Quantifying the effects of the division of labor in metabolic pathways - PubMed (original) (raw)
Quantifying the effects of the division of labor in metabolic pathways
Emily Harvey et al. J Theor Biol. 2014.
Abstract
Division of labor is commonly observed in nature. There are several theories that suggest diversification in a microbial community may enhance stability and robustness, decrease concentration of inhibitory intermediates, and increase efficiency. Theoretical studies to date have focused on proving when the stable co-existence of multiple strains occurs, but have not investigated the productivity or biomass production of these systems when compared to a single 'super microbe' which has the same metabolic capacity. In this work we prove that if there is no change in the growth kinetics or yield of the metabolic pathways when the metabolism is specialized into two separate microbes, the biomass (and productivity) of a binary consortia system is always less than that of the equivalent monoculture. Using a specific example of Escherichia coli growing on a glucose substrate, we find that increasing the growth rates or substrate affinities of the pathways is not sufficient to explain the experimentally observed productivity increase in a community. An increase in pathway efficiency (yield) in specialized organisms provides the best explanation of the observed increase in productivity.
Keywords: Chemostat; Cross-feeding; Mathematical modeling; Microbial ecology; Syntrophic consortia.
Copyright © 2014 Elsevier Ltd. All rights reserved.
Figures
Fig. A1
Qualitative sketches of the growth and inhibition curves and positions of equilibria for the case where there are two co-existence equilibrium points. Panel (a) shows the function m(A) showing Amax and the relative positions of ASb and AUb. Panel (b) shows the function I(A) with the relative positions of ASb and AUb. Panel (c) shows function f(G) and the relative positions of GSb and GUb. The position of G_0 defined by f(G) = 1 is shown as well. Panel (d) shows the curves f(G)I(A) = 1 (solid black curve) and m(A) = 1 (dashed black lines). The co-existence equilibrium points projected to (A, G) plane are at the two intersections of these curves. The dotted black line G = 1−_A (together with the solid black curve f (G)I(A) = 1) determines the position of the boundary (_x_2 = 0) equilibrium point projected into the (A, G) plane. The position of _G_0 defined by f(G) = 1 is shown as well.
Fig. A2
Qualitative sketches of the six possible cases based on assumptions [A1]–[A4] in Section 2.3. Equilibrium points are marked with a circle for stable equilibria and a triangle for unstable equilibria. In the shaded gray region (G_>1−_A) any solutions are physically unrealistic as _x_2 <0. The trivial (_x_1 = 0, _x_2 = 0) equilibrium point at (0,1) is shown. For all cases, there is a boundary (_x_2 = 0) equilibrium point (_A_1, G_1) which, when projected into the (A, G) plane, lies at the intersection of the dotted black line G = 1−_A together with the solid black curve f (G)I(A) = 1. The curves f (G)I(A) = 1 (solid black curve) and m(A) = 1 (dashed black lines) are shown, the position of co-existence equilibrium points projected to (A, G) plane is at the intersections of these curves. The vertical asymptote of the curve f (G)I(A) = 1 is labeled Acritb. The position of _G_0 defined by f (G) = 1 is shown as well.
Fig. B1
(a) Qualitative sketch of the curve _G_1 implicitly defined by (45). (b) Qualitative sketch of the curve _G_2 defined by (46).
Fig. B2
Qualitative sketch of the curves _G_1 (dashed black curve) and _G_2 (solid black curve). The position of non-trivial equilibrium points projected to (A, G) plane is at the intersections of these curves. Equilibria are marked with a circle for a stable equilibrium point and a triangle for an unstable saddle. The trivial (_x_=0) equilibrium point at (0,1) is shown. In the shaded gray region (G_>1−_A/(1+γ)) any solutions are physically unrealistic as x<0. The vertical asymptote of the curve f(G)I(A)+m(A) = 1 is labeled Acritm.
Fig. B3
Qualitative sketch of the curves m(A) (solid black curve) and 1 − γ/(1+ γ −A) (dot-dashed black curve). The case with only one intersection A1t is shown in panel (a). In panel (b) the case with three intersections is shown. The A1v and A2v values which are solutions to m(A) = 1/(1+ γ) are also shown.
Fig. 1
Diagram showing the metabolic pathways in (a) the monoculture system with the microbes x and (b) the binary culture system with the primary (glucose) consumer _x_1 and the scavenger (acetate consumer) _x_2, where G is the primary substrate (glucose) and A is the inhibitory intermediate substrate (acetate). The yields _Y_11 and _Y_22 fix the rates of biomass production per gram of substrate consumed, the yield _Y_21 is the ratio of biomass production to byproduct production, and the growth rates are given by the functions _μ_1(G, A) and _μ_2(A).
Fig. 2
Qualitative sketch of the growth and inhibition functions: (a) f(G), (b) I(A), and (c) m(A) in nondimensional variables. The solutions to the equation m(A) = 1, ASb and AUb, are shown in panels (b) and (c).
Fig. 3
Qualitative sketch of the curves m(A) (solid black curve) and _μ_1,maxI(A) (dot-dash black curve) that satisfy the sufficient condition in Theorem 2c ruling out the upper monoculture equilibria. The curve _μ_1,maxI(A) lies in the shaded region defined by the value m(A2v), where A2v is the upper solution of m(A) = 1/(1 + γ).
Fig. 4
The functions f(G), I(A), and m(A) (20, 21, and (22), respectively), in the original (dimensional) variables, for the chosen parameter values.
Fig. 5
The curves _G_1 and _G_2, which determine the position of the monoculture equilibrium point(s) in the nondimensionalized system are shown in black solid lines, with the intersection labeled (Am, Gm). The curves f(G)I(A) = 1 and m(A) = 1, which determine the position of the binary culture equilibrium point(s) are shown in gray dot-dash lines, with the intersection labeled (ASb,GSb). The lines of constant biomass for the monoculture and binary culture are black dashed and dotted lines, respectively.
Fig. 6
Panel (a) shows the position of the monoculture equilibrium point in the (A, G)-plane as β varies as a solid black line. The coexistence equilibrium point of the binary culture (ASb,GSb) is shown as a filled black circle. The dotted line shows the line of constant biomass for the binary culture at the coexistence equilibrium point, equilibrium points to the left of the curve (the shaded area) have a higher biomass than the binary culture. Important β values are labeled in the figure. Saddle-node bifurcations are labeled _SN_1,2,3, but only _SN_2 is within the range of this figure. Panel (b) shows the biomass (x value) of the monoculture equilibrium point as β varies which is shown as a solid black line; thick where it is stable, thin where it is unstable. The dashed line shows the biomass of the comparative binary culture system, x1,Sb+x2,Sb, and the shaded region represents the β values where the monoculture has higher biomass than the binary culture system.
References
- Abrams P. The functional responses of adaptive consumers of two resources. Theor Popul Biol. 1987;32:262–288.
- Aota Y, Nakajima H. Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton. Ecol Res. 2001;16:289–299.
- Brown TDK, Jones-Mortimer MC, Kornberg HL. The enzymic interconversion of acetate and acetyl-coenzyme A in Escherichia coli. J Gen Microbiol. 1977;102:327–336. - PubMed
- Burchard A. Substrate degradation by a mutualistic association of two species in the chemostat. J Math Biol. 1994;32:465–489.
MeSH terms
Substances
LinkOut - more resources
Full Text Sources
Other Literature Sources