Quantifying the effects of the division of labor in metabolic pathways

Transcription

1 Quantifying the effects of the division of laor in metaolic pathways Emily Harvey,a,,, Jeffrey Heys,Tomá s Gedeon a a Mathematical Sciences, Montana State University, Bozeman, Montana 5975 Chemical and Biological Engineering, Montana State University, Bozeman, Montana 5975 stract Division of laor is commonly oserved in nature. There are several theories that suggest diversification in a microial community may enhance staility and roustness, decrease concentration of inhiitory intermediates, and increase efficiency. Theoretical studies to date have focused on proving when the stale co-existence of multiple strains occurs, ut have not investigated the productivity or iomass production of these systems when compared to a single super microe which has the same metaolic capacity. In this work we prove that if there is no change in the growth kinetics or yield of the metaolic pathways when the metaolism is specialised into two separate microes, the iomass (and productivity) of a inary consortia system is always less than that of the equivalent monoculture. Using a specific example of Escherichia coli growing on a glucose sustrate, we find that increasing the growth rates or sustrate affinities of the pathways is not sufficient to explain the experimentally oserved productivity increase in a community. n increase in pathway efficiency (yield) in specialised organisms provides the est explanation of the oserved increase in productivity. Key words: microial ecology, mathematical modelling, syntrophic consortia, cross-feeding, chemostat. Introduction From the earliest oservations of microial organisms, it has een apparent that microial consortia are uiquitous in nature. In fact, naturally occurring ecosystems are almost exclusively organized as consortia. Recent metagenomic studies from the soil [9], to the ocean [66], to the human gut [22], have found Corresponding author. Telephone: extn addresses: e.p.harvey@massey.ac.nz (Emily Harvey), jeffrey.heys@coe.montana.edu (Jeffrey Heys), gedeon@math.montana.edu (Tomá s Gedeon) Present address: Institute of Natural and Mathematical Sciences, Massey University lany, uckland, New Zealand. Preprint sumitted to Elsevier July, 204

2 that microial communities are incredily diverse, often consisting of thousands of interacting species. Susets of these communities form consortia that act together to enhance their capailities and survival [4]. Early theoretical ecology studies led to the development of the competitive exclusion principle (CEP), which states that the maximum numer of species that can coexist in a system is equal to the total numer of limiting (essential) resources [26, 37, 49]. However, in nature we see many examples of multiple microial species staly coexisting. Frequent explanations for coexistence in a natural population can e categorized into three main types: the development of multiple niches due to spatial heterogeneity or self organized segregation, the system not eing in equilirium due to environmental fluctuations or external forcing, and the presence of inter- and intra-species interactions. Despite the limited direct applicaility of CEP to natural systems, the clear mathematical formulation of CEP allows for significant insight y identifying which conditions of the CEP have een violated to lead to the oserved coexistence of species. For example, y applying the CEP to the dynamics in a chemostat, we see that due to the consistent flow of nutrients and continuous mixing, the environment is kept constant and the development of different niches due to spatial heterogeneity is not possile; therefore, it must e some form of inter- or intra-species interactions such as crowding, chemical signaling, cooperativity, or mutual inhiition that lead to any oserved coexistence. Studying these interactions in simple chemostat systems gives us a etter understanding of the factors that maintain the diversity in naturally occurring microial consortia. Natural consortia are often found to form syntrophic systems, where the microes depend on each other for survival, either y the production of required metaolic sustrates or y the maintenance of chemically advantageous conditions [54]. It is often oserved that this syntrophic cooperation within microial consortia increases their productivity and can allow the consortia to perform advanced functions that the microial species are not capale of individually. These microial interactions are known to e important in diverse areas including chronic medical infections (e.g. diaetic ulcers [2, 29]), iofuel synthesis (e.g. iodiesel production [44, 70]), environmental nutrient cycling (e.g. CO 2 sequestering, nitrification [8]), ioprocessing [60], and wastewater treatment [53, 57, 58]. frequently oserved syntrophic system is a cross-feeding chain where microes work together to perform the sequential degradation of complex compounds like lingocellulosic material [54]. In these syntrophic cross-feeding systems a single sustrate must e roken down in many steps, with one species cataolizing the availale sustrate and oxidizing it to produce a yproduct that the next species in the chain can consume. The intermediate yproducts in these systems are often found to e inhiitory. In this work we will consider the case where the intermediate yproduct inhiites growth. However, this more complicated system can e reduced to the case where there is no inhiition y taking the appropriate limit. It has een oserved experimentally that these syntrophic chain systems where the metaolic pathways are split among separate organisms (known as 2

