Results obtained from application of the three computational algorithms to two-member models T1-T5 will be separately discussed based on the type of microbial interactions. For each interaction, the effect of conditions (see Table 1) on community behavior will be investigated. Furthermore, the results obtained by application of each algorithm will be compared with those of the other two algorithms under the same conditions. A complete set of results for runs in Table 1 is provided in supplementary material.
3.1. Commensal Community (T1)
Commensalism, a beneficiary interaction for one member in community, can be found in various natural microbial ecosystems such as biogas-producing microbial communities or the gut microbiome. Toy model T1 consisting of members m1 and m2 with specialized and limited substrates A and E, respectively, has been designed as a community that exclusively features the commensal interaction with growth of m2 relying on m1 through the shared metabolite C. This commensal community was analyzed using FBA, cFBA and SteadyCom under six different conditions (runs T1-1 to T1-6 in Table 1). Comparisons of the results obtained via application of these algorithms showed identical results for each run as given in Table 2. In all runs of commensal community, the cFBA results indicate that there is a unique optimal solution, and for other xm1 values, the system becomes infeasible.
Table 2. Identical results of commensal community obtained by FBA, cFBA and SteadyCom algorithms
However, alterations were observed in the results of different runs due to the dissimilar conditions. In case of T1-1 with equal fluxes of A and E and equal biomass yields of 1, results show equal fluxes of F and G, community growth rate of 2 h-1 and equal mass fraction of members (0.5). Two fold increase in uptake of A in T1-2 (compared to T1-1) resulted in higher flux of F alongside constant flux of G due to the unchanged flux of E, triggering higher community growth rate (3 h-1). To counterbalance the higher growth rate at constant flux of G, mass fraction of m2 lessened and this in turn increased the mass fraction of m1 (0.67). Since the increased production flux of C by m1 in case T1-2 could not be taken up by m2, C was exchanged with exterior as compared to T1-1 showing no exchange of C. This shows that higher supply of a dedicated substrate to a member with other conditions remaining constant will result in higher community growth rate and increased mass fraction of that member to comply with the equality of members’ growth rates.
Cases T1-3,4 consider commensal communities with one member having higher biomass yield on its dedicated substrate than the other member while other conditions remained the same as case T1-1. Of these two cases, explanations are only given for case T1-4 where yield of biomass formation for m2 on E was increased by two fold by setting the coefficient of G equal to 2 compared to case T1-1. This resulted in a higher flux of G with no change in flux of F and hence a higher community growth rate of 3 h-1. To satisfy the equality of growth rates, mass fraction of m2 was enlarged while that of F was reduced. One can therefore conclude that increasing biomass yield of one member results in the enlargement of its mass fraction provided that other conditions are unchanged. This also applies to the case T1-3 where biomass yield of m1 was doubled and its mass fraction was increased.
Case T1-5 was a case similar to T1-3 with respect to biomass yield and other conditions apart from uptake flux of A by m1 was doubled. Figure 5 depicts the flux distribution of T1-5 where fluxes of F and C were doubled while flux of G remained unchanged due to unaltered flux of E, all compared to fluxes in T1-3. To comply with the higher community growth rate of 5 h-1 as a result of higher overall production of F and G, mass fraction of m2 decreased and this resulted in an increase in mass fraction of m1. Since consumption of C by m2 was constrained by the limited availability of E, a part of produced C by m1was released to exterior.
The effect of variation of uptake rate of A at constant biomass yields on flux distribution can be analyzed by comparison of the results of cases T1-3 and T1-5. It can be seen that doubling the flux of A in T1-5 compared to T1-3 resulted in increased mass fraction of m1 alongside a higher community growth rate. This also applies to the cases of T1-1 and T1-2 as well as cases T1-4 and T1-6 with equal values of biomass yields and doubled supply of A. As can be seen in Table 2, mass fraction of m1 and community growth rates both increased.
Altogether in commensal community, different biomass yields of members and different supply of substrates significantly affected the flux distribution, member abundances and community growth rate. However, the performance of the three algorithms used for flux analysis were reasonable and the same results were obtained for each case.
