The main determinants of evolution rate

To understand the main drivers of evolution we must first understand the protein properties that mostly determine evolutionary rates in proteins on longer time scales. This rate is also used to classify and predict the functional impact of human variants e.g. in relation to disease (Glaser et al., Reference Glaser, Pupko, Paz, Bell, Bechor-Shental, Martz and Ben-Tal2003; Capra and Singh, Reference Capra and Singh2007; Thusberg et al., Reference Thusberg, Olatubosun and Vihinen2011; Tang et al., Reference Tang, Dehury and Kepp2019). Table 1 provides an overview of the most important relationships between a protein's properties and its evolution rate. As easily verified from sequence alignment, active sites in proteins are highly conserved due to strong purifying selection, because random deleterious mutations impair fitness more in highly optimized parts of the protein. Related to this, solvent-exposed sites in contrast evolve faster than average, consistent with their typically smaller functional and structural effects on the overall protein (Overington et al., Reference Overington, Donnelly, Johnson, Šali and Blundell1992; Goldman et al., Reference Goldman, Thorne and Jones1998; Ramsey et al., Reference Ramsey, Scherrer, Zhou and Wilke2011).

Table 1.

Important correlators of the evolution rate and size of proteins

The strongest descriptor of evolutionary rate is protein abundance or equally, mRNA levels, as these correlate (Gygi et al., Reference Gygi, Rochon, Franza and Aebersold1999); it typically spans 5–7 orders of magnitude in eukaryotes (Jansen and Gerstein, Reference Jansen and Gerstein2000; Ghaemmaghami et al., Reference Ghaemmaghami, Huh, Bower, Howson, Belle, Dephoure, O'Shea and Weissman2003; Beck et al., Reference Beck, Schmidt, Malmstroem, Claassen, Ori, Szymborska, Herzog, Rinner, Ellenberg and Aebersold2011; Milo Reference Milo2013). High expression is associated with slower protein evolution in both prokaryotes (Sharp, Reference Sharp1991; Rocha and Danchin, Reference Rocha and Danchin2004) and eukaryotes (Pál et al., Reference Pál, Papp and Hurst2001), including mammals (Jordan et al., Reference Jordan, Mariño-Ramírez, Wolf and Koonin2004; Zhang and Li Reference Zhang and Li2004), a phenomenon known as the expression-rate (E-R) anti-correlation (Drummond et al., Reference Drummond, Bloom, Adami, Wilke and Arnold2005; Bloom et al., Reference Bloom, Drummond, Arnold and Wilke2006a). Protein expression may explain half of the evolutionary rate variation in yeast (Drummond et al., Reference Drummond, Raval and Wilke2006) indicating a universal driving force of evolution. This remarkable relationship has been studied using many biophysical models focusing on protein stability, misfolding avoidance, and flexibility (Lobkovsky et al., Reference Lobkovsky, Wolf and Koonin2010; Geiler-Samerotte et al., Reference Geiler-Samerotte, Dion, Budnik, Wang, Hartl and Drummond2011; Wylie and Shakhnovich, Reference Wylie and Shakhnovich2011; Liberles et al., Reference Liberles, Teichmann, Bahar, Bastolla, Bloom, Bornberg‐Bauer, Colwell, De Koning, Dokholyan, Echave and Elofsson2012; Serohijos et al., Reference Serohijos, Rimas and Shakhnovich2012; Yang et al., Reference Yang, Liao, Zhuang and Zhang2012; Kepp and Dasmeh, Reference Kepp and Dasmeh2014; Sikosek and Chan, Reference Sikosek and Chan2014). All-else-being-equal, a protein's fitness impact should be proportional to its cellular abundance regardless of the specific selection pressure. Thus, any fitness function that scales with protein abundance may seem reasonable. Such models can explain about 60% of site-variations in the evolutionary rate (McInerney, Reference McInerney2006; Echave et al., Reference Echave, Spielman and Wilke2016). Protein stability has mainly been related to fitness via the copy number of misfolded proteins, assuming one-step unfolding (Serohijos et al., Reference Serohijos, Rimas and Shakhnovich2012; Dasmeh et al., Reference Dasmeh, Serohijos, Kepp and Shakhnovich2014a). These ideas are expanded further below. To summarize the tendencies of Table 1, compared to the average protein, the slowly evolving protein tends to be highly expressed, intracellular, smaller than average, and have a higher functional density, i.e. more important sites relatively to its size.

The E-R anti-correlation has been explained (Drummond and Wilke, Reference Drummond and Wilke2008, Reference Drummond and Wilke2009) as a selection against inefficient translation leading to toxic misfolded proteins, a theory originally proposed by Kurland and Ehrenberg (Ehrenberg and Kurland, Reference Ehrenberg and Kurland1984; Kurland and Ehrenberg, Reference Kurland, Ehrenberg, Cohn and Moldave1984, Reference Kurland and Ehrenberg1987). Protein synthesis is inherently error-prone, and translation operates with typical missense error rates of 1/1000 to 1/10 000 (Kurland and Ehrenberg, Reference Kