bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
Microeconomics of metabolism: Overflow metabolism as Giffen behavior Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama
Department of Basic Science, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
Abstract metabolism rather than human society. Living organisms optimize their survivability through evolu- Indeed, an analogy between biology and microeconomics has tionary processes. In particular, intracellular metabolic systems been sometimes argued (7–9). Here, we push beyond an anal- are rationally regulated to maximize the cellular growth rate. ogy to exact mapping of metabolic systems onto the theory of Correspondingly, the field of microeconomics investigates the consumer choice in microeconomics. Specifically, we reveal behavior of individuals assumed to act rationally to maximize the conditions required for overflow metabolism and Giffen their utility. Therefore, microeconomics can be applied to an- behavior, long-standing mysteries in biology and economics, alyze the metabolic strategies of cells. Toward this end, we de- respectively. veloped a microeconomics-based theory of cellular metabolism Overflow metabolism is a seemingly wasteful and yet ubiq- by precisely mapping the regulation of metabolic systems onto uitous behavior by which fermentation is favored over res- the theory of consumer choice in microeconomics. As a repre- piration even in the presence of abundant oxygen. This is sentative example, we focus on overflow metabolism, a seem- ingly wasteful strategy in which cells utilize fermentation in- an apparent contradiction, given that the respiratory path- stead of the more energetically efficient respiration (so-called way yields approximately 15-fold more ATP than the fermen- Warburg effect in cancer). To resolve this apparent contra- tation pathway, and yet aerobic organisms generally utilize diction, we formulate overflow metabolism as an optimization the less energetically efficient fermentation strategy (10–16). problem of the allocation of carbon fluxes under the guidance of This phenomenon is often observed in fast-growing cells with microeconomic theory. Accordingly, we demonstrate that over- abundant carbon, resulting in the overflow of end products of flow metabolism corresponds to Giffen behavior in economics, the fermentation pathway, such as acetate and ethanol, lead- the strange consumer behavior by which greater amounts of ing to the designation of the term overflow metabolism. This goods are consumed as their price increases. We reveal the behavior is ubiquitously observed across a variety of cells, general conditions required for both overflow metabolism and ranging from bacteria to eukaryotes, e.g., yeasts (known as Giffen goods: trade-off and complementarity, i.e., the impos- the Crabtree effect) (12) and mammalian cells, including can- sibility of substitution for different goods, among multiple ob- jectives. Based on the correspondence with Giffen behavior, a cer cells (known as the Warburg effect) (13, 14), stem cells counterintuitive response of metabolism against the leakage and (15), and immune cells (16). Many experimental and the- degradation of intermediate metabolites, which corresponds to oretical studies have proposed hypotheses to explain over- the change in the price of a consumer good, is predicted. Over- flow metabolism. One of the main hypotheses put forward all, this demonstration highlights that application of microeco- is that fermentation and respiration compete for limited re- nomics to metabolic systems will offer new predictions and po- sources such as the limited amount of proteome (11, 17– tentially new paradigms for both biology and economics. 19), empty space on the cellular membrane (so-called “mem- metabolic systems theory of consumer choice Warburg effect optimiza- brane real estate”) (20, 21), and the volume of cytoplasm | | | tion trade-off (13). Although a variety of mechanisms underlying overflow | DRAFT Correspondence: [email protected] metabolism have been considered in specific organisms and surrounding environments, there is still little progress made to reveal the universal structure of this common phenomenon. Introduction From a similar perspective in microeconomics, Giffen behav- iving organisms are considered to optimize their survivability ior represents the mysterious phenomenon by which the de- through evolutionary processes. In particular, the metabolic mand for a good increases when the price increases. This systems of cells need to be optimized to improve the growth is in stark contrast to the general pattern of human economic rate or biomass yield. Indeed, several biological theories such activities in which demand decreases when the price of a nor- as flux balance analysis (FBA) have succeeded in predicting mal good increases, and vice versa. Thus, Giffen behavior is the metabolic behavior under the assumption that cells ratio- also referred to as Giffen’s paradox (6, 22). Although the ex- nally regulate their metabolism (1–5). istence of Giffen goods was theoretically predicted more than Correspondingly, microeconomics is the study of the behav- a century ago, their practical existence remains controversial. ior of individuals who are assumed to have perfect rationality However, a few examples have recently been considered to to maximize their “utility” (6). However, in reality, humans represent Giffen goods in practice (22, 23). do not behave completely rationally, and it is difficult to de- In this study, we discovered a strong link between the termine the precise forms of utility functions. Thus, microe- above two biological and microeconomics phenomena. By conomics might actually be more applicable to understanding mapping resource allocation models for metabolic systems
Yamagishi et al. | bioR‰iv | April 18, 2019 | 1–7 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. on the theory of consumer choice in microeconomics, we A B1.5 u formulated overflow metabolism as an optimization prob- 2.5 lem, and demonstrated that the utility function for overflow 2.0 1.0
metabolism has the same universal structure required for Gif- 2 1.5 _ fen behavior. Through this correspondence, we revealed two x 2 key conditions required for both overflow metabolism and u 0.5 1.0 Giffen behavior: (i) a trade-off and (ii) “complementarity” x2 0.5 among multiple objectives. The latter is autonomously sat- 0.0 0 isfied in metabolic systems due to the nature of stoichiom- 0.0 0.5 1.0 1.5 x1 x1 etry. The correspondence also allows predicting a counter- x_ 1 intuitive response of metabolic systems to the leakage and Fig. 1. Representation of optimization problems in microeconomics. (A) Landscape
0.6 and (B) contour map0.6 of the utility function. Here we adopted0.6 u(x1,x2)=Ôx1 + degradation of intermediate metabolites. We further discuss0.5 0.5 0.5 0.4 f
0.4 0.4 f f x as the utility function, where x and x denote the demand for goods and .
Ô , 1 2 ,
, 2 1 2 0.3
0.3C
0.3 C C J J the possibility of observing strange behaviors similar to over-J 0.2 The background color0.2 shows the value of u(x1,x2). Gray0.2 lines are contour lines, 0.1 0.1 0.1 0.0 0.0 0.0 flow metabolism in other metabolic systems. 0.0 0.1J0.2 0.3which0.4 are known as0.0 0.1 indifferenceJ0.2 0.3 0.4 curves in microeconomics.0.0 0.1J0.2 Red0.3 0.4 and green lines C,r C,r J C,r are budget constraintr lines, whose slopes dependC on,in ther price of each good. The A optimal(I) solution f is determined(II) B from the tangent(I) pointr of thef budget(II) constraint line 0.5 0.5 0 Model to an indifference curve, and thus depends on the price of goods. Here, the optimal 0.4 point moves from red to green0.4 points when the price p increases.