3.2. Competitive Community (T2)
Competition is a prevalent interaction in various communities, particularly in industrial applications of microbial consortia, such as wastewater treatment. Competition for limited substrate is a challenging interaction for metabolic analysis and algorithms have been developed to confront the existing difficulties. Model T2 (see Figure 4) represents a community with competition as the sole interaction in which growth of both members rely on the limited substrate A.
Community growth rates obtained by cFBA algorithm at varying mass fraction of m1 for case T2-1 plotted in Figure 6A show the same community growth rate of 1 h-1 across all abundances. In other words, the maximum growth rate of competitive community with equal biomass yields can be achieved at all mass fractions. This is a case where the corresponding linear programming problem show multiple optimal solutions resulting in the same optimized value of the objective function despite showing different solutions (fluxes and mass fractions). By application of FBA and SteadyCom algorithms, identical fluxes as given in Figure 6D were obtained. Actually, this solution was one of the multiple solutions obtained via cFBA in which substrate A was completely consumed by m1, leading to zero production rate of G (biomass of m2).
This at first may seem contradictory with equal growth rates of members (μ = μ1 = μ2) assumed in SteadyCom. However, it can be explained by the fact that the variables in SteadyCom are aggregate fluxes. Considering Vbiomass, m2 = μ xm2 as aggregate biomass formation flux of m2, one can perceive that zero value of Vbiomass, m2 could be as a result of zero xm2, not zero value of μ2 and hence μ1 = μ2 can be satisfied even in this special solution. In fact, this is only one of the multiple solutions of case T2-1 leading to maximal community growth rate. Actually,
SteadyCom terminates by finding the first optimal solution and the reasonable optimal answer might sometimes lay in the middle of the range of multiple optimal solutions. It should however be mentioned that the SteadyCom in COBRA toolbox of Matlab has provided an option called "BMcon" allowing preset values of members biomass which can be used to obtain a more realistic solution.
Flux distribution of case T2-2(3) with members having unequal biomass yields given in Figure 6E(F) show that all three algorithms promoted only the growth of member with the higher biomass yield (m1 for T2-2 and m2 for T2-3), leading to the complete consumption of substrate A by this member. Figure 6B(C) presenting the cFBA results for T2-2(3), shows the monotonic escalation of community growth rate with mass fraction of m1(m2) until reaching its highest value at xm1 = 1(xm2=1) beating one member. Similar to the previous case, more realistic suboptimal solutions can be found by using "BMcon" command at fixed biomass values.
The occurrence of zero abundance in SteadyCom had also been previously reported by some researchers [25, 27]. In simulation of gut microbiota using a 28-species community by SteadyCom, results indicated that only five/six species had non-zero abundances in different diet scenarios while other members were omitted [25] The members with low biomass yields exhibited zero abundances, similar to our cases T2-2,3. Mutualism and commensalism were reported as the dominant interactions of remaining members while the interactions of omitted species were not specified. According to our results, competition in their case might be a contributing factor in zero abundances of most members. In another study on chronic wound microbiota, a 12-species community was simulated by SteadyCom and five members with zero abundances were reported [27]. However, the specific type of interactions for omitted members were not reported.
Competitive communities with equal biomass yields are susceptible to showing multiple optimal solutions which can only be visualized by plotting the growth rate vs. mass fraction (such as Figure 6D) using cFBA. SteadyCom and FBA algorithms terminate by finding the first feasible solution and may sometimes produce less reasonable answers, such as the elimination of a member. As in natural competitive communities, it is expected that all members would coexist even with varying biomass levels. Incorporation of experimental data/general knowledge of member abundances into simulations can help refine the results and enhance their biological relevance. To achieve a more realistic solution in SteadyCom, the "BMcon" option can be utilized.
When the biomass yields of members are unequal, all three algorithms generate a result where only one member would remain. This outcome aligns with the objective function of maximizing growth, which results in the member with a larger biomass yield monopolizing the substrate. Since maximizing growth might not accurately represent the goals of competitive communities in reality, a suboptimal solution could offer a more realistic outcome or alternatively experimental data on member abundances may be incorporated to allow better mirror real-world results.
