over 1050 gene knockouts out of 1260 metabolic genes with the cell still surviving! The smallest minimal genome was found to be made up of 178 genes.

Algorithm details for these results:

The initial growth rate threshold was set to 95% of the initial growth rate. This threshold was decreased by 17% every iteration (deletion cycle) until the final growth rate that was 27.15% of the initial growth rate. 76 minimal genomes were found. The smallest was 282 genes in size and and the biggest was 312. (Every list differed from every other list by an average of around 50 genes - even lists of the same size.) After media changes, this process was repeated again. The initial growth rate threshold was set to 91.2% of the initial growth rate. This threshold decreased by about 15.1% for every deletion cycle until the final growth rate was 0.57% of the initial growth rate. The smallest genome found after this was executed was 178 genes in size. The largest was 214.

Our two most important codes are:

FindMinimalGenomes.m

FindEssentialGenes.m

Start.m - this will set the solver to glpk (a way of solving the vast amounts of matrix equations involved with flux balance analysis), and load the model that the researcher will be working with.

FindEssentialGenes.m - Uses single gene deletion analysis to find a list of essential genes. A deletion of just one of these genes will cause cell death.

CheckForRepeats.m - Checks for repeats in a gene list (and can also remove the repeats if wanted).

InvertList.m - "Inverts" the input List1 with respect to the input List2. The output is a list of genes in List2 that did not appear in List1. It can be used if one wants to find the inessential genes that need to be deleted from a model given an essential gene list (either from a database or compiled from many literature sources). This helps the researcher analyze the effects of deleting genes that were previously thought to be inessential.

FindMatches.m - Finds the matches between two lists of genes. Can be used to compare the researched literature essential genes with the model's predicted essential genes. It outputs the number of matches and the list of matches.

MediaChange.m We reduced the amount of genes that were essential by changing the in silico media. In the end, we added to the glucose minimal media:

• 20 Standard Amino Acids
• Adenosine, Guanosine, Uridine, Thymidine, Cytidine
• Thiamine
• Pantothenic Acid
• Xylose
• Shikimate
• NMN
• Meso Diaminopimelic Acid
• Biotin

FindMinimalGenomes.m - This code outputs an array of minimal genome lists, the genes that were deleted for every list, and the resulting models with the genes deleted. Input arguments include the number of minimal genomes to be found, the final growth rate wanted, and two other factors relating to how the algorithm is carried out which will affect the time it takes, and how small the minimal genomes will be.

MaxMinAvg.m - This is a simple code which lets the user know the largest minimal genome, the smallest minimal genome, and the average size of the minimal genomes created in the FindMinimalGenomes.m code. It can be used on other gene lists (for instance, on an array of gene lists that represent what genes are missing from the researched literature list with respect to each minimal genome found).

InvertListArray.m - "Inverts" input2 with respect to every gene list in input1. input1 may include any number of gene lists. This code is typically used when comparing the researched literature gene list with each of the model's minimal genome lists (produced by FindMinimalGenomes).

FindMatchesArray.m - Typically only used with the array of minimal genomes and the researched literature essential genes. Outputs a list of matches for every minimal genome creating the output MatchListArray. The code also outputs the number of matches for each list and the position of the smallest difference between the model's minimal genome and the designed minimal genome (designed from research/literature). This is very helpful when trying to find what genes are missing in the researched literature essential genes list when compared to many minimal genomes produced by our code.

CellToMatrix.m - Code that makes it more convenient for you to copy and paste the cell array data into Excel.