Supplementary Note 1. Substitution Effect and Income Effect

1 Supplementary Note 1. Substitution Effect and Income Effect

2 The influence of a change in price on the demand for goods can be decomposed into an income effect and a substitution effect 3 as mentioned in the main text. This is known as Hicksian decomposition or the Slutsky equation (1). 4 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) 5 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 6 smallest demand for good i necessary to achieve the given utility value u. We can differentiate this with respect to pj and 7 obtain

∂hi(p, uˆ(p,I)) ∂xi(p,I) ∂xi(p,I) ∂E(p, uˆ(p,I)) 8 = + , ∂pj ∂pj ∂I ∂pj

9 where uˆ(p,I) is the maximal utility with the given price p and income I. Note that the last term, ∂E(p, u(p,I))/∂pj , is simply 10 equal to the demand xj under price p and income I. 11 Accordingly, the change in xi(p,I) due to a change in pj is given by the Slutsky equation:

∂xi(p,I) ∂hi(p, uˆ(p,I)) ∂xi(p,I) 12 = − xj (p,I) . [S1] ∂pj ∂pj ∂I

13 The first term, ∂hi(p, u)/∂pj , is the substitution effect, which is caused by relative changes in the combination of price values 14 (2). The case of i = j represents the self-substitution effect, which is proven to be always non-positive (1), i.e., the substitution ∂xi(p,I) 15 effect never increases the demand for a good when its own price increases. In contrast, the second term, xj (p,I) ∂I , is the 16 income effect, which can be either positive or negative. This effect reflects the change in the demand for goods due to the 17 effective decrease of income that is caused by increasing the price of a good. According to Eq. [S1], these two effects determine 18 the dependence of the demand for goods on the price (see also the Box Table in the main text).

19 The substitution effect is represented as movement of the combination of the demand for goods along an indifference curve 20 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 21 is given as a Leontief utility function, the substitution effect is zero within a certain range of the price change due to the 22 indifferentiability of each indifference curve at the kink. It follows that the substitution effect of the utility is zero in the case of 23 metabolic systems. Consequently, whether or not a metabolic pathway is a Giffen good depends only on the sign of its income 24 effect; namely, a metabolic pathway behaves as a Giffen good if its income effect is negative. In the case of overflow metabolism, 25 when the budget constraint line intersects with the ridgeline (Eq. [2] in the main text), the income effect is negative for the 26 utility λ(JC,e,JC,g), and thus Giffen behaviour is observed. 27 Note that although the demand for branded goods also increases with their price, they are not Giffen goods. This is because 28 the demand for branded goods increases with the income, in other words, their income effect is positive. Such goods are called 29 Veblen goods in microeconomics (3,4).

30 Supplementary Note 2. The theory of consumer choice for overﬂow metabolism

31 The general theory for two goods and two complementary “objectives” elucidates the minimal requirements for Giﬀen behaviour, 32 which is applicable to a variety of phenomena in biology and economics.

33 Let us deﬁne a Leontief utility function 34 u(x1, x2) ≡ min(A, B), [S2]

35 where two complementary objectives, A and B, are deﬁned as A(x1, x2) = a1x1 + a2x2 + A0, 36 B(x1, x2) = b1x1 + b2x2 + B0.

37 The demand for and price of two goods are represented by x1, x2 and p1, p2, respectively. The budget constraint line is thus 38 I = p1x1 + p2x2. 39 With respect to utility [S2], the model proposed for overﬂow metabolism in the main text corresponds to the case where the a1 a2 + + 40 signs of the parameters are given as sgn = , while other sets of parameters also evoke overﬂow metabolism b1 b2 − −

41 and Giffen behaviour. For instance, a trade-off between the efficiencies of producing different molecules such as ATP and a1 a2 + + 42 NADPH (5) can also cause overflow metabolism, where the parameters are given by sgn = (Supplementary b1 b2 + + a1 a2 + + 43 Figure2). Indeed, the Leontief utility function with sgn = was recently reported to demonstrate Giffen b1 b2 + 0

44 behaviour in the context of microeconomics (6,7), which is a special case of the utility function [S2].

45 In this section, we demonstrate that the optimization problem of the above utility [S2] with every set of parameters will 46 always be reduced to the optimization problem with an identical structure to that shown in the main text, as long as it shows 47 Giffen behaviour. Hence, the two conditions, i.e., complementarity and trade-off, are required in all cases. Moreover, we expand 48 the trade-off to include the effect of price.

Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 1 of9 49 First, we consider the simple case where p1 = p2 = 1. We here assume a trade-off between goods 1 and 2 as a1 > a2 and 50 b1 < b2, whereas without the trade-off, the optimal strategy is to use either good 1 or 2 only. Owing to the trade-off, the utility 51 is maximized on the ridgeline a1 − b1 A0 − B0 52 x2 = − x1 − [S3] a2 − b2 a2 − b2

53 where A(x1, x2) = B(x1, x2) holds. Giffen behaviour can be observed when this ridgeline has a negative slope and exists on the 54 first quadrant, i.e., when −(a1 − b1)/(a2 − b2) < 0 and (B0 − A0)/(a2 − b2) > 0 hold. 55 From symmetry between A(x1, x2) and B(x1, x2), it is sufficient to consider the situation where

56 a1 > b1, a2 > b2,B0 > A0 [S4]

57 are satisfied. Note that this is the condition for a comparative advantage (8). 58 If both b1 and b2 are non-positive and a1 and a2 are positive, the condition [S4] is autonomously satisfied. Then, the income 59 effect is negative and Giffen behaviour is observed, as shown in the main text (see Fig. 1). 60 Even if b1 or b2 is positive, the condition [S4] can be satisfied as long as the condition sgn(a1 − b1) = sgn(a2 − b2) is satisfied, 61 as in Supplementary Figure2. As discussed below, even such cases can be reduced to an optimization problem with the same 62 universal structure demonstrated in the main text. 63 When Giffen behaviour can occur (i.e., the condition [S4] is satisfied), we can take a real number c such that a2 > c > b2, 64 and the utility [S2] can be represented as

65 u(x1, x2) = min(A, B) = c(x1 + x2) + A0 + min ((a1 − c)x1 + (a2 − c)x2, (b1 − c)x1 + (b2 − c)x2 + B0 − A0) 0 0 66 = cI + A0 + min(A ,B ), [S5]

67 where 0 0 0 A (x1, x2) = (a1 − c)x1 + (a2 − c)x2 = a1x1 + a2x2, 68 0 0 0 0 B (x1, x2) = (b1 − c)x1 + (b2 − c)x2 + B0 − A0 = b1x1 + b2x2 + B0.

0 0 0 0 0 69 Here, a1 = a1 − c > a2 − c = a2, b1 = b1 − c < b2 − c = b2 < 0, and B0 = B0 − A0 > 0 are satisﬁed from the condition [S4]. 0 0 70 Because only the last term, min(A ,B ), depends on allocation of the income to goods, the generalized model is reduced to the 71 same optimization problem as proposed in the main text as long as it shows Giﬀen behaviour.

72 A similar argument remains valid even when considering changes in price, although in this case the definition of trade-offs 73 needs to be expanded to include the effect of price. In this case, a real number c can be taken such that a2/p2 > c > b2/p2. 74 Then, the utility [S2] can be represented as

75 u(x1, x2) = c(p1x1 + p2x2) + A0 + min ((a1 − cp1)x1 + (a2 − cp2)x2, (b1 − cp1)x1 + (b2 − cp2)x2 + B0 − A0) 0 0 76 = cI + A0 + min(A ,B ), [S6]

77 where 0 0 0 A (x1, x2) = (a1 − cp1)x1 + (a2 − cp2)x2 = a1x1 + a2x2, 78 0 0 0 0 B (x1, x2) = (b1 − cp1)x1 + (b2 − cp2)x2 + B0 − A0 = b1x1 + b2x2 + B0.