3 microial specialisation ora division of laor ) are more productive than a single organism with the equivalent metaolic capailities [3, 67]. For example, if we compare a single organism that metaolizes B C to a pair of organisms that metaolize B (organism ) and B C (organism 2), experimental oservation has found the pair of organisms to e more productive than the single organism. Productivity is defined here as total iomass production per unit of input. n example of this division of laor that has een found to evolve repeatedly in different experiments occurs when E. coli is grown on a glucose sustrate. The original population of E. coli can fully metaolize glucose (glucose acetate CO 2 (TC cycle)), ut when grown on glucose for many generations the population splits into two main supopulations: microes that preferentially consume glucose and produce acetate (glucose acetate) and microes that preferentially consume acetate (acetate CO 2 ) [50, 5, 64]. In this paper we use standard chemostat modeling techniques to investigate whether the division of laor (splitting the pathways into two separate organisms) alone is sufficient to explain the oserved increase in iomass, and if not, what other changes may e required. It is important to note that in contrast to the past work which has investigated the evolution of such cross-feeding systems, we will not consider the two systems in direct competition, instead comparing the maximum iomass (productivity) of the systems in isolation. This is motivated y industrial applications where the total productivity of the consortia, which is usually proportional to the iomass, is of primary interest. This type of system has een studied mathematically, and it can e shown that for n species in a simple syntrophic cross-feeding chain, there is a stale coexistence steady state [33, 46, 47, 48]. This simple system has een modified to include other forms of inhiition, external toxins, multiple sustrates, and other forms of mutualism; in all cases, a stale, stationary, coexistence steady state is found[2,5,5,3,52].previousresearch has focused on proving the existence and staility of coexistence steady-states. There has een no investigation of the productivity of these syntrophic chain systems. Some recent work [0, 6, 45] investigated the evolution of cross-feeding in microial populations and found that there are a wide range of parameter values for which cross-feeding is seen to evolve. The aim of these evolution studies was to identify conditions for stale coexisting syntrophic chain systems to evolve and outcompete equivalent monocultures. They did not explicitely investigate the productivity of the systems that are found to evolve. In our initial model, we assume that the metaolic dynamics of the pathways do not change when eing split into a separate microes. In addition, we initially assume that the growth rate for the monoculture is a linear comination of the growth rates of the two pathways. This formulation allows us to otain theoretical results and is effectively an upper ound on the growth rate of the monoculture microe. Realistically, there are costs to utilizing oth pathways at once for the monoculture, and there are changes that are known to occur to the metaolic pathway dynamics when the microes specialise to a single sustrate. For example, Pfeiffer and Bonhoeffer [45] theorize that the evolution 3

4 of cross-feeding could e due to the system minimizing the concentration of inhiitory intermediates and minimizing the concentration of enzymes it must produce, while maximising the rate of TP-production. nother explanation comes from Johnsonet al. [30] who find, y considering the iochemical conflicts which constrain the relationships etween two metaolic processes, that the division of metaolic pathways could e advantageous as it allows microes to focus on producing a smaller numer of enzymes and optimizing a smaller suset of pathways. By starting with the assumption of no adaptation for the specialists and the maximal growth rate for the monoculture, we get a strong theoretical result. Then, y varying the growth kinetics (growth rates and sustrate affinities) and yields etween the monoculture and the inary culture systems, we are ale to use our model to test possile explanations for the oserved increase in productivity and to quantitatively investigate what changes are required for the division of laor to e advantageous (increased iomass). Considering the specific example of E. coli grown on a glucose sustrate [50, 5, 64], we use standard Monod kinetics and measured experimental parameters to determine the conditions under which the model results match the experimentally oserved increase in iomass. The main result of this work is a comparison of the iomass production of a single microe with full metaolic capacity to a syntrophic consortium of two specialized microes each with a unique suset of the full metaolic chain, where the intermediate yproduct may e inhiitory. We prove that the monoculture system will always have higher iomass production (i.e., higher productivity) if there are no changes to the growth kinetics or yield of the pathways etween the two systems. In a specific example, we show that increasing the growth rates or sustrate affinities of the consortial pathways, a change that might e expected due to specialisation, is not sufficient to generate the oserved higher iomass production in the inary consortia. However, y varying the yields for the consortial pathways, higher iomass production can e achieved for the inary consortia relative to the monoculture. Yield changes are an expected result of microial specialisation ecause utilizing a smaller numer of specialized pathways leads to reduced energy requirements. This result provides evidence for the argument that an increase in efficiency (yields) is a key factor underlying the oserved productivity enefit of microial consortia. dditionally, the finding that changing yields can lead to higher iomass production through specialisation has important implications for industrial settings where higher iomass production is often highly desirale. 2. Model Construction To investigate the effects of separating metaolic pathways, we will analyse the simplest case of a syntrophic chain where there are only two microial species. For clarity we will use a specific example in developing our notation, in this case the metaolism of glucose y E. coli. To construct the model we assume that the metaolism of a single provided sustrate (glucose, G) 4

5 G Y x μ (G,) μ 2() Y Y 2 22 B G Y μ (G,) x x 2 μ () 2 Y Y 2 22 Figure : Diagram showing the metaolic pathways in (a) the monoculture system with the microes x and () the inary culture system with the primary (glucose) consumer x and the scavenger (acetate consumer) x 2,whereG is the primary sustrate (glucose) and is the inhiitory intermediate sustrate (acetate). The yields Y and Y 22 fix the rates of iomass production per gram of sustrate consumed, the yield Y 2 is the ratio of iomass production to yproduct production, and the growth rates are given y the functions µ (G, ) andµ 2 (). can e split into two pathways. The first step (pathway) takes sustrate G and produces an intermediate y-product (acetate), creating new iomass at a growth rate µ (G, ). The intermediate y-product acetate serves as a sustrate for the second step (pathway), which has a growth rate µ 2 (). s well as eing a sustrate, acetate lowers the ph and is detrimental to oth metaolism pathways in the cell, which slows the growth rates µ i and thus slows the uptake of sustrate. The lower ph caused y acetate production could also increase the energy required for cell homeostasis, which would affect the yield parameters. We do not consider this effect here, as if yield did decrease at higher values, this would further disadvantage the inary consortia, and thus would not qualitatively affect our results ut would greatly complicate the analysis. We will consider the situation where the microes are grown in continuous culture (chemostat). We compare the total iomass produced in two different systems: a inary culture system where the two pathways are in two separate microes, and a monoculture system where the pathways are comined into a single microe. These two systems are shown in Figure. For the monoculture system, acetate that is produced through the first pathway is then availale to e consumed through the second pathway. If the acetate consumption rate is 5

6 lower than the acetate production rate, acetate will accumulate in the system, see equations (4) and (7). In our model of the monoculture, we consider the case where acetate is consumed (metaolized) y the individual microial cell that produced it, and the case where acetate is released from that microe into the environment and then consumed y distinct microes, to e equivalent. This is possile ecause this model does not include any cost involved in transporting acetate into or out of the cell. Including a cost for transporting acetate would disadvantage the inary consortia compared to the monoculture, and thus it would not qualitatively affect our results. 2.. Model formulation To model the dynamics in a chemostat we construct a system of ordinary differential equations (ODEs), following standard techniques [6], that descrie the evolution over time of the microial iomass and sustrate concentrations. In oth systems we will track the concentration of primary sustrate, G and the concentration of the intermediate y-product,. In inary culture, the iomass of a microe that consumes G and produces to grow at rate µ (G, ) will e denoted y x, and the iomass of the second microe that consumes and grows at rate µ 2 () will e x 2. For the inary chemostat this situation can e descried y the following system of ordinary differential equations: dx =(µ (G, ) D)x, dt () dx 2 =(µ 2 () D)x 2, dt (2) dg dt =(G in G)D µ (G, ) x, Y (3) d dt = D + µ (G, ) x µ 2() x 2, Y 22 (4) Y 2 where D>0 is the dilution rate (with units /hr, andd = V/Q where V is the volume and Q is the volumetric flow rate) and G in > 0 is the input concentration of primary sustrate. The parameters D and G in are set y experimental conditions. The yields, constants Y ij, are determined y the stoichiometry of the metaolic pathways involved, which dictates the energy produced per unit of sustrate consumed, and the proportion of energy produced that goes towards iomass production, which depends on additional factors including the energy required for cell homeostasis and for the production of enzymes. In this work we have chosen to define the yield terms Y and Y 22 as the amount of iomass made y one unit of sustrate, G and respectively, and Y 2 is the amount of iomass produced when one unit of intermediate sustrate is produced through the G-metaolism pathway. The parameter, r = Y /Y 2 defines the grams of the inhiitory intermediate sustrate produced per gram of primary sustrate consumed, and is fixed for a specific metaolic pathway. 6

7 To model the monoculture system, we assume that the two pathways are comined in a single microe, whose iomass is represented y the variale x. The growth rate is assumed to e the sum of the metaolic pathways of the two microes (µ WT = µ (G, ) +µ 2 ()). This provides the maximal possile growth rate for the monoculture, without making further assumptions on whether monoculture can actually use this potential. In reality, using oth pathways concurrently comes at a cost, as microes have finite resources for the transport of nutrients into the cell, and the production of enzymes. We will later (Section 4.3) consider modelling the monoculture growth rate as a convex comination of the pathways ( µ WT = βf(g)i() +( β)m()). However, we start the linear comination (µ WT = µ (G, )+µ 2 ()), which is the upper ound for the wildtype growth rate, in order to e ale to otain theoretical results. This gives the system of rate equations for the monoculture system: x =[µ (G, )+µ 2 () D] x, (5) G =(G in G)D µ (G, ) x, (6) Y [ µ (G, ) = D + µ ] 2() x. (7) Y 2 Y 22 In this formulation we assume that there is no change in pathway efficiency (yield), growth rate, sustrate affinity, or inhiition when the pathways are split into two separate microes, thus the functions and parameters for the monoculture system (5)-(7) are identical to those in the inary culture system ()-(4). In this work we will assume that the inhiition y on the first pathway is non-competitive, so that the growth rate of pathway can e descried y µ (G, ) =f(g)i(), where the function f(g) descries the growth of x on G and the function I() descries the inhiition of x y. The growth of x 2 due to the consumption of we descrie y the function µ 2 () =m(), where m() increases at low values ut decreases when gets too high Model non-dimensionalization In order to simplify the calculations, we non-dimensionalize the original system using the scalings: ˆx = x G in Y, ˆx = x G in Y, ˆx 2 = x 2 G in Y, Ĝ = G G in, Â = Y 2 G in Y, ˆt = td, = Y 2 Y 22. (8) This gives us the dimensionless variales ( ˆx, ˆx 2, Ĝ, Â) for the inary culture and (ˆx, Ĝ, Â) for the monoculture. Both systems of equations depend upon the new dimensionless time variale ˆt, and the only remaining parameter in the system is the dimensionless parameter. 7

8 Removing the hats for convenience and making the sustitution µ (G, ) = f(g)i(), µ 2 () =m(), gives us a dimensionless version of the inary culture system (equations ()-(4)) x = (f(g)i() )x, x 2 = (m() )x 2, (9) G = G f(g)i()x, = + f(g)i()x m()x 2, and a dimensionless version monoculture system (equations (5)-(7)) x = [f(g)i()+m() ] x, G = G f(g)i() x, (0) = +[f(g)i() m()] x Model assumptions Before we descrie our main result, it is important to specify our assumptions on the functions and parameters in the model (in the dimensionless setting).. The functions f(g), I() and m() are the same in oth models, i.e. that the separation of the pathways does not modify the metaolic dynamics. 2. The function f(g) is a monotonically increasing function of sustrate concentration, G, with limiting rate, µ,max (see Figure 2(a)). The properties of f(g) are: f(0) = 0, lim G f(g) = µ,max, df (G) () > 0 for G 0, dg df (G) lim G dg = 0. These conditions on f(g) are readily satisfied y an increasing Hill function µ,max G n /(k n + G n ) for any n. 3. The function I descries the inhiition of the G-metaolism pathway y the intermediate sustrate (see Figure 2()). The properties of I() are: I(0) =, I() > 0 for 0, lim I() = 0, (2) I() < 0 for >0. These conditions on I() are satisfied y a decreasing Hill function k n /(k n + n ) for any n. 8

9 μ,max f (G) B 0 G I () I( S ) I( U ) C 0 0 μ 2,max S U m () 0 0 S max U Figure 2: Qualitative sketch of the growth and inhiition functions: (a) f(g), () I(), and (c) m() in nondimensional variales. The solutions to the equation m() =, S and U, are shown in panels () and (c). 9

10 4. The function m() descries the growth rate on the inhiitory sustrate () (see Figure 2(c)). m() has the following properties: m(0) = 0, lim m() = 0, max(m()) = m( max) =µ 2,max, dm() d > 0 for 0 max, dm() d = 0 at = max, dm() d 0 for max, dm() lim d = 0. (3) These conditions on m() are satisfied y commonly used sustrateinhiition functions [24]. 5. We consider the case where the growth rates of each microe are sufficiently large that they can survive at the chosen dilution rate; i.e., µ,max >, µ 2,max >. For µ 2,max > the equation m() = has two solutions, which we define as S and U,where S < U. 6. We consider the case where the sustrate inflow is in excess, y which we mean f(g in ) µ,max and consequently, f(g in ) >D(in the original variales). These conditions correspond to f() µ,max and f() > in the dimensionless setting. We use the product form of the function µ (G, ) = f(g)i() since it captures the growth and non-competitive inhiition of pathway and simplifies the analysis. However, a more general function µ (G, ) with the properties µ (G, ) > 0 for all G>0and µ (G, ) < 0 for all >0mayeusedif G inhiition is known to e competitive. 3. Main results In this section we present our main result, the proof of which can e found in the ppendix. The implications of this technical result are then demonstrated with an example from a realistic situation in the following sections. 0

11 3.. Binary culture system equiliria and staility For the inary culture (9) we differentiate etween three different types of equilirium points: trivial equiliria where x =0,x 2 =0,oundary equiliria where x 0,x 2 = 0 (note there are no x =0,x 2 0 equiliria, as G is the only sustrate provided to the chemostat system), and co-existence equiliria where x 0,x 2 0. We consider only equiliria in the positive cone (x,x 2,G,) R 4 +. Theorem. Given ssumptions [.]-[6.] in Section 2.3, the inary culture system (9) has the following equiliria: a. There exists a trivial equilirium when there are no microes in the system at (x,x 2,G,)=(0, 0,, 0), for all functions satisfying [2.]-[4.]. This trivial equilirium point is an unstale saddle for all parameter values satisfying [5.]-[6.].. There exists a oundary equilirium point, where only x survives, at (x,x 2,G, )=( G, 0,G, G ),whereg is implicitly defined y f(g )I( G )=. The staility depends on the specific functions and parameters. c. We define a value = crit,where crit is implicitly defined y I( crit )= /µ,max. (i) If crit < S,orif S < crit ut G S + S >, there are no coexistence equiliria. The oundary equilirium point is a stale node. (ii) If S < crit < U,andG S + S <, orif U < crit,andg S + S <, ut <G U + U, there is a single, stale, co-existence equilirium point S := (x,s,x 2,S,G S, S ). The oundary equilirium point is an unstale saddle. (iii) If U < crit,andg U + U <, there are two co-existence (x 0,x 2 0) equiliria, S := (x,s,x 2,S,G S, S ) and U := (x,u,x 2,U,G U, U ). The point U is an unstale saddle, and the point S is a stale node. The oundary equilirium point is a stale node. Proof. Remark. The conditions related to crit ensure that the equation f(g)i() = has a solution for the given value. The conditions G + < ensure that the equilirium iomass values (x and x 2 ) are strictly positive; if G + > then x 2,U < 0 and the equilirium point is non-physical. Remark 2. Note that in case (iii) of Theorem c. the system is istale, with the stale manifold of the saddle point U forming a separatrix that separates the asins of attraction of the two stale equiliria, S and the oundary equilirium point.

12 3.2. Monoculture system equiliria and staility For the monoculture system (0), there are two types of equiliria: trivial equiliria where x = 0andnon-trivial equiliria where x 0. Theorem 2. Given ssumptions [.]-[6.] in Section 2.3, the monoculture system (0) has the following equiliria: a. There exists a trivial equilirium point at (x, G, ) =(0,, 0), which is an unstale saddle.. There are 2n + non-trivial (x 0) equiliria E i := (x m i,gm i,m i ) in the region < S,wheren 0, ordered y increasing m i. If there is a unique internal equilirium (n =0) this equilirium point is a stale node. If there are multiple internal equiliria, the odd equiliria (i =, 3,.., 2n +) are stale nodes, and the even equiliria (i =2, 4,.., 2n) are unstale saddles. c. If the curve G = /( + /( m())) intersects the curve f(g)i() +m() =, in the region > U, then there is an even numer of equiliria K j,j =,...,2m. If we order these equiliria y increasing value of, then the odd equiliria are saddles and even equiliria are nodes. These curves can only intersect if m() = /( + ) has more than one solution. This condition is illustrated in Figure 3. If µ,max I( v 2 ) < +, (4) these equiliria do not exist, where v 2 is the upper solution of m() = /( + ). This condition is illustrated in Figure 3. Proof. B Remark 3. The equiliria K j only exist in specific circumstances when the monoculture growth is less inhiited y acetate than the acetate consuming specialist (see (4) for exact statement), while at least one lower equilirium E i exists for all parameter values satisfying [.]-[6.]. The high (low iomass) stale equiliria, when they exist, have a small asin of attraction, which is characterized y high levels of acetate and low levels of glucose. Only initial conditions with high initial acetate and low glucose can lead to a monoculture that evolves to such stale equilirium. For this reason, we do not consider these equiliria in the results elow Biomass comparison We wish to compare the productivity of the inary culture and monoculture systems, using their total iomass as a measure of productivity. The iomass for the inary system is largest at the stale co-existence equilirium point S, since it can e easily shown that x,s +x 2,S >x (see ppendix 2

13 μ,max μ,max I() m() /(+) v 2 μ I( ),max /(+) 2 v Figure 3: Qualitative sketch of the curves m() (solid lack curve) and µ,max I() (dotdash lack curve) that satisfy the sufficient condition in Theorem 2c. ruling out the upper monoculture equiliria. The curve µ,max I() lies in the shaded region defined y the value m( v 2 ), where v 2 is the upper solution of m() =/( + ). C). The monoculture system has (at least) one stale non-trivial equilirium point E i, with high iomass. Since we do not consider the possile upper equiliria K j with low iomass to e physical, we will compare the iomass at the inary system equilirium S with an aritrary monoculture equilirium E i. Theorem 3. Given ssumptions [.]-[6.] in Section 2.3, and the existence of at least one (stale) co-existence equilirium point, S, in the inary system (conditions in Theorem (c.). Then the iomass of the monoculture, x m i,at any of the stale non-trivial equilirium points E i for i =, 3,...2n + is always higher than the total iomass of the inary culture system x,s +x 2,S at its stale co-existence equilirium point S: x m i >x,s + x 2,S. Proof. C Outline of the proof. We show in C that the monoculture iomass is given y ( x m = + ) ( G m ) m, (5) and the comined iomass of the inary culture is ( x tot = x,s + x 2,S = + ) ( G S) S. (6) We can also show that for all E i monoculture equiliria: G S >Gm i and S >m i. (7) 3

14 Thus, ( + ) ( ( G S ) S < + ) ( G m ) m and the iomass of the inary culture is always less than the iomass of the monoculture: x,s + x 2,S <x m. Remark 4. If there are no upper equiliria in the monoculture, E i for i =, 3,...2n + are the only stale equilirium points of the system, and the monoculture iomass will always e higher than the total iomass of the inary culture, for all initial conditions (G, ) R Effect of modifications to the model on relative iomass In chemostat experiments, a inary culture of the type analysed here shows a iomass increase of around 20% over a wild-type monoculture system [3]. s we have shown in Theorem 3, under our assumptions the monoculture system always has a higher iomass. Clearly this means that one or more of the assumptions that we make in Section 2 are incorrect. This is useful, as y examining the assumptions we made, we can investigate which of the effects that we did not include are most relevant. One of the assumptions we made in formulating the initial model is that there is no change in the growth dynamics or yields of the metaolic pathways when the microes specialise to a single sustrate. However, as discussed already, there is evidence that the growth dynamics and pathway efficiencies will change, and we will investigate the effects of incorporating these changes in this section. 4.. Increasing the growth rates in the inary culture system Experiments have oserved that growing E. coli on a single nutrient source can increase its growth rate 5-20% [2, 28, 3], as the microe adapts to the single nutrient and can up regulate the transport for that sustrate [42] and down regulate the transport for the pathways it is not using (cataolite repression [3, 35]). We can incorporate the effect of increasing the growth rate of the species in the inary culture system y introducing multiplicative parameters α and α 2 in front of growth rates f(g) andm(), thus making the replacements: f(g) α f(g), m() α 2 m(). (8) Note that this increases the growth rates at all concentrations of G and, not just for high growth regions of G or. For the original systems (with no change in growth rate) we have α =,α 2 =. To simulate an increase in growth rate when the metaolic pathways are split up we simply choose α,α 2 > inthe inary culture system. Increasing α decreases G S 0, while increasing α 2 in the inary culture moves S 0, which in turn decreases G S (ounded elow y G 0). We find 4

15 that it is always possile to get higher iomass for the inary culture than the monoculture (G S <Gm, S <m ) if there are no ounds on the increases for oth α and α 2. If we can only vary one pathway s growth rate, i.e., keeping α = or α 2 =, then it is not always possile to get higher iomass in the inary culture. Whether or not it is possile to surpass the monoculture iomass, and the increases in growth rate required, will depend on the specific functions and parameters for the system of interest Increasing the sustrate affinity in the inary culture system When growing on a single caron source, it has een found that some microes, including E. coli, can increase their affinity when the concentration of the sustrate remains low for an extended period of time [27, 59]. This is thought to e achieved through up-regulating higher affinity transporters [8]. Decreasing K G,movesG S 0, while decreasing K moves S 0, which in turn decreases G S (ounded elow y G 0). This matches the effects of increasing the growth rates y a factor α or α 2 in Section 4.. We can explain this mathematically as follows: in the region where G (or ) islow,increasing the sustrate affinity acts to increase the growth rate for that G (or ) value. By considering f(g) atlowg, we find that f(g) µ,max K G G for small G, so increasing µ,max or decreasing K G y the same factor should have the same effect Varying the form of the monoculture growth rate It could e argued that the monoculture out-performs the inary culture in our formulation due to the way we have descried the monoculture growth rate (µ WT (G, ) =f(g)i() +m()). In this formulation we are assuming that the microe can use oth pathways concurrently, with oth functioning at their maximal rates for the specific G and. This acts as an upper ound on the maximal possile growth rate for the monoculture. In reality, there are limitations due to the transport of nutrients into the cell and the energy required for the synthesis and maintenance of metaolic enzymes. It has een oserved experimentally that often a microe preferentially consumes the sustrate sustaining the higher growth rate (cataolite repression). In this process, the synthesis of enzymes required for alternate pathway is down-regulated, which would decrease the maximal growth rate of these other pathways. If we assume that the cell must choose to allocate resources etween the two pathways, we can incorporate this into our model y making the monoculture s growth rate a convex comination of the two pathways growth rates: µ WT = βf(g)i()+( β)m(). (9) Where β could e some function of the relative growth rates of the two pathways at the given G and concentrations, following [, 3, 35]. ccording to cataolite repression the microe will only consume the preferred sustrate. However, if the intermediate sustrate is inhiitory there is 5

16 often a switch that overrides this effect at high to consume the excess intermediate sustrate and decrease the inhiitory effects [68]. In our model, we could incorporate this y using (9), with β eing a function of, switching to low β (increased consumption) when reached a critical value, even in the presence of high G. Using µ WT, we cannot get theoretical results for the relative productivy of the monoculture and inary culture systems in general, as the iomass will depend on the specific situation and the β values (or functions) chosen. However, we can easily show that for sufficiently high or low β the inary culture system would now out-perform the monoculture system: t β =, the monoculture growth rate is given solely y f(g)i(), which is the equivalent of the inary culture system with x 2 = 0. In ppendix C we showed that this x -only equilirium point has a lower iomass than the inary culture system coexistence equilirium point, thus the inary culture outperforms the monoculture. s β 0, the monoculture growth rate is increasingly dependent on the second pathway, m(), which consumes exclusively acetate. Since glucose G is the only sustrate provided to the chemostat, we expect that the monoculture will not survive. Based on optimal foraging theory and experiments, it is reasonale to assume that E. coli feeding on multiple sustrates would allocate resources to pathways such that the growth rate is maximized [, 3, 35], and this is indeed what is found with cataolite repression. In a chemostat at equilirium, the growth rate is always fixed to the dilution rate D, so we cannot determine the optimum β value at equilirium. One option could e to first find the β values required for the monoculture system to have lower iomass (for a given system), then to use the optimal foraging arguments to determine whether these β values are realistic y looking at the ratio of µ to µ 2 for the given G and values. nother option would e to look at what β value maximizes iomass, and consider whether this leads to higher iomass in the monoculture than the inary culture, and whether this is a reasonale β value. We will consider oth in section for the specific example of E. coli growing on glucose Increasing the efficiencies (yields) in the inary culture system The yield terms Y and Y 22 descrie the amount of iomass produced per unit of sustrate consumed, and are a measure of the microe s efficiency. The yields are determined y only two factors: how much energy the pathway can produce (given y the stoichiometry of the pathway), and how much of the energy produced is allocated to iomass production. There are many ways that microes can adapt to optimize their productivity, ut only some of these will increase the yield terms in our formulation. For example, it is oserved that when provided with multiple sustrates, cells will consume the most productive sustrate preferentially (also known as cataolite repression). In this way, the microes are utilizing the most efficient pathways 6

17 and thus increasing their efficiency. In our model the inary culture microes only have one sustrate, and thus cataolite repression will not affect their dynamics. In the monoculture, cataolite repression would lead to the most efficient pathways eing prioritised. This increase in resources would increase the sustrate uptake and growth rate of the preferred pathway, and could e incorporated into our model through changing the parameter β as descried aove, where β = descries consumption of the primary sustrate only. However, as the stoichiometry of the pathways does not change, this effect would not directly change Y or Y 22 in our model. In order to increase the yield, a microe must reduce the energy that is required for functions other than iomass production, such as cell homeostasis and enzyme production. Microes growing on a single source are thought to down-regulate the production of enzymes for the metaolism of other sustrates [30]. Since the synthesis and maintenance of metaolic enzymes requires energy, if a microe can reduced its net enzyme production in this way, this would reduce the energy required and its yield would increase. Thus it is possile that y specialising to a single sustrate, the yields Y and Y 22 would increase. nother factor that could affect the energy requirements, is the cost of transporting sustrates into the cell. So far we have not included this cost. If we were to include the cost of transporting the primary sustrate G into the cell, this would decrease the term Y in oth inary and monoculture systems equally. Including the cost of transporting the inhiitory intermediate sustrate into the cell, would decrease Y 22. The inary culture needs to transport all of the consumed into the scavenger specialist x 2, whereas the monoculture can metaolize intracellular as it is produced, as well as transporting into the cell, thus deflecting some of this cost. Therefore, Y 22 would decrease more in the inary culture than in the monoculture system. Examining the equations for steady-state iomass we find that changing Y does not affect the nondimensionalized inary iomass, ut from equation (8) it will affect the actual (dimensional) iomass. The total iomass of the inary culture increases when Y increases (the G-metaolism pathway ecomes more efficient). When Y 22 increases (the -metaolism pathway ecomes more efficient), or when Y 2 decreases (the amount of inhiitory metaolite produced increases), decreases, which will increase the total iomass of the inary culture. s discussed in Section 2.,the ratio r = Y /Y 2, which descries the grams of the inhiitory intermediate sustrate produced per gram of primary sustrate consumed, is fixed for a specific metaolic pathway. Therefore, varying Y 2 or Y independently is not realistic, and in the latter analysis we will vary Y and Y 2 together, keeping r constant. When making increases to Y or Y 22 we must keep in mind that there is an upper ound on the yield terms that given y the stoichiometry of the utilized metaolic pathway. Whether the yield increase required to match the oserved increase in iomass for the inary culture is aove that ound will depend on the system of interest. 7

18 4.5. Varying G in and D Some earlier work has found that the inary culture is favoured over a monoculture for certain sustrate inflow concentrations G in or dilution rates D [0, 45]. However, we find that for all G in and D, satisfying assumptions [5.] and [6.], the monoculture has higher iomass than the inary culture. That eing said, the difference in total iomass etween the two systems will vary as G in and D change, so there may e some experimental conditions where the inary culture is less disadvantaged, and where the aforementioned modifications would e more effective. 5. Specific example using Monod kinetics and typical values for E. coli In this section, the results of this paper will e applied to the specific example of E. coli growing on glucose as the sole caron source. Experimentally oserved parameters for E. coli growth on glucose or acetate are summarized in Tale. In the majority of the literature, E. coli were grown in aeroic conditions in which oth pathways would e utilized. This will lead to overestimation of the parameters: µ,max, Y, Y 2, K IP, and underestimation of the parameter r. Parameter Experimental values Maximum growth rate on glucose a (h ) [3], 0.57 [7], 0.43 (anaeroic) [7], 0.76 [], 0.53 [7], 0.39 [20],.2 [23], 0.9 [32], 0.77 [40], 0.57 [43], 0.92 [59], [59], 0.68 [65], 0.43 (anaeroic) [65], [69] Half-saturation glucose concentration a (mg/l) 7.2 [], [8], 00 [23], 4 [32], 0.68 [40], [59], [59], 50 [69] cetate product inhiition constant a (g/l) 4 [23], 4-5 [69], 9 [69] Maximum growth rate on acetate (h ) c [3], 0.3 [2], 0.9 [23], 0.33 [42], 0.22 [43], 0.05 [69], [69] Half-saturation acetate concentration (g/l) 0.4 [23], 0.05 [69] cetate sustrate inhiition constant (g/l) 3.8 [23] Biomass yield (gbiomass/gglucose consumed) a 0.23 [3], 0.26 [7], 0.8 (anaeroic) [7], 0.3 [7], 0.30 [20], 0.43 [32], 0.50 [43], 0.36 [65], 0.3 (anaeroic) [65], 0.52 [69], [69], 0.5 (anaeroic) [69] cetate yield (gbiomass/gcetate produced) a 0.4 [23], [32] Ratio (gcetate produced/gglucose consumed) a 0.25 [7], 0.38 [32] Biomass yield (gbiomass/gcetate consumed) 0.38 [2], [43], 0.4 [69] Tale : Typical parameter values for E. Coli grown on glucose or acetate as the caron source. See Tale 4 for further details of the strains used and the specific experimental conditions. a the growth and yield of wildtype E. coli on glucose includes contriutions from the acetate metaolism pathway, except in anaeroic conditions. growth rate of mutant with acetate consumption pathway knocked out. c growth rate of mutant with glucose consumption pathway knocked out. We can quantify the discrepancy etween the actual yield of each pathway and the reported yield in wild-type aeroic conditions. If we assume that the 8

19 growth rate from the first (anaeroic) pathway is given y µ and the growth from the aeroic metaolism of acetate (acetyl Co) is given y µ 2, then the discrepancy etween the actual yield of each pathway and the reported yield in wild-type aeroic conditions will e: The reported yield Y WT (gbiomass/gglucose consumed) will e overestimate y Y WT = Y (µ + µ 2 )/µ. The reported acetate production yield Y2 WT (gbiomass/gcetate produced) will e overestimated y Y2 WT = Y 2 (µ + µ 2 )/(µ µ 2 ). The oserved ratio r WT (gcetate produced/gglucose consumed) will e underestimated y r WT = r( µ 2 /µ ). From the equation for Y WT it can e seen that if µ 2 = 0 (i.e. there is no metaolism of acetate, which occurs in anaeroic conditions), then Y WT = Y. Thus, in order to estimate the parameters of just the first pathway (max f(g) and Y ), we use the yield and growth parameters from anaeroic experiments [7, 65, 69]. To estimate Y 2 we use the ratio r =0.667 determined y the reaction stoichiometry and calculate Y 2 = Y /r. Sustituting reasonale values for growth rates, we find that our choice of parameters leads to oserved Y2 WT r WT that match experimental data (Tale ). The oserved growth and yield parameters for E. coli grown on acetate as the sole caron source provide good estimates for max(m()) and Y 22. The chosen yields are given in Tale 2. Parameter Selected value Notes Y 0.52 from anaeroic growth experiments r = Y /Y reaction stoichiometry Y calculated, Y 2 = Y /r Y Tale 2: Selected yield values for monoculture and inary culture systems. It is important to note that for the yield terms Y and Y 22, which are the grams of iomass produced when one gram of sustrate is consumed, only some of the energy from the the sustrate metaolism will go towards iomass synthesis. Depending on the efficiency of the pathway, the energy produced per gram of sustrate will vary, and the energy required for cell homeostasis (i.e. not availale for iomass production) also varies. Estimates for the ratio of energy produced that goes towards iomass production and the efficiencies of the pathways used vary depending on the E. coli strain and the environmental conditions [6, 25]. We choose to use measured values for the total yield from the literature. These yields could vary with strain and condition, and we will consider this in Section In all cases the yields are ounded aove y the stoichiometry of the most efficient iomass production pathways assuming zero energy required for cell homeostasis, and our choices for yield parameters are elow this ound and consistent with experimental findings. 9

20 To fit growth parameters, we must choose specific functions for f(g), I(), and m(). Here we select Monod type kinetics for the growth and inhiition functions: G f(g) =µ,max, (20) G + K G I() = K IP K IP +, (2) m() =µ 2,max. (22) + K + 2 K IS which satisfy assumptions [2.]-[4.] and are found to e appropriate for descriing growth and inhiition rates of E. coli in chemostats or atch growth. With the exception of the half-saturation of glucose, K G, experimental findings (Tale ) are reasonaly consistent and the chosen parameters for these functions are given in Tale 3. Parameter Selected value Notes µ,max 0.6h max(f(g)) = 0.6h K G 0.05g/L K IP 0.5g/L µ 2,max 0.8h max(m()) = 0.2h K.5g/L half-saturation at =0.23g/L K IS 0.7 g/l inhiited to half maximum at =4.6g/L D 0. h chosen to prevent wash-out G in 0g/L chosen to ensure excess sustrate Tale 3: Selected parameter values for growth and inhiition functions given in equations (20), (2), (22), dilution rates, and sustrate inflow concentrations. t low glucose concentrations it is known that glucose affinity increases [8, 27], and the parameter K G varies from O(0.0) to O(00) mg/l (Tale 4) through the upregulation of higher affinity (and often slower or more energy intensive) pathways. In our model, K G is a fixed parameter, and to demonstrate the results of this paper we choose an intermediate K G value (see Tale 3). This choice does not affect our result, as for equal sustrate affinities the monoculture will outperform the inary culture for any choice of K G (Theorem 3). We will investigate the effect of varying the glucose affinities etween the monoculture and inary culture in Section However, as increases in glucose affinity are found to depend mostly on the growth conditions, we would expect any increase in glucose affinity (K G ) to affect oth the monoculture and the inary culture similarly. In E. coli there are two pathways for acetate assimilation: the phosphotransacetylaseacetate kinase (PT-CK) pathway and the acetyl-co synthetase (acs) pathway. The PT-CK pathway has lower sustrate affinity, with K m g/L [4, 68], ut is more efficient requiring only one TP for acetate assimilation. The PT-CK pathway is found to e active until acetate gets to 20

21 low concentrations ( 0.5g/L) [34]. In contrast, the acs pathway has higher sustrate affinity, with K m 0.0g/L [4, 68], ut is more energy intensive, requiring two TP for acetate assimilation. Kumari et al. [34] find that oth pathways are required for optimal growth on acetate across a range of concentrations, with the acs pathway eing required for <0.6g/L and the PT-CK pathway eing required for >.5g/L. In our model, we use a single function m() and single yield term Y 22 to descries the growth kinetics and efficiency of growth on acetate; we do not include the two pathways separately. We choose growth parameters µ 2,max, K, and K IS that match the experimentally oserved growth in chemostat cultures which are utilizing oth pathways. For example, the comined growth kinetics has a sustrate affinity of K m =0.23g/L which is midway etween the K m values of each pathway. We will investigate the effect of varying the acetate affinities etween the monoculture and inary culture in Section The yield parameter Y 22 is fixed, so we have chosen Y 22 =0.4 whichisatthe lower end of the yields oserved across a range of growth conditions [32]. This is ecause we find that in our model acetate concentrations stay low and the yield will e determined predominantly y the lower efficiency, higher affinity acs pathway. The dilution rate D and glucose inflow concentration G in are set y experimental conditions. We must select a dilution rate that satisfies non wash-out conditions (from assumption [6.]: max(m()) >D, µ,max >D). This leads to a low dilution rate (D =0.) due to the low value of max(m()) >Dseen experimentally. We also choose a high sustrate inflow G in G 0, so that the system is in glucose-excess conditions (assumption [5.]). For the selected functions and parameter values, the growth and inhiition curves are depicted in Figure 4. We graph them in terms of the original variales, glucose concentration, G (g/l), and acetate concentration, (g/l), for ease of comparison to the experimental literature (Tale ). However, for the remainder of the analysis we will use the dimensionless variales and parameter values unless otherwise stated. We use the nondimensionalization in equations (8) and the parameter scalings: where r = Y /Y 2. ˆµ,max = µ,max D ˆ K = K rg in,, ˆµ 2,max = µ 2,max D, ˆ K G = K G G in, (23) ˆ K IS = K IS rg in, ˆ K IP = K IP rg in. 5.. Monoculture and inary culture with no changes In the monoculture system (0), for growth and inhiition functions (20)- (22) and parameter values in Tale 2 and Tale 3, we find a unique non-trivial equilirium at (x m,gm,m )=(2.72, , 0.097). 2

22 0.6 f (G) B D G (gl ) I () 0.5 C (gl ) m () D (gl ) Figure 4: The functions f(g), I(), and m() (equations (20),(2), and (22), respectively), in the original (dimensional) variales, for the chosen parameter values. 22

23 In the inary culture system (9), there is one stale co-existence equilirium point (x,s,x 2,S,G S, S )=(0.998,.70, , ), with x S =2.70. There is also a stale equilirium point where only x survives at (x,x 2,G, )=(0.37, 0, 0.629, 0.37). Theiomassofthemonoculture,x =2.72, is greater than the total iomass in the inary culture at the stale co-existence equilirium, x = x + x 2 =2.70, as given y Theorem G G 0 m m (,G ) (,G ) S S Figure 5: The curves G and G 2, which determine the position of the monoculture equilirium point(s) in the nondimensionalized system are shown in lack solid lines, with the intersection laelled ( m,g m ). The curves f(g)i() =andm() =, which determine the position of the inary culture equilirium point(s) are shown in grey dot-dash lines, with the intersection laelled ( S,G S ). The lines of constant iomass for the monoculture and inary culture are lack dashed and dotted lines, respectively Quantitative investigation of how to increase the inary culture s relative productivity In chemostat experiments, a inary culture of this type shows a iomass increase of around 20% over the wild-type monoculture system [3]. In Figure 5 we plot the curves that determine the equilirium points of the systems. The iomass is determined y the values of G and, given in equations (5) and (6), thus in the (, G)-plane, the curves G = c /( + ), for a constant c = x c /( + ), are the lines of constant iomass, x c. In order for the inary culture iomass to exceed that of the monoculture, the inary culture system must e modified such that the point (,G )crossesthedashedlinein Figure 5, or, alternatively, the monoculture system must e modified such that the point ( m,g m ) crossed the dotted line in Figure 5. (The only exception to this is if Y or G in are varied etween the two systems, as these parameters appear in the non-dimensionalization of the iomass variales.) 23