1
The theory of consumer choice explains how the price0.3 C,f
^ 0.3
J p of goods and income I determine consumption behav- λ i , 0.2 of the metabolic pathways0.2 increases. ior, i.e., how a consumer determines the demand for eachC,r ^ 0.1J In the case of overflow0.1 metabolism, we consider the carbon good (xi for the ith good) under the given utility function source as the nutrient and the carbon fluxes allocated for res- 0.00 0.00 u(x1,x2, ,xn). When the number of goods n is two,1.0 1.2 1.4 1.6 1.8 2.0 ··· 1 pirationpr/p andf fermentation,1.01 J1.2C,r and1.4pr/pJ1.6C,ff ,1.8 respectively,2.0 as the utility function is represented as a curved surface (Fig. the demand of two goods. The budget constraint line is thus 1A). If the consumer is sufficiently rational in their decisions, given by the carbon balance as the decision-making process can be mapped onto an opti- mization problem of the utility under the budget constraint JC,in = prJC,r + pf JC,f , (1) n p x I. i=1 i i Æ Changes in a good’s price or the income of the consumer can where J is the intake flux of the carbon source, and p q C,in r both alter the consumption behavior so as to maximize utility and pf are the inefficiencies of respiration and fermentation (Fig. 1). To understand such changes intuitively, microe- to metabolize the carbon source to downstream metabolites, conomists usually use contours of the utility function, which especially ATP, respectively. For simplicity, we set pr and pf are termed indifference curves, representing sets of demand to 1 (i.e., assume that the carbon source is perfectly converted from which the utilities take the same value (Fig. 1B). Then, to the energy molecules with neither leakage nor degrada- the demand for each good is basically determined from a tan- tion), unless otherwise stated. gent point of the budget constraint line to the indifference The utility function is given by the growth rate, curve on which the utility takes the largest value. When the ⁄(JC,r,JC,f ). For successful division, cells have to price of either good changes, the budget constraint line tilts build their components from precursors of biomass and in the x1-x2 plane, and thus the optimal demand changes. In energy molecules. Hence, ⁄(JC,r,JC,f ) is determined from the example of the utility function u(x1,x2)=Ôx1 + Ôx2 the production rates of the energy molecules JE and biomass shown in Fig. 1, the demand for each good increases when precursors JBM. its price decreases, and vice versa. Such goods are referred DRAFTThe energy production flux, JE, is the sum of the fluxes of to as normal goods in microeconomics. ATP synthesis via the respiratory and fermentation pathways, In this paper, we adopt the above framework for the theory JE,r and JE,f, which are proportional to JC,r and JC,f with of consumer choice to understand metabolic systems (see SI different coefficients, respectively: Appendix, Table S1). We consider a simple metabolic sys- tem that consists of a single flux to obtain a nutrient and mul- JE JC,r,JC,f = JE,r + JE,f = ‘rJC,r + ‘f JC,f . (2) tiple fluxes to metabolize the nutrient to energy molecules. ! " If the rate of growth or biomass synthesis is given as a func- The positive constants satisfy ‘r >‘f because respiration can tion of these fluxes, we can map the metabolic system onto a produce a greater amount of ATP than fermentation using the problem of consumer choice in which the intake flux of the same amount of the carbon source (24). nutrient is the income, and the multiple metabolic pathways In a cell, respiratory, fermentation, and biomass precur- correspond to multiple goods. Hence, the demand for goods sor production pathways compete for nutrients as well as are the rates to metabolize the nutrient in the pathways. In other limited resources such as the total amount of pro- microeconomics, the price quantifies the inefficiency of con- teins (11, 17–19) and empty space on the cellular membrane version from money to goods; accordingly, the price of each (20, 21). If we denote the total amount of a given resource metabolic pathway is interpreted as the inverse efficiency allocated to growth by fltot, competition for the limited re- to metabolize the nutrient. For instance, when leakage and source can be described as flr + flf + flBM = fltot, where degradation of intermediate metabolites increases, the price flr, flf , and flBM are the resources allocated to respiration,
2 | bioR‰iv Yamagishi et al. | Microeconomics of metabolism bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. fermentation, and production of biomass precursors, respec- case of the substitution effect for a good due to a price change tively. For example, in the proteome allocation model (11), of the good itself, has to be non-negative (25); i.e., the de- fltot is the total fraction of enzymes allocated to growth. In mand for a good cannot be increased by the substitution effect the membrane real estate model (20, 21), fltot is the total when its price increases. By contrast, a change in the price of empty space in the cellular membrane, and flr and flf re- a good affects the budget to spend freely; that is, an increase spectively correspond to the occupied membrane areas by in a good’s price can be interpreted as an effective decrease membrane proteins used for respiration and fermentation; in the consumer’s income. Such an effective change in the flf is close to zero, and flBM is the membrane area used income alters the demand for each good, which is termed the to take in the compounds for building biomass precursors. income effect. The income effect of a good can be either posi- Based on the law of mass action, each flux is proportional tive or negative, by which an increase in the income increases to the allocated resources: JE,r = ‘rÕ flr, JE,f = ‘fÕ flf , and or reduces the demand for the good, respectively. The goods JBM = ‘BMÕ flBM. It follows that with a negative income effect are known as inferior goods. The overall change in the demand for goods according to a JBM JC,r,JC,f = price change is given by the sum of the substitution and in- ‘r ‘f come effects (see SI Appendix, Eq. [S1]). ‘!BMÕ fltot ‘"BMÕ JC,r ‘BMÕ JC,f . (3) ≠ ‘rÕ ≠ ‘fÕ A Giffen good, which is a particular type of inferior good, shows a negative income effect that is larger than its substi- Empirical observations have shown that respiration requires tution effect (see SI Appendix, Sec. S1 for details). more resources than fermentation, e.g., enzymes for respira- Indifference curves can allow for considering both the substi- tion function in bigger protein complexes than those used for tution and income effects. If the growth rate takes the value fermentation (11, 13) and occupy a larger area of the cellu- ⁄˜, the indifference curve is given as a two-valued function lar membrane (20, 21). That