0 0 0 0 0 79 If a1/a2 > p1/p2 > b1/b2 holds, a1 = a1 − cp1 > a2 − cp2 = a2, b1 = b1 − cp1 < b2 − cp2 = b2 < 0, and B0 = B0 − A0 > 0 are 80 satisfied. Again, the generalized models showing Giffen behaviour are reduced to the optimization problem with the same 81 universal structure as demonstrated in the main text. 82 Of note, if the difference of the price between goods 1 and 2 is so large that p1/p2 is larger than a1/a2 or smaller than b1/b2, 83 Giffen behaviour disappears even when the condition [S4] and the trade-off of efficiencies to produce A and B (i.e., a1 > a2 84 and b1 < b2) hold. In this case, we need to consider a trade-off including the price. If p1/p2 is larger than a1/a2, a unit of 85 demand for good 2 produces more A than that for good 1 even when a1 is higher than a2. That is, there is no longer a trade-off 86 between the production of A and B. Hence, the condition for a trade-off is rewritten as a1/p1 > a2/p2 and b1/p1 < b2/p2 (or 87 a1/p1 < a2/p2 and b1/p1 > b2/p2). 88 Intuitively, the trade-off including the price corresponds to rescaling of the utility landscape so that the slope of the budget p1 89 constraint line, − , becomes equal to −1. If there is a trade-off between A and B in the rescaled landscape (i.e., a1/p1 > a2/p2 p2 90 and b1/p1 < b2/p2, or a1/p1 < a2/p2 and b1/p1 > b2/p2), Giffen behaviour can be observed.

91 Supplementary Note 3. Estimation of parameters 0 0 0 92 In the main text, we used an arbitrarily chosen set of parameters (e = 0.75, g = 0.45, e = 0.5, g = 0.7, ρtot = 0.6, BM = 93 1, sE = sBM = 0.5) in order to demonstrate a mathematical structure of the landscape clearly (see also Supplementary Table1 94 for meaning of the symbols). However, our results do not depend on the precise values of the parameters as long as the relative 0 0 95 values of parameters satisfy the conditions described in the main text: e > g and e < g. 96 One can estimate the parameter set in actual cells and show that the estimated parameters indeed satisfy the conditions for 97 overﬂow metabolism. The details are given below (see also Supplementary Table2).

98 First, the intake rate of the carbon source (e.g., glucose) is the order of 1 [mMol glucose/gDW] (9).

2 of9 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 99 According to stoichiometry, mitochondrial respiration can generate 36 ATP molecules per 1 molecule of glucose at most, 100 while in reality it generates about 30 − 32 ATP molecules per 1 molecule of glucose (10, 11). In contrast, aerobic glycolysis 101 generates 2 ATP and 4 NADH, in total, from 1 glucose, and fermentation generates 2 ATP more. Besides, in the aerobic 102 environment, 1 NADH molecule can be converted into, at most, 2.5 ATP molecules; thus, at most, 12 or 14 ATP molecules 103 can be generated from 1 glucose by aerobic glycolysis or fermentation, respectively (10, 12). Then, the values of e and g 104 are dependent on the organism in question, but a requirement for overflow metabolism, e > g, is always satisfied: e ' 32 105 [mMol ATP/ mMol glucose] and g = 2 ∼ 12 [mMol ATP/ mMol glucose]. Notably, the decrease in the optimal flux JˆC,e 106 against nutrient supply JC,in can be very gentle (whereas glycolysis JˆC,g increases steeply), e.g., in the situation where e/g is 107 relatively large (i.e., NADH produced in glycolysis is not converted to ATP consuming oxygen), as experimentally observed 108 (11, 13). 0 0 109 ρtot, e, and g depend on the nature of the limited resource in question. When the limited resource ρtot is the total volume 110 of mitochondria or solvent capacity of mitochondria in cancer cells, muscle cells, yeasts, et al.:the intracellular volume for −3 0 3 111 proteins per cellular dry weight is ρtot ' 3 × 10 [L/gDW] and the approximate values of the other parameters are e ' 3 × 10 0 4 0 −3 112 [mMol ATP/L/hour], g ' 9 × 10 [mMol ATP/L/hour], and sBM /BM ' 10 [hour × L/gDW] (11, 14). Alternatively, when −1 113 ρtot is the fraction of enzymes for growth in E. coli: the approximate values of the parameters are estimated as ρtot ' 2 × 10 , 0 2 0 2 0 −1 114 e ' 4 × 10 [mMol ATP/gDW/hour] and g ' 8 × 10 [mMol ATP/gDW/hour], and sBM /BM ' 10 [hour] (12). + 115 When the balance between cellular redox state and energy demand is considered (15), ρtot is the maximal flux of NAD 2 + 0 116 produced by the other cellular processes and is estimated as ρtot ' 3 × 10 [mMol NAD /gWW/hour]. Then, ρg = JE,g/g 0 0 0 + 117 is 0 (i.e., g and e/g can be regarded as infinity and zero, respectively) because glycolysis does not consume NAD . In + 0 + 118 contrast, respiration consumes 1 NAD per 1 glucose, or per e ATP molecules; thus, e equals e[mMol ATP/mMol NAD ]. + 0 2 + 119 In addition, the amount of required NAD per cellular biomass is estimated as sBM /BM ' 2 × 10 [mMol NAD /gWW]. 0 0 120 Again, the other requirement for overflow metabolism, e < g, is satisfied in all the above cases. 121 Finally, pe and pg approximately equal 1 without the administration of drugs because, at most, only ∼ 10% carbon is usually 122 leaked as intermediates of central metabolism (16). In contrast, under the existence of drug such as uncouplers of respiration, 123 pe is substantially larger than 1, quantified as the inverse of the ATP yield, i.e., the fraction of the non-dissipated proton.