3.3. Mutualistic Community (T3)
Mutualistic interaction refers to a symbiotic relationship between two members relying on each other for growth. This interaction often coexists with other types of interactions in microbial communities. The toy model T3, depicted in Figure 4, exemplifies the mutualism interaction facilitated through the exchange of metabolites C and D as the sole form of interaction within this toy model while A or E functions as specialized substrate for each member. The interdependence of these members resulted in distinct outcomes. Five different conditions were examined in cases of T3-1 to T3-5 to consider the effect of variations on biomass yields and substrates uptake rates. Coexistence of the two members in this model is expected since the shared metabolites D and C are required for growth of m1 and m2, respectively, in addition to the dedicated substrates A and E.
FBA results are firstly described and discussed for the examined cases as these were inconsistent with the results obtained via cFBA and SteadyCom. In all cases, FBA showed the growth of only one member (m1 in most cases and m2 in T3-3) while the other member showed no growth. This means that the growing member obtains the required shared metabolites (C or D) by the non-growing member (see Figures S11-S15 of supplementary material). Actually, the member showing no growth consumed its dedicated substrate to solely provide the metabolite required for the other member’s growth. Consequently, the community biomass was solely derived from the biomass reaction of the growing member. Community growth rate of 1 1/h was obtained for case T3-1 with biomass yields of 1, while this was 2 1/h for all other runs in which biomass yield of one member was 2. In case T3-4 (T3-5) where extra substrate A (E) were taken up, release of shared metabolite C (D) by community to exterior environment was observed. Although donation of the required metabolites by a non-growing member in a microbial community to other members has previously been reported when using FBA algorithm, the type of interaction leading to this situation was not specified [18].
Results obtained by cFBA and SteadyCom algorithms for mutualistic community under different conditions are illustrated in Table 3. Where more reasonable results for this type of interaction were predicted by these two algorithms.
In run T3-1 with equal biomass yields and equal dedicated substrate availabilities, both cFBA and SteadyCom algorithms demonstrated non-zero abundances for the two members, though one was much lower. Similar to previously discussed cases, cFBA allowed visualization of the existing multiple optimal solutions at xm1 (mass fraction of m1 in community) ranging from 0.001 up to 0.999 (see Figure 7A) and outside this range the system becomes infeasible, while SteadyCom gave only one optimal solution at xm1 = 0.001.
Table 3. Comparison of the results of mutualistic community (T3) predicted by cFBA and SteadyCom algorithms under different conditions
For cases T3-2,3,4,5 with unequal biomass yields, identical unique solutions were obtained by cFBA and SteadyCom, while FBA predicted different results. Community growth rate obtained by FBA was slightly higher compared to other algorithms as FBA led to no growth of the member with lower biomass yield while balanced growth assumption of the other algorithms enforced growth of that member, though at a small rate, lowering the community growth rate. This reveals that coexistence of both members in community may not result in a higher community growth rate, but it aligns more closely with reality. Although the same maximized community growth rate of 1.998 h-1 was obtained for cases T3-2,3, abundance of the member with higher biomass yield was higher than that of the other member (see Figure 7B,C). Results of case T3-4 were identical to those of T3-2, apart from C being released by community to exterior at a rate of 1 mmol/h/gCom in case T3-4 due to doubling the uptake of A. Case T3-5 having double uptake rate of E with other conditions being unchanged compared to case T3-2 led to the decreased community growth rate alongside the decreased xm1, the release of D as community product and higher and lower exchange rates of C and D by m2 and m1, respectively.
In summary, FBA was found an inappropriate choice for analyzing the metabolic network of a mutualistic community due to providing the required shared metabolite by the non-growing member. SteadyCom and cFBA were shown to provide reasonable identical results and hence appear to be suitable methods for simulating mutualistic interaction.
Comparing our findings to a study described in the introduction section that reported favorable outcomes using the SteadyCom method [18], reveals interesting insights. The community investigated in that study involved both mutualistic and competitive interactions. Similar to our results in a mutualistic community, they obtained reasonable outcomes when compared to FBA. It is noteworthy that in their study, competition did not revolve around a limited external substrate. Consequently, the reported results in that context were considered plausible.
3.4. Commensal-Competitive Community (T4)
In real cases, a combination of interactions as opposed to single interactions, with the simplest being the dual interaction, could occur which needs consideration. Commensal-competitive community is an important case among the communities with dual interactions where members compete for the same substrate while a metabolite formed by one member is necessary for the growth of another member. Designed community T4 (see Figure 4) represents a simple commensal-competitive community with both members requiring substrate A and metabolite C, produced by m1, being necessary for growth of m2. This community was examined under three different conditions, the results of which are presented in Figure 8.
