There are 1261 genes in the model of E. coli_iAF1260. If a single gene is associated with multiple reactions, the deletion of that gene will result in the removal of all associated reactions. On the other hand, a reaction that can be catalyzed by multiple non-interacting gene products will not be removed in a single gene deletion. By the aid of the COBRA toolbox 11, we can calculate the impact of their deletion. We define the impact of one gene deletion on the whole metabolic flux redistribution as p

p = ∑ i R ( v ′ i − v i ) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaaNaeyypa0ZaaabCaeaacqGGOaakcuWG2bGDgaqbamaaBaaaleaacqWGPbqAaeqaaOGaeyOeI0IaemODay3aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGPbqAaeaacqWGsbGua0GaeyyeIuoaaaa@3D27@

Where v_i and v′i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmODayNbauaadaWgaaWcbaGaemyAaKgabeaaaaa@2EDD@ are respectively the flux value of i-th reaction of the model of E. coli_iAF1260 before and after a single gene deleting and R is the whole reaction set. In most of the researches on gene deletion 456789101516171819202122, the change of growth rate f is often used to describe the impact of gene deletion. The reason why we define p as the impact of gene deletion is that we believe it is more detailed in describing the change of organism's metabolism. p has considered the flux change taking place at every reaction, and it uses the square sum of the difference between v_i and v′i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmODayNbauaadaWgaaWcbaGaemyAaKgabeaaaaa@2EDD@. Otherwise, f is just a whole measure and it does not distinguish the flux change taking place at every reaction.

Figure 2 shows the deletion impact of these 1261 genes. Table 1 gives p scopes, gene numbers falling within these scopes and their corresponding percentages that these genes take.

If a gene catalyzes a reaction which belongs to a certain subsystem, we say that the gene belongs to the subsystem. Functional subsystems about important genes in the metabolic system of micro-organism are seldom reported. We have hereinabove defined those genes with p>9800 as key genes. We now list the functional subsystems to which every key gene belongs, 23 subsystems in total, and several genes appear in more than one subsystem, shown in Table 2. The 23 functional subsystems are "Threonine and Lysine Metabolism (TLM), Cofactor and Prosthetic Group Biosynthesis (CPGB), Valine Leucine and Isoleucine Metabolism (VLIM), Cell Envelope Biosynthesis (CEB), Lipopolysaccharide Biosynthesis Recycling (LBR), Methionine Metabolism (MM), Arginine and Proline Metabolism (APM), Glycerophospholipid Metabolism (GM), Membrane Lipid Metabolism (MLM), Purine and Pyrimidine Biosynthesis (PPB), Nucleotide Salvage Pathway (NSP), Citric Acid Cycle (CAC), Glycine and Serine Metabolism (GSM), Tyrosine Tryptophan and Phenylalanine Metabolism (TTPM), Transport Inner Membrane (TIM), Alanine and Aspartate Metabolism (AAM), Folate Metabolism (FM), Alternate Carbon Metabolism (ACM), Histidine Metabolism (HM), Cysteine Metabolism (CM), Inorganic Ion Transport and Metabolism (IITM), Oxidative Phosphorylation (OP), Unassigned (U)".

We can find that the distribution to subsystems of these 195 key genes is not even and most of them are related to "Cofactor and Prosthetic Group Biosynthesis", "Cell Envelope Biosynthesis" and "Purine and Pyrimidine Biosynthesis" subsystems, especially CPGB. We can also find that all of the important but not essential (INE) genes, 8 in total, appear in "Oxidative Phosphorylation".

The reason for many high-p genes just belonging to several metabolic subsystems maybe is in that these subsystems involve many reactions and provide supports for other subsystems; The reason for INE genes just belonging to "Oxidative Phosphorylation (OP)" subsystem probably is in that the permissibility which E. coli use other reaction ways to carry out this kind of metabolism, under the given media condition, takes place on OP subsystem.

Figure 4 is the scatter diagram (p, f), total 1261 data pairs. Many data pairs are superposition and locate at the same place, so there aren't lots of points in the figure. From the diagram, we can easily find that the relationship between p and f is of highly linear correlation. High p corresponds to low f.

We compute out the related reaction number d of every gene in those 1261 genes of the E. coli_iAF1260 model, as illustrated in Figure 5. From the figure, we can find that some but not many genes have high d value, but we don't know whether they affect metabolic flux distribution greatly.

Figure 6 is the scatter diagram (d, p), 1261 data pairs in total. Still many data pairs are superposition and locate at the same place. From the diagram, we can easily find that the relationship between d and p is not of linear correlation. So high-d genes and low-d genes are equally important to the metabolism of E. coli_iAF1260.

We define the flux sum controlled by every gene as

v g e n e = ∑ j R g e n e | v j | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aaSbaaSqaaiabdEgaNjabdwgaLjabd6gaUjabdwgaLbqabaGccqGH9aqpdaaeWbqaamaaemaabaGaemODay3aaSbaaSqaaiabdQgaQbqabaaakiaawEa7caGLiWoaaSqaaiabdQgaQbqaaiabdkfasnaaBaaameaacqWGNbWzcqWGLbqzcqWGUbGBcqWGLbqzaeqaaaqdcqGHris5aaaa@44BB@

Where v_j is the flux value of j-th reaction of the model of E. coli_iAF1260 before a single gene deleting and R_gene is the reaction set controlled by the given gene. We can easily compute out the flux sum v_gene of every gene in those 1261 genes of the E. coli_iAF1260 model, as illustrated in Figure 7. From the figure, we can find that some but not many genes have high v_gene value, but will they affect metabolic flux distribution greatly?

Figure 8 is the scatter diagram (v_gene, p), 1261 data pairs in total, and many data pairs are superposition. From the diagram, we can also find that the relationship between v_gene and p is not of linear correlation as well. So genes with high v_gene and genes with low v_gene are equally important to the metabolism of E. coli_iAF1260.

There are 2382 reactions in the in silico model of E. coli_iAF1260. We define Rg to measure the impact that a reaction is gotten in the whole 1261 gene deletion.

R g = ∑ k G | v k − v 0 | MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuai1aaSbaaSqaaiabdEgaNbqabaGccqGH9aqpdaaeWbqaamaaemaabaGaemODay3aaSbaaSqaaiabdUgaRbqabaGccqGHsislcqWG2bGDdaWgaaWcbaGaeGimaadabeaaaOGaay5bSlaawIa7aaWcbaGaem4AaSgabaGaem4raCeaniabggHiLdaaaa@3E4D@

Where v₀ and v_k are respectively the flux value of a certain reaction of the model of E. coli_iAF1260 before and after k-th gene deleting, and G is the set of whole 1261 genes.

Figure 9 provides each Rg of 2382 reactions and Table 3 shows Rg scopes, corresponding reaction number within these scopes and the percentages that these reactions take. In the following section, we will determine what the highly-affected reactions are.

In silico modelling and simulation of genome-scale biological systems are different from that practiced in the physicochemical sciences. A network can fundamentally have many different states or many different solutions. Which states (or solutions) are picked is up to the cell and based on the selection pressure experienced, and such choices can change over time. Therefore, constraint-based approaches 23 to the analysis of complex biological systems have proven to be very useful. The differences between the physicochemical sciences and the physical sciences or engineering are illustrated in Ref. 14. All theory-based considerations (i.e., engineering and physics) lead one to attempt to seek an "exact" solution, and typically computed based on the laws of physics and chemistry. However, constraint-based considerations (as in biology) are useful. Not only can a network have many different behaviors that are picked based on the evolutionary history of the organism, but also these networks can carry out the same function in many different and equivalent ways 14.

Before calculation and simulation, the reconstructed metabolic network must be represented mathematically. The stoichiometric matrix, S, is the centerpiece of a mathematical representation of genome-scale metabolic networks. It represents each reaction as a column and each metabolite as a row, where each numerical element is the corresponding stoichiometric coefficient.

An upper and lower bound for the allowable flux through each reaction also requires defining. This represents the lowest and highest reaction rate possible for each reaction. The set of upper and lower bounds is represented as two separate vectors, each containing as many components as there are columns in S, and in the same order. In many cases, reversible reactions are defined to have an arbitrary large upper bound and an arbitrarily large negative lower bound. Irreversible reactions have a lower bound that is nonnegative, usually zero.

In order to predict meaningful fluxes, setting upper and lower bounds is especially important for exchange reactions which serve to uptake compounds to the cell or secrete compounds from the cell. The lower bound of exchange reaction column must be a finite negative number using this orientation (e.g., glucose). The upper bound of exchange reaction column must be greater than zero. At least one of the reactions in the model must have a constrained lower/upper bound, and typically, the substrate (e.g., glucose or oxygen) uptake rates are set to experimentally measured values. The upper and lower bounds for exchange reactions are quantitative in silico representations of the growth media environment.

The constraint-based approach is based on the assumption that cells strive to maximize their growth rate. This assumption which provides an acceptable starting point for many types of computations is satisfied by simulating maximal production of the molecules required to make new cells (biomass precursor molecules). In spite of their limitations, the predictive power of genome-scale models of metabolic networks has been demonstrated in diverse situations through careful experimentation 11.

The biomass objective function (the function v_growth, see below) is a special reaction taking as substrates of all biomass metabolites, ATP and water and producing ADP, protons, and phosphate (as a result of the non-growth associated ATP maintenance requirement) 6.

The minimal media is determined computationally with the systematic testing of distinct inputs. Different combinations of molecules are allowed to enter the reaction network until the minimal group that allowed biomass production, or non-zero Z (see below), was found 6. It is only concerned that some amount of biomass production is calculated but do not discriminate between extremely slow, inefficient growth and rapid growth.

In genome-scale metabolic networks, the fluxes within a cell usually cannot be uniquely calculated because a range of feasible values exist when fluxes are subjected to known constraints. Flux balance analysis (FBA) is used to find optimal growth phenotypes. Briefly, a large-scale linear programming is used to find a complete set of metabolic fluxes (v) that are consistent with steady-state condition (eq. 4) and reaction rate bounds (eq. 5), and at the same time maximize the biomass objective function in the defined ratio. This corresponds to the following linear programming problem 6:

max Z = v_growth

Subject to

S·v = 0

α_i <v_i <β_i

Where S is the stoichiometric matrix, and α_iand β_i define the bounds through each reaction v_i. The flux range was set arbitrarily high for all internal reactions so that no internal reaction restricted the network, with the exception of irreversible reactions, which have a minimum flux of zero. The inputs to the system were restricted to a minimal media.

The value of Z computed with the above procedure can either be zero (predicting no growth) or greater than zero (corresponding to cellular growth) depending on the inputs and outputs that are allowed, according to the nutrients provided in the media.

The effect of a gene deletion experiment on cellular growth can be simulated in a manner similar to linear optimization of growth 511. Gene-reaction associations model the logical relationship between genes and their corresponding reactions. If a single gene is associated with multiple reactions, the deletion of that gene will result in the removal of all associated reactions, i.e. to simultaneously restrict the fluxes (upper and lower flux bounds) of these reactions to zero prior to computing maximal biomass objective function. On the other hand, a reaction that can be catalyzed by multiple non-interacting gene products will not be removed in a single gene deletion. The possible results from a simulation of a single gene deletion are unchanged maximal growth (non-lethal), reduced maximal growth or no growth (lethal). Those genes were considered essential if no biomass could be produced without their usage.