124 Supplementary Note 4. Dependence of the optimal strategy on the income and price

125 The growth rate λ(JC,e,JC,g) takes the value λ˜, the contour (i.e., indiﬀerence curve in microeconomics) is given as a two-valued 126 function, as shown in Fig. 1c in the main text:

( e sE 1 1 − JC,e + λ,˜ if JE ≤ JBM g g sE sBM 127 JC,g = 0 0 [S7] e g g sBM ˜ 1 1 − 0 JC,e + ρtot − 0 λ . if s JE ≥ s JBM g e g BM E BM

128 The growth rate is maximized at the tangent point of the budget constraint line (Eq. [1]) to the contour with the largest 129 growth rate.

130 In the case of pe = pg = 1, the dependence of the optimal strategy (JˆC,e, JˆC,g) on JC,in, called the Engel curve in 131 microeconomics, is calculated (see also Fig. 1c in the main text):

(J , 0) , if J ≤ e  C,in C,in 0 ˆ ˆ e0 g0 132 JC,e, JC,g = g −e (g0 − JC,in), g −e (JC,in − e0) , if g0 ≥ JC,in ≥ e0 [S8]  0 0 0 0 (0, g0) . if JC,in ≥ g0

133 Changes in price alter the demand for goods, and thus the optimal strategy depends on the price pe and pg as well as on the 134 income JC,in. Accordingly, if the price pe or pg takes a value larger than 1, the Engel curve is generalized as follows.  (JC,in/pe, 0) , if JC,in ≤ pee0  JC,in g0 pe JC,in pg e0 135 JˆC,e, JˆC,g = g0 − − , − e0 − , if pgg0 ≥ JC,in ≥ pee0 [S9] pg e0 pg pe pe g0  (0, g0) , if JC,in ≥ pgg0

sBM e e sBM g g 136 with e0 = ρtot/( 0 s + 0 ) and g0 = ρtot/( 0 s + 0 ). Here, the optimal (JC,e,JC,g) for the case pgg0 ≥ JC,in ≥ pee0 is BM E e BM E g 137 obtained from the intersecting point of the budget constraint line with the ridgeline (Eq. [2] in the main text). Note that Eq. 138 [S8] is identical to Eq. [S7] if pe = pg = 1 is satisﬁed.

139 Supplementary Note 5. Giffen behaviour requires complementarity but not perfect complementarity

140 Based on biological considerations, we introduced the Leontief utility function λ(JC,e,JC,g) that has perfect complementarity 141 in the main text. However, Giffen behaviour can be observed for utility functions with only partial complementarity, as long as 142 the substitution effect is sufficiently small.

Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 3 of9 143 An example of such utility functions is −1 e g 0 144 u(x1, x2) ≡ sE / (ex1 + gx2) + sBM / ρtot − 0 x1 − 0 x2 BM . [S10] e g

145 The landscape of this utility function (Supplementary Figure3A) is similar to that of λ(JC,e,JC,g) (Fig. 1a in the main text). 146 As shown in Supplementary Figure3B, x1 with the utility [S9] showing Giﬀen behaviour within a range of the price of the 147 good 1, p1. 148 Another example of utility functions showing Giﬀen behaviour is u(JC,e,JC,g) ≡ JE H(JBM − JBM,min), where H(·) is the 149 Heaviside step function.

4 of9 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama Supplementary Table 1. Biological and economic meanings of symbols.

Symbol Biological meaning Economic meaning

JC,in Intake flux of carbon source Income JC,e,JC,g Fluxes of carbon in oxidative phosphorylation and glycolysis Demand for goods pe, pg Inefficiency of metabolism in oxidative phosphorylation and glycolysis Price of goods λ Growth rate Utility JE Total flux of ATP production An objective JE,e,JE,g Flux of ATP production by oxidative phosphorylation and glycolysis JBM Total flux of biomass precursors production Another objective ρtot Total amount of the limited resource ρe, ρg, ρBM Fraction of the limited resource used for oxidative phosphorylation, glycolysis, and biomass synthesis e, g Stoichiometric efficiency of ATP production in oxidative phosphorylation and glycolysis 0 0 0 e, g, BM Occupancy rate of the limited resource for oxidative phosphorylation, glycolysis, and biomass synthesis sE , sBM Stoichiometric constants

Supplementary Table 2. Estimated parameters

Parameter Value Reference

e ∼ 30 [mMol ATP/mMol glucose] Refs. (10) g 2 ∼ 14 [mMol ATP/mMol glucose] Refs. (10) pe & 1 Ref. (16) pg ∼ 1 Ref. (16) 1 sE 6 × 10 [mMol ATP/gDW] Ref. (17) Solvent capacity hypothesis Ref. (11, 14) −3 ρtot 3 × 10 [L/gDW] 0 4 e 2 × 10 [mMol ATP/L/hour] 0 4 g 9 × 10 [mMol ATP/L/hour] 0 −3 sBM /BM 10 [hour × L/gDW] Proteome allocation hypothesis Ref. (12) −1 ρtot 2 × 10 0 2 e 4 × 10 [mMol ATP/gDW/hour] 0 2 g 8 × 10 [mMol ATP/gDW/hour] 0 −1 sBM /BM 10 [hour] Redox balance hypothesis Ref. (15) 2 + ρtot 3 × 10 [mMol NAD /gWW/hour] 0 + e e [mMol ATP/mMol NAD ] 0 ρg(∝ 1/g) 0 0 2 + sBM /BM 2 × 10 [mMol NAD /gWW]

Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 5 of9 JC,in<εee0/εg 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4

0.3 0.3C,g 0.3 C,g C,g J

0.2 0.2J

J 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.1JC,e0.2 0.3 0.4 0.0 0.1J0.2C,e0.3 0.4 0.0 0.1JC,e0.2 0.3 0.4 0.5 0.4 J＾C,g ＾λ 0.3 ＾JC,e 0.2 0.1 JC,in/e0 εe/εf 0.00 1.01 1.2 1.4pe/p1.6g 1.8 2.0

Supplementary Figure 1. Dependence of the optimal strategy (Jˆ , Jˆ ) on the price of electron transport chain p with J < e e . J > e and p = 1 are C,e C,g e C,in g 0 C,in 0 g ﬁxed here. The cyan, magenta, and green curves depict JˆC,e, JˆC,g , and λˆ ≡ λ(JˆC,e, JˆC,g ), respectively (scaled with different units). Top panels depict the contour maps and the budget constraint lines for regime (I) pe < e/g (light-green area) and JC,in ≤ pee0 and pe < e/g (light-blue area) and (II) pe > e/g (yellow area).

6 of9 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 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 Supplementary Figure 2. Example of a generalized model with the coefficients a1, a2, b1, b2 > 0. (A) Contour map. Indifference curves are shown as grey lines, and the dashed line depicts the ridgeline [S3] on which A(x1, x2) = B(x1, x2) holds. The background colour exhibits the value of the Leontief-type utility u(x1, x2) = min(A, B) = min(a1x1 + a2x2 + A0, b1x1 + b2x2 + B0) (Eq. [S2]). (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 a1 > a2 > b2 > b1 and B0 > A0: a1 = 1, a2 = 0.5, b1 = 0.25, b2 = 0.3,A0 = −0.15,B0 = 0. Since there is a tradeoff between the production efficiencies of A and B and the condition [S4] is satisfied, Giffen behaviour is observed.

Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 7 of9 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

Supplementary Figure 3. Example of utility functions showing Giffen behaviour without perfect complementarity. The utility function u(x1, x2) is given by Eq. [S9]. (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 0 0 0 parameters e = 0.75, g = 0.4, e = 0.5, g = 1,JC,in = 0.4; all other parameters (i.e., ρtot, BM , sE , sBM , and p2) were set at unity.

8 of9 Jumpei F. Yamagishi and Tetsuhiro S. Hatakeyama 150 Supplementary References

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