Figure 8A illustrates the variation of community growth rate with xm1 obtained via cFBA for case T4-1 having equal yields. System was feasible only for a range of xm1 ≥ 0.5 where a unique maximized community growth rate of 1 h-1 was attained, suggesting the existence of multiple optimal solutions. By increasing xm1 within this range, metabolite C was concurrently released as product of m1. The same optimized community growth rate of 1 h-1 was found by SteadyCom and FBA however different values of 1 and 0.5 for xm1 were predicted, respectively. Comparison of these results with cFBA plot (Figure 8A) shows that the results obtained by SteadyCom and FBA for case T4-1 lies at the limits of optimal range of xm1 obtained by cFBA and hence only represent one of the optimal solutions.
In case T4-2 where biomass yield of m1 was doubled, cFBA resulted in feasible solutions for xm1 ≥ 0.667 with an increasing trend of community growth rate with xm1, leading to the highest rate of 2 h-1 by dominance of m1 at xm1=1. Flux distribution predicted by FBA and SteadyCom were as predicted by cFBA which is given in Figure S16. As mentioned in Section 3-2, the balanced growth assumption was satisfied despite observing zero mass fraction of m2. By doubling biomass yield of m2 in run T4-3, all algorithms led to a community growth rate of 1.5 (1/h) with substrate A being equally shared between members as presented in Figure 8B. The same xm1 (0.33) was obtained by SteadyCom and cFBA however the community growth rate was lower compared to case T4-2. This can be explained by the fact that commensalism demands the growth of m1 to provide metabolite C for growth of m2 with higher biomass yield. This in turn diminishes the maximized growth rate of community as the member with lower yield should also grow alongside the other member.
Overall, competitive interaction in commensal-competitive community supports the dominance of independent member provided that it has the higher biomass yield on the shared substrate. On the contrary, when dependent member has higher biomass yield, commensalism forces this member to obtain a shared metabolite produced by the other member and hence both members coexist.
3.5. Mutualistic-Competitive Community (T5)
Among the communities with dual microbial interaction, mutualistic-competitive community is another important type observed in various natural and industrial microbial communities, including yogurt-producing microbiomes. To investigate this type of dual interactions, community T5, as depicted in Figure 4 was designed in which growth of m1 depends on m2 and vice versa in addition to competence of m1 and m2 for the common limited substrate A.
Three cases T5-1,2,3 with equal and unequal biomass yields were considered. Overall, results were similar to those of mutualistic community (T3). In all examined cases, FBA predicted growth of only one member obtaining its essential metabolite by the non-growing member. However, SteadyCom and cFBA both predicted the growth of both members, though at various extents.
cFBA plot of case T5-1 (Figure S18) with equal yields showed the existence of multiple optimal solutions at optimal community growth rate of 0.5 h-1 and xm1 ranging from very low values (0.0005) up to values close to 1 (0.9995). By applying SteadyCom, one optimal solution was obtained at xm1= 0.9995 which lies within the multiple optimal solutions. For cases with unequal yields (T5-2,3), cFBA plots (Figures S19,20) showed increasing and decreasing trends of the community growth rate with xm1 for cases with doubled biomass yields of m1 (case T5-2) and m2 (case T5-3), respectively. Therefore, cFBA predicted a unique optimal solution for case T5-2 (T5-3) at xm1 = 0.9995 (xm1 = 0.0005). The same results were obtained by SteadyCom due to the existence of unique solutions for cases T5-2,3. Compared to results obtained by FBA, optimal community growth rate obtained by SteadyCom and cFBA were slightly lower (0.9995 1/h compared to 1 1/h) due to metabolites C and D being produced by growing member as opposed to non-growing member in FBA.
Overall, the performance of the three algorithms for simulating competition alongside mutualism were almost similar to that of mutualistic community. SteadyCom algorithm showing zero aggregate fluxes for one member under equal yields in community with solely competitive interaction, led to the more realistic coexistence of both members for competition alongside mutualism.