is, ‘rÕ is smaller than ‘fÕ . Since from Eqs. (2)-(4), as shown in Fig. 2A and 2B: respiration is a much more efficient method of metabolizing ‘r sE 1 1 ATP than fermentation, there is a trade-off between the pro- JC,r + ⁄˜, if JE JBM ≠ ‘f ‘f sE Æ sBM duction of energy molecules and occupancy of the limited JC,f = ‘ ‘ Y ‘r fÕ fÕ sBM ˜ 1 1 resources. ‘ ‘ JC,r + ‘ fltot ‘ ⁄ . if s JE s JBM ]≠ f rÕ f ≠ BMÕ E Ø BM Here, metabolic reactions, including biomass synthesis, fol- 1 2 low the rules of stoichiometry. Each component of a cell is The growth[ rate is maximized at the tangent point of the bud- built by biochemical reactions involving multiple compounds get constraint line (Eq. (1)) to an indifference curve. whose amounts are determined by the law of mass conserva- If the influx of carbon sources JC,in is lower than r0 = sBM ‘r ‘r tion. In general, the compounds cannot be replaced by each fltot/( + ‘ ), the budget constraint line has no tangent ‘BMÕ sE rÕ other; thus, the total amount of the product is determined by point to any indifference curve, and the maximum growth rate the least abundant component. This property of stoichiome- is achieved at (JC,r,JC,f )=(JC,in,0) (see the light-blue try is identical to the concept of (perfect) complementarity in area in Fig. 2C and 2D). In this regime, the occupancy of the microeconomics, in which perfect complementary utility is limited resources does not limit cell growth, and cells only represented by a Leontief utility function, i.e., the least avail- use respiration to produce energy molecules more efficiently. able element of all goods (22). Hence, the growth rate is When the intake JC,in is sufficiently high, the budget con- represented by a Leontief utility function as straint line has a tangent point to an indifference curve. The set of such tangent points with various JC,in is given as the 1 1 1 1 ⁄(J ,J )=min J , J , (4) line on which JE = JBM (see the ridgeline in Fig. C,r C,f s E s BM sE sBM 3 E BM 4 DRAFT2A and 2B represented by the dashed lines): where sE and sBM are the stoichiometric coefficients for f0 energy molecules and biomass precursors to synthesize J = J + f , (5) C,f ≠ r C,r 0 biomass, respectively. 0 The biological meanings of the variables and parameters in sBM ‘f ‘f where f0 = fltot/( ‘ s + ‘ ). Thus, the growth rate is the model, and their correspondence to microeconomics are BMÕ E fÕ summarized in Table 1. maximized at the intersection between the above ridgeline and the budget constraint line (Eq. (1)). The slope of this ridgeline is negative, as the growth rate rises Results when JC,f increases and JC,r decreases along the ridgeline Overflow metabolism as Giffen behavior. In microeco- because of the trade-off between the efficiency of energy pro- nomics, the effect of a change in price can be decomposed duction and resource occupancy. Accordingly, with greater into two distinct effects (6, 25): a substitution effect and an amounts of carbon sources a cell takes in, there will be more income effect (see SI Appendix, Sec. S1 for details). The flux flowing to the fermentation pathway, and less flowing substitution effect is caused by relative changes in the combi- to the respiratory pathway (see the light-green area in Fig. nation of demand for goods, and is represented as a change of 2C and 2D). That is, overflow metabolism occurs. In other the tangent point along the indifference curve with the same words, the income effect for respiration is negative. Here, the utility value. The self-substitution effect, which is a special Leontief utility function has no substitution effect (26, 27),
Yamagishi et al. | Microeconomics of metabolism bioR‰iv | 3 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
0.6 0.6 0.6 0.5 0.5 0.5 2.5 0.4 0.4 0.4 A f f f 0.3, 0.3, 0.3, C C C
2.0 0.2J 0.2J 0.2J 0.1 0.1 0.1 1.5 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 λ JC,r JC,r JC,r 1.0 0.6 0.5 0.5 C 00
0.4C,f
JC,f ^ 0 0 JC,r J 0.3 Indifference Curves , Fig. 2. Overflow metabolism and Giffen behavior as an B Budget Constraint Line 0.2C,r optimization problem. (A) Landscape of the growth rate ^ 0.6 J ⁄(JC,r ,JC,f ) and (B) corresponding contour map. In- high 0.1 difference curves (Eq. (4)) and a budget constraint line f0 2.5 0.5 0.70.0 (Eq. (1)) are represented by black and red solid lines, re- 00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 spectively. The gray dashed line is the ridgeline of the 2.0 0.6 utility function (Eq. (5)). The background color represents 0.4 BM D ↑ 0.5 the growth rate ⁄(JC,r ,JC,f ). (C) Dependence of the /s 1.5 optimal strategy (JˆC,r , JˆC,f ) on JC,in (Engel curve; 0.3C,f 0.4 BM J λ Eq. (6)). JC,r (cyan line) and JC,f (magenta line) in the J 0.3 optimal solutions are plotted against JC,in. (D) JE /sE 1.0↓ , 0.2 (blue line) and J /s (dark-red line) in the optimal 0.2E BM BM solutions are plotted against JC,in; ⁄(JˆC,r , JˆC,f ) also 0.1 0.5 /s 0.1E r f corresponds to the blue line. Top panels depict the con- 0.0J 0 0 tour maps for the cases JC,in r0 (light-blue area), 0.0 0 0 Æ 0 low f0 JC,in r0 (light-green area), and JC,in f0 0.0 0.1 0.2 r 0.3 0.4 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Ø Ø Ø 0 JC,r 0 JC,in (pink area). J_{C, in} and thus overflow metabolism immediately corresponds to tion. Giffen behavior in which the respiratory pathway is a Giffen Note that if there is no trade-off, i.e., both ‘r >‘f and good. ‘rÕ >‘fÕ hold, the maximal point of the utility function is When the carbon intake increases further and exceeds f0, the not (JC,r,JC,f )=(0,f0) but rather (JC,r,JC,f )=(r0,0). JC,f -intercept of the ridgeline, (JC,r,JC,f ) in the optimal Hence, when the intake of carbon sources increases, only solution always takes the value (0,f0) (see the pink area in JC,r increases and saturates at JC,r = r0, while JC,f remains Fig. 2C and 2D) because the global maximum of the utility at zero; namely, overflow metabolism cannot be observed. A function is achieved at this point. In this scenario, the effi- trade-off is thus required for overflow metabolism and Giffen ciency of producing energy molecules is no longer a primary behavior. concern given the excess carbon sources available, and occu- pancy of the limited resources is the only selection criterion Leakage of intermediate metabolites drastically influ- for the two metabolic pathways. Hence, cells allocate the ence overflow metabolism. This clear correspondence be- carbon intake to only fermentation until reaching the upper tween overflow metabolism and Giffen behavior leads to bound determined by the available resources. a new prediction. Although we have considered the case Based on these results, we calculated the dependence of pr = p =1, the price pr and p can be larger than 1 due the optimal strategy Jˆ ,Jˆ on J , called the Engel f f ( C,r C,f ) C,in to inefficiencies in carbon metabolization, i.e., the leakage curve in microeconomics, as shown in Fig. 2C: DRAFTand degradation of intermediate metabolites will increases the price of metabolic pathways. For example, leakage in ˆ ˆ JC,r,JC,f = the tricarboxylic acid cycle or electron transport chain will 1 2 increase pr. (JC,in,0), if JC,in r0 Æ Such changes in price alter the slope of the budget constraint r0 f0 (f0 JC,in), (JC,in r0) , Y f0 r0 f0 r0 line (1) along with its intersection with the ridgeline (5). _ ≠ ≠ ≠ ≠ (6) _1 if f J r 2 Since the property of each metabolic pathway depends on ]_ 0 Ø C,in Ø 0 (0,f ). if J f how the budget constraint line intersects with the indiffer- 0 C,in Ø 0 _ ence curves, the form of the Engel curve (Eq. (6)) is modified _ This Engel[ curve is in good agreement with experimental ob- when the price pr or pf is not equal to 1 (SI Appendix, Sec. servations (10, 11). We also calculated the dependence of S2). the optimal growth rate ⁄ on JC,in (see the blue curve in Fig. We here set an initial condition at a point where overflow 2D). When the intake of carbon sources increases, the growth metabolism is observed (i.e., f J r holds), and 0 Ø C,in Ø 0 rate ⁄ increases linearly on JC,in initially, and then the slope consider the influence of increases in pr. Notably, the prop- of the increase in ⁄ becomes gentle at JC,in = r0 due to the erty of the optimal carbon allocation qualitatively changes trade-off between the energy efficiency and resource occu- depending on whether the price ratio pr/pf is higher or lower pancy, where overflow metabolism is observed. Finally, the than ‘r/‘f . growth rate reaches the maximum due to the resource limita- (I) pr/pf <‘r/‘f (the green and blue areas in Fig. 3): the
4 | bioR‰iv Yamagishi et al. | Microeconomics of metabolism bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 ditions generally show overflow metabolism and Giffen be- f f f , , 0.3 0.3 ,
C 0.3 C C J J 0.2 0.2 0.2J 0.1 0.1 0.1 havior regardless of the origin of the trade-off (e.g., protein 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 JC r JC,r JC r allocation and membrane allocation). 0.5 , 0.5 , 0.4A 0.4B Of note, another mechanism of overflow metabolism has been proposed in terms of the Warburg effect in cancer, which 0.3 C,f
^ 0.3 J λ hypothesizes that overflow metabolism is caused by a trade- 0.2 , 0.2
C,r off between the production efficiencies of ATP and some ^ r J r 0.1J 0.1 C,in f r f other metabolites required for growth such as nucleosides 0.00 0.00 0 and amino acids (14). However, this mechanism is also for- 1.01 1.2 1.4pr/p1.6f 1.8 2.0 1.01 1.2 1.4pr/p1.6f 1.8 2.0 mulated as an optimization problem with the same univer- ˆ ˆ Fig. 3. Dependence of the optimal strategy (JC,r , JC,f ) on the price of respiration sal structure as shown in Fig. 2 (see SI Appendix, Fig. S1 ‘r ‘r pr . Here pf is set to 1. (A) JC,in > r0 and (B) JC,in < r0. The cyan, ‘f ‘f and Sec. S3). Hence, the systems satisfying the two con- ˆ ˆ ˆ ˆ magenta, and green curves depict JC,r , JC,f , and ⁄(JC,r , JC,f ), respectively. ditions, i.e., trade-off and complementarity, generally show Top panels depict the contour maps of the utility function for the cases J C,in Ø p r and p <‘ /‘ (light-green area), J p r and p <‘ /‘ (light- Giffen behavior even if the precise mechanism is different r 0 r r f C,in Æ r 0 r r f blue area), and pr >‘r /‘f (yellow area). Indifference curves and the budget from that based on limited resource allocation. constraint line are represented by black and red solid lines, respectively. Although our study was based on the spirit inherited from constraint-based modeling in that evolutionary processes budget constraint line intersects with the ridgeline, and the force metabolic systems to become optimized (1–4), we here respiratory pathway behaves as a Giffen good, i.e., the de- adopted a reductionist approach. We considered a model mand J increases along with the price p . In other words, C,r r with only a few variables so as to best uncover the univer- if there is more leakage and degradation of intermediates sal and essential structure, whereas FBA comprises numer- in respiration pathways, i.e., the carbon metabolization ef- ous variables with the aim of predicting the metabolic states ficiency for respiration decreases, the carbon flux toward res- of actual cells. Recent studies have successfully reproduced piration will increase counterintuitively. Note that in the case overflow metabolism with FBA modified in an ad hoc man- of J <‘ r /‘ , the carbon flux toward fermentation goes C,in r 0 f ner (19, 29, 30). We believe that such modified versions of to zero at p /p = J /r , and thus respiration decreases r f C,in 0 FBA could be reduced to the universal structure we have re- against an increase in p (see the blue area in Fig. 3B). r vealed herein. Moreover, our framework will provide a de- (II) p /p >‘ /‘ (yellow areas in Fig. 3): the budget r f r f sign principle for constraint-based modeling to handle over- constraint line does not intersect with the ridgeline, and the flow metabolism across various species. point (JC,r,JC,f )=(0,JC,in/pf ) is optimal in terms of the growth rate. In this regime, considering the imbalance in The analogy between overflow metabolism and Giffen goods price, the fermentation pathway becomes more efficient for offers new predictions on the relationship between optimal carbon allocation and the inefficiency of carbon metabolism. both JE and JBM, and there is no longer a trade-off (see also SI Appendix, Sec. S3 for details). Hence, the carbon Specifically, we demonstrated that if the carbon metabolism intake is allocated to only fermentation and not respiration. efficiency of respiration decreases due to leakage of the inter- The optimal allocation of carbon fluxes shows a jump from mediates, the carbon allocation for respiration will counter- intuitively increase, as in Giffen goods. Indeed, a shift from regime (I) to (II) in a discontinuous manner when pr in- creases (see the cyan and magenta curves in Fig. 3). Nev- fermentation to respiration was reported in a line of cancer ertheless, the optimal growth rate changes continuously (see cells that dissipates energy by leakage of proton required to the green dashed curves in Fig. 3). produce ATP (31). We expect that more quantitative exper- DRAFTiments will further prove our predictions by increasing the price of respiration. For example, uncouplers of oxidative Discussion phosphorylation such as 2,4-dinitrophenol and CCCP pro- We here developed the microeconomics of metabolism, mote the dissipation of the proton gradient (32). which revealed the clear correspondence between overflow Besides overflow metabolism, Giffen behavior will be ob- metabolism in biology and Giffen behavior in economics as served in a variety of metabolic systems as long as the two a prominent example. Such correspondence indicates their requirements are satisfied. Since perfect complementarity common requirements: (i) a trade-off and (ii) complemen- should hold in metabolic systems owing to the law of mass tarity among multiple objectives. In metabolism, cells have conservation, metabolic systems with a trade-off are gener- to produce biomass from precursors and energy molecules, ally expected to show Giffen behavior. One representative ex- which introduces a trade-off: higher efficiency for ATP pro- ample is the Embden-Meyerhoff-Parnass (EMP) and Entner- duction places a heavier burden on biomass precursor pro- Doudoroff (ED) pathways for the glycolytic strategy. The duction. Complementarity between biomass precursors and EMP pathway is more efficient for ATP production but re- energy molecules is autonomously achieved because of the quires a greater amount of enzymes than the ED pathway law of mass conservation. Hence, although respiration has (19, 33, 34), which imposes a heavier burden for the produc- higher efficiency for ATP production, cells also use fermen- tion of biomass precursors. As this is the same trade-off that tation to catabolize carbon sources in a nutrient-enriched en- occurs in overflow metabolism, the EMP pathway is expected vironment. The metabolic systems satisfying these two con- to behave as a Giffen good. Indeed, at high growth rates, the
Yamagishi et al. | Microeconomics of metabolism bioR‰iv | 5 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
ED pathway is utilized to catabolize glucose (33, 34). More- 6. Mankiw NG (2016) The Theory of Consumer Choice. Principles of economics, eighth edition (Cengage Learning, Boston), pp 425–450. over, an identical trade-off can be observed with respect to 7. Carlson RP, Taffs RL (2010) Molecular-level tradeoffs and metabolic adaptation to simulta- the usage of the mixed-acid or lactic-acid fermentation path- neous stressors. Curr Opin Biotechnol 21(5):670–676. 8. Aidelberg G, et al. (2014) Hierarchy of non-glucose sugars in Escherichia coli. BMC Syst ways (35, 36). Biol 8:133. Furthermore, different types of trade-offs are present in a 9. Shoval O, et al. (2012) Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space. Science 336(6085):1157–1160. wide range of metabolic systems, including trade-offs for the 10. Vemuri GN, Altman E, Sangurdekar DP, Khodursky AB, Eiteman MA (2006) Overflow production of different metabolites from resources taken up Metabolism in Escherichia coli during Steady-State Growth: Transcriptional Regulation and Effect of the Redox Ratio. Appl Environ Microbiol 72(5):3653–3661. by extracellular carbon, nitrogen, and phosphorus-acquiring 11. Basan M, et al. (2015) Overflow metabolism in Escherichia coli results from efficient pro- enzymes (37); a trade-off between carbon fixation and light teome allocation. Nature 528(7580):99–104. 12. De Deken RH (1966) The Crabtree effect: a regulatory system in yeast. J Gen Microbiol harvesting in nitrogen allocation toward rubisco and chloro- 44:149–156. phyll (38, 39); a trade-off between energy production and 13. Alexei Vazquez A (2017) Overflow Metabolism: From Yeast to Marathon Runners (Academic Press; Cambridge). resistance to external stresses in nitrogen allocation toward 14. Heiden MGV, Cantley LC, Thompson CB (2009) Understanding the Warburg Effect: The photosynthesis and cell walls (40); and a trade-off between Metabolic Requirements of Cell Proliferation. Science 324(5930):1029-1033. 15. Higuera GA, et al. (2011) Patterns of Amino Acid Metabolism by Proliferating Human Mes- ATP synthesis and occupancy of the biomembrane in the co- enchymal Stem Cells. Tissue Eng Part A 18(5–6):654–664. utilization of photosynthesis and glycolysis (41). We expect 16. O’Neill LAJ, Kishton RJ, Rathmell J (2016) A guide to immunometabolism for immunologists. Nat Rev Immunol 16(9):553–565. that these examples in metabolism can also be described by 17. Peebo K, et al. (2015) Proteome reallocation in Escherichia coli with increasing specific the same universal utility landscape and will show Giffen be- growth rate. Mol Biosyst 11:1184–1193. 18. Molenaar D, Van Berlo R, De Ridder D, Teusink B (2009) Shifts in growth strategies reflect havior. tradeoffs in cellular economics. Mol Syst Biol 5(323):1–10. From the perspective of microeconomics, we have provided 19. Mori M, Hwa T, Martin OC, De Martino A, Marinari E (2016) Constrained Allocation Flux Balance Analysis. PLoS Comput Biol 12(6):e1004913. a concrete example of Giffen goods. These results are con- 20. Zhuang K, Vemuri GN, Mahadevan R (2011) Economics of membrane occupancy and sistent with and contribute toward resolving the mechanism respiro-fermentation. Mol Syst Biol 7(500):1–9. 21. Szenk M, Dill KA, de Graff AMR (2017) Why Do Fast-Growing Bacteria Enter Overflow for an economic experiment (23) in which Giffen behavior Metabolism? Testing the Membrane Real Estate Hypothesis. Cell Syst 5(2):95–104. occurs when the income is not high, but cannot be observed 22. Heijman WJM, et al. (2012) New Insights into the Theory of Giffen Goods, eds Heijman WJM, van Mouche P (Springer, Heidelberg). in the case of extreme poverty. In our model, Giffen behav- 23. Jensen BRT, Miller NH (2008) Giffen Behavior and Subsistence Consumption. The Ameri- ior was observed when the income was in the intermediate can Economic Review 98(4):1553–1577. 24. Lehninger AL, Nelson DL, Cox MM (1993) Principles of Biochemistry, second edition (Worth, range at which point the trade-off is operating (see Fig. 2). New York). However, when the income is too low, a good (e.g., the respi- 25. Varian HR (1992) Microeconomic analysis, third edition (W. W. Norton, New York). 26. Lancaster KJ (1966) A new approach to consumer theory. J Polit Econ 74:132–157. ratory pathway) behaves as a perfect substitute. Although one 27. Sørensen PN (2007) Simple Utility Functions with Giffen Demand. Econ Theory can construct other types of utility functions that are plausi- 31(2):367–370. 28. Paczia N, et al. (2012) Extensive exometabolome analysis reveals extended overflow ble for economic behavior, which will show the similar En- metabolism in various microorganisms. Microb Cell Fact 11:1–14. gel curve and Giffen behavior, they are reduced into the same 29. Zeng H, Yang A (2019) Modelling overflow metabolism in Escherichia coli with flux balance analysis incorporating differential proteomic efficiencies of energy pathways. BMC Syst Biol optimization problem as formulated herein (see SI Appendix, 13(1):1–18. Sec. S3). Note that although perfect complementarity does 30. Pfeiffer T, Bonhoeffer S (2004) Evolution of Cross-Feeding in Microbial Populations. Am Nat 163(6):E126–E135. not always exist outside of metabolic systems, Giffen behav- 31. Winer LSP, Wu M (2014) Rapid analysis of glycolytic and oxidative substrate flux of cancer ior can still be observed without perfect complementarity as cells in a microplate. PLoS One 9(10):e109916. 32. Murphy MP, et al. (2008) Mitochondrial uncouplers with an extraordinary dynamic range. long as a trade-off exists and the form of the utility landscape Biochem J 407:129–140. is similar to that of the Leontief type (see SI Appendix, Sec. 33. Flamholz A, Noor E, Bar-Even A, Liebermeister W, Milo R (2013) Glycolytic strategy as a tradeoff between energy yield and protein cost. Proc Natl Acad Sci 110(24):10039–10044. S4 and Fig. S2). 34. Fuhrer T, Fischer E, Sauer U (2005) Experimental identification and quantification of glucose We have paved the road for the field of the microeconomics of metabolism in seven bacterial species. J Bacteriol 187(5):1581–1590. 35. Thomas TD, Ellwood DC, Longyear VM (1979) Change from homo- to heterolactic fermen- metabolism using overflow metabolism and Giffen behavior tation by Streptococcus lactis resulting from glucose limitation in anaerobic chemostat cul- as stepping stones. We hope that close links will beDRAFT revealed tures. J Bacteriol 138:109–117 36. Teusink B, et al. (2006) Analysis of growth of Lactobacillus plantarum WCFS1 on a complex for a variety of other metabolic phenomena and theories in medium using a genome-scale metabolic model. J Biol Chem 281:40041–40048 microeconomics, as shown in this paper, and we expect that 37. Sinsabaugh RL, Moorehead DL (1994) Resource allocation to extracellular enzyme produc- tion: a model for nitrogen and phosphorus control of litter decomposition. Soil Biol Biochem further development in the microeconomics of metabolism 26(10):1305–1311. will bring about a deeper understanding of both biology and 38. Henry HAL, Aarssen LW (1997) On the Relationship between Shade Tolerance and Shade Avoidance Strategies in Woodland Plants. Oikos 80(3):575–582. economics. 39. Zavrelˇ T, et al. (2019) Quantitative insights into the cyanobacterial cell economy. eLife 8:e42508. ACKNOWLEDGEMENTS 40. Feng Y-L, et al. (2009) Evolutionary tradeoffs for nitrogen allocation to photosynthesis versus We would like to acknowledge Chikara Furusawa for helpful discussions and careful cell walls in an invasive plant. Proc Natl Acad Sci 106(6):1853–1856. reading of the manuscript, and Kunihiko Kaneko and Atsushi Kamimura for useful 41. Amthor JS (1995) Higher plant respiration and its relationships to photosynthesis. Ecophys- comments. iology of Photosynthesis, eds Schulze ED, Caldwell MM (Springer, Berlin), pp 71-101.
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6 | bioR‰iv Yamagishi et al. | Microeconomics of metabolism bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
Table 1. Biological and economic meanings of symbols.
Symbol Biological meaning Economic meaning
JC,in Intake flux of carbon source Income JC,r,JC,f Fluxes of carbon in respiratory and fermentation pathways Demand for goods pr,pf Inefficiency of carbon metabolism in respiration and fermentation Price of goods ⁄ Growth rate Utility JE Total flux of ATP production An objective JE,r,JE,f Flux of ATP production by respiration and fermentation JBM Total flux of biomass precursors production Another objective fltot Total amount of the limited resource flr,flf ,flBM Fraction of the limited resource used for respiration, fermentation, and biomass synthesis ‘r,‘f Efficiency of ATP production in respiration and fermentation ‘rÕ ,‘fÕ ,‘BMÕ Occupancy rate of the limited resource for respiration, fermentation, and biomass synthesis sE,sBM StoichiometricDRAFT constants
Yamagishi et al. | Microeconomics of metabolism bioR‰iv | 7 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
12 Supporting Information Text
13 S1. Substitution Effect and Income Effect
14 The influence of a change in price on the demand for goods can be decomposed into an income e ect and a substitution e ect, 15 as mentioned in the main text. This is known as Hicksian decomposition or the Slutsky equation (1). 16 We here define xi(p,I) as the (optimal) demand for good i as a function of the price of goods p and income I. E(p,u) 17 represents the minimum income required for a given utility value u under price p, and thus hi(p,u) xi(p,E(p,u)) is the © 18 smallest demand for good i necessary to achieve the given utility value u. We can di erentiate this with respect to pj and 19 obtain
ˆhi(p,u(p,I)) ˆxi(p,I) ˆxi(p,I) ˆE(p,u(p,I)) 20 = + , ˆpj ˆpj ˆI ˆpj
21 where u(p,I) is the maximal utility with given price p and income I. Note that the last term, ˆE(p,u(p,I))/ˆpj , is simply 22 equal to the demand xj under price p and income I. 23 Accordingly, the change in xi(p,I) due to a change in pj is given by the Slutsky equation:
ˆxi(p,I) ˆhi(p,u(p,I)) ˆxi(p,I) 24 = xj (p,I) . [S1] ˆpj ˆpj ≠ ˆI
25 The first term, ˆhi(p,u)/ˆpj , is the substitution e ect, which is caused by relative changes in the combination of the price values 26 (2). The case of i = j represents the self-substitution e ect, which is proven to be always non-positive (1), i.e., the substitution ˆxi(p,I) 27 e ect never increases the demand for a good when its own price increases. In contrast, the second term, xj (p,I) ˆI ,is 28 the income e ect, which can be either positive or negative. This e ect reflects the change in the demand for goods due to 29 the e ective decrease of the income that is caused by increasing the price of a good. According to Eq. [S1], these two e ects 30 determine the dependence of the demand for goods on the price (Table S2). 31 The substitution e ect is represented as movement of the combination of the demand for goods along an indi erence curve 32 to a point at which the tangent line has a slope that is equal to the ratio of the altered price of goods. Hence, if the utility 33 is given as a Leontief utility function, the substitution e ect is zero within a certain range of the price change due to the 34 indi erentiability of each indi erence curve at the kink. It follows that the substitution e ect of the utility is zero in the case of 35 metabolic systems. Consequently, whether or not a metabolic pathway is a Gi en good depends only on the sign of its income 36 e ect; namely, a metabolic pathway behaves as a Gi en good if its income e ect is negative. In the case of overflow metabolism, 37 when the budget constraint line intersects with the ridgeline (Eq. (5) in the main text), the income e ect is negative for the 38 utility ⁄(JC,r,JC,f ) (Eq. (4) in the main text), and thus Gi en behavior is observed. 39 Note that although the demand for branded goods also increase with their price, they are not Gi en goods. This is because 40 the demand for branded goods increase when the income increase, in other words, their income e ect is positive. Such goods 41 are called Veblen goods in microeconomics (3, 4).
42 S2. Dependence of the optimal strategy on the price
43 The optimal strategy depends on the price pr and pf as well as on the income JC,in. This is because changes in price alter the 44 demand for goods, as described above. Accordingly, if the price pr or pf takes a value larger than 1, the Engel curve (Eq. (6) 45 in the main text) is modified as follows.
(JC,in/pr, 0) , if JC,in prr Æ 0 J J p ˆ ˆ C,in f0 pr C,in f r0 46 JC,r, JC,f = Y f0 p r p , p r0 p f , if pf f0 JC,in prr0 [S2] _ ≠ f 0 ≠ f r ≠ r ≠ 0 Ø Ø ]_3 ? ? 4 ! " (01,f0) , if JC,in2 1pf f0 2 1 2 ! " Ø _ sBM ‘r _ ‘r sBM ‘f ‘f 47 with r0 = fltot/( ‘ s [+ ‘ ) and f0 = fltot/( ‘ s + ‘ ). Here, the optimal (JC,r,JC,f ) for the case pf f0 JC,in prr0 is BMÕ E rÕ BMÕ E fÕ Ø Ø 48 obtained from the intersecting point of the budget constraint line with the ridgeline (Eq. (5) in the main text). Note that Eq. 49 [S2] is identical to Eq. (6) in the main text if pr = pf =1.
50 S3. Generalization of the theory of consumer choice for overflow metabolism
51 Generalization of the proposed model elucidates the necessary and su cient conditions for Gi en behavior, which is applicable 52 to a variety of biological and economic phenomena. Let us define a Leontief utility function
53 u(x ,x ) min(A, B), [S3] 1 2 ©
54 where A(x1,x2)=a1x1 + a2x2 + A0, 55 B(x ,x )=b x + b x + B . ; 1 2 1 1 2 2 0
2 of 7 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.
56 The demand for and price of two goods are represented by x1,x2 and p1,p2, respectively. The budget constraint line is thus 57 I = p1x1 + p2x2. 58 With respect to the utility [S3], the model proposed for overflow metabolism in the main text corresponds to the case
a1 a2 ++ 59 where the signs of the parameters are given as sgn = , while other sets of parameters also evoke overflow b1 b2 3 4 3≠≠4 60 metabolism and Gi en behavior. For instance, a trade-o between the production e ciencies of ATP and other molecules a1 a2 ++ 61 required for growth (5) can cause overflow are given by sgn = (Fig. S1). Indeed, the Leontief utility b1 b2 ++ 3 4 3 4 a1 a2 ++ 62 function with sgn = was recently reported to demonstrate Gi en behavior in the context of microeconomics b1 b2 +0 3 4 3 4 63 (6, 7), which is a special case of the utility function [S3]. 64 In this section, we demonstrate that the optimization problem of the above utility [S3] with every set of parameters will 65 always be reduced into the optimization problem with an identical structure to that shown in the main text, as long as it 66 shows Gi en behavior. Hence, the two conditions, i.e., complementarity and trade-o , are required in all cases. Moreover, we 67 expand the trade-o to include the e ect of the price. 68 First, we consider the simple case where p1 = p2 =1. We here assume a trade-o between goods 1 and 2 as a1 >a2 and 69 b1
75 a1 >b1,a2 >b2,B0 >A0 [S5]
76 are satisfied. Note that this is the condition for a comparative advantage (8). 77 If both b1 and b2 are non-positive and a1 and a2 are positive, the condition [S5] is autonomously satisfied. Then, the income 78 e ect is negative and Gi en behavior is observed, as shown in the main text (see Fig. 2). 79 Even if b or b is positive, the condition [S5] can be satisfied as long as the condition sgn(a b )=sgn(a b ) is satisfied, 1 2 1 ≠ 1 2 ≠ 2 80 as in Fig. S1. As discussed below, even such cases can be reduced to an optimization problem with the same universal structure 81 as demonstrated in the main text. 82 When Gi en behavior can occur (i.e., the condition [S5] is satisfied), we can take a real number c such that a2 >c>b2, 83 and the utility [S3] can be represented as
84 u(x ,x )=min(A, B)=c(x + x )+A +min((a c)x +(a c)x , (b c)x +(b c)x + B A ) 1 2 1 2 0 1 ≠ 1 2 ≠ 2 1 ≠ 1 2 ≠ 2 0 ≠ 0 85 = cI + A0 +min(AÕ,BÕ), [S6]
86 where AÕ(x1,x2)=(a1 c)x1 +(a2 c)x2 = a1Õ x1 + a2Õ x2, 87 ≠ ≠ BÕ(x ,x )=(b c)x +(b c)x + B A = bÕ x + bÕ x + BÕ . ; 1 2 1 ≠ 1 2 ≠ 2 0 ≠ 0 1 1 2 2 0 88 Here, aÕ = a c>a c = aÕ , bÕ = b c 0 are satisfied from the condition [S5]. 1 1 ≠ 2 ≠ 2 1 1 ≠ 2 ≠ 2 0 0 ≠ 0 89 Because only the last term, min(AÕ,BÕ), depends on allocation of the income to goods, the generalized model is reduced to the 90 same optimization problem as proposed in the main text as long as it shows Gi en behavior. 91 A similar argument remains valid even when considering changes in the price, although in this case the definition of trade-o s 92 needs to be expanded to include the e ect of the price. In this case, a real number c can be taken such that a2/p2 >c>b2/p2. 93 Then, the utility [S3] can be represented as
94 u(x ,x )=c(p x + p x )+A +min((a cp )x +(a cp )x , (b cp )x +(b cp )x + B A ) 1 2 1 1 2 2 0 1 ≠ 1 1 2 ≠ 2 2 1 ≠ 1 1 2 ≠ 2 2 0 ≠ 0 95 = cI + A0 +min(AÕ,BÕ), [S7]
96 where AÕ(x1,x2)=(a1 cp1)x1 +(a2 cp2)x2 = a1Õ x1 + a2Õ x2, 97 ≠ ≠ BÕ(x ,x )=(b cp )x +(b cp )x + B A = bÕ x + bÕ x + BÕ . ; 1 2 1 ≠ 1 1 2 ≠ 2 2 0 ≠ 0 1 1 2 2 0 98 If a /a >p/p >b/b holds, aÕ = a cp >a cp = aÕ , bÕ = b cp 0 1 2 1 2 1 2 1 1 ≠ 1 2 ≠ 2 2 1 1 ≠ 1 2 ≠ 2 2 0 0 ≠ 0 99 are satisfied. Again, the generalized models showing Gi en behavior are reduced to the optimization problem with the same 100 universal structure as demonstrated in the main text. 101 Of note, if the di erence of the price between goods 1 and 2 is so large that p1/p2 is larger than a1/a2 or smaller than b1/b2, 102 Gi en behavior disappears even when the condition [S5] and the trade-o of e ciencies to produce A and B (i.e., a1 >a2 103 and b1 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 3 of 7 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 105 between the production of A and B. Hence, the condition for a trade-o is rewritten as a1/p1 >a2/p2 and b1/p1 110 S4. Giffen behavior requires complementarity but not perfect complementarity 111 Based on biological considerations, we introduced the Leontief utility function ⁄(JC,r,JC,f ) (Eq. (4) in the main text) that has 112 perfect complementarity. However, Gi en behavior can be observed with utility functions showing only partial complementarity 113 as long as the substitution e ect is su ciently small. 114 An example of such utility functions is 1 ‘r ‘f ≠ 115 u(x1,x2) sE / (‘rx1 + ‘f x2)+sBM / fltot x1 x2 ‘BMÕ . [S8] © ≠ ‘Õr ≠ ‘Õ 5 3 f 4 6 116 The landscape of this utility function (Fig. S2A) is similar to that of ⁄(JC,r,JC,f ) (Fig. 2B in the main text). As shown in Fig. 117 S2B, x1 with the utility [S8] shows Gi en behavior within a range of the price of good 1, p1. 118 Another example of utility functions showing Gi en behavior is u(JC,r,JC,f ) JE H(JBM JBM,min), where H( ) is the © ≠ · 119 Heaviside step function. Table S1. General correspondence between metabolism and microeconomics Metabolism Microeconomics Intake of nutrient Income Metabolic pathways Goods Nutrient allocation Demand for goods Loss of intermediate metabolites Price of goods (inefficiency of metabolism from nutrient to products) (inefficiency of conversion from income to goods) Growth rate (Biomass synthesis rate) Utility Table S2. Microeconomic properties of goods. Type of goods Self-substitution effect Income effect Income ↑ Price ↑ Normal good non-positive positive Demand ↑ Demand ↓ Inferior good non-positive (slightly) negative Demand ↓ Demand ↓ Giffen good non-positive negative Demand ↓ Demand ↑ 4 of 7 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted 0.4April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) isA the author/funder, who has granted bioRxivB a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 0.3 2 ^ x 0.2 u , 1 0.1 x ^ x2 0.0 1.0 1.2 1.4 1.6 1.8 x1 p1 Indifference Curves A u 0.45 0.8 0.8 B 0.39 2 0.6 x ^ 0.33 , , 0.4 0.6 0.27 1 x ^ 0.2 0.21 2 0.40.0 x x2 0.15 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.3 C 0.09 B 0.2 , 0.03 0.1 0.2 -0.03 A 0.0 -0.09 -0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 J_{IC, in} x1 x1 Fig. S1. Example of a generalized model with the coefficients a1,a2,b1,b2 > 0. (A) Contour map. Indifference curves are shown as gray lines, and the dashed line depicts the ridgeline [S4] on which A(x1,x2)=B(x1,x2) holds. The background color exhibits the value of the Leontief-type utility u(x1,x2)=min(A, B)= min(a1x1 + a2x2 + A0,b1x1 + b2x2 + B0) (Eq. [S3]). (B) The Engel curve: x1 (cyan line) and x2 (magenta line) in the optimal solutions are plotted against the income I. (C) A (blue line) and B (dark-red line) in the optimal solutions are plotted against the income I. In (B-C), the price p1 and p2 are set at unity. In this example, the coefficients are set such that a >a >b >b and B >A : a =1,a =0.5,b =0.25,b =0.3,A = 0.15,B =0. Since there is a tradeoff between the 1 2 2 1 0 0 1 2 1 2 0 ≠ 0 production efficiencies of A and B and the condition [S5] is satisfied, Giffen behavior is observed. Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 5 of 7 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. A B 0.4 0.3 2 ^ x 0.2 u , 1 0.1 x ^ x2 0.0 1.0 1.2 1.4 1.6 1.8 x1 p1 Fig. S2. Example of utility functions showing Giffen behavior without perfect complementarity. The utility function u(x1,x2) is given by Eq. [S8]. (A) Utility landscape of u(x1,x2) and (B) demand curves for goods 1 (light-blue dots) and 2 (pink dots). The optimal strategies (xˆ1, xˆ2) were numerically calculated with the parameters ‘r =0.75, ‘f =0.4, ‘rÕ =0.5, ‘fÕ =1,JC,in =0.4; all other parameters (i.e., fltot, ‘BMÕ , sE , sBM , and p2) were set at unity. Indifference Curves A u 0.476 0.8 0.8 B 0.420 2 0.6 x ^ 0.364 , , 0.4 0.6 0.308 1 x 6 of 7 Jumpei^ F.0.2 Yamagishi and Tetsuhiro S. Hatakeyama 0.252 2 0.40.0 x x2 0.196 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.3 C 0.140 0.2 B 0.084 , 0.2 0.1 0.028 A 0.0 -0.028 -0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 J_{IC, in} x1 x1 bioRxiv preprint doi: https://doi.org/10.1101/613166; this version posted April 18, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 120 References 121 1. Varian HR (1992) Microeconomic analysis, third edition (W. W. Norton, New York). 122 2. Mankiw NG (2016) The Theory of Consumer Choice. Principles of economics, eighth edition (Cengage Learning, Boston), 123 pp 425–450. 124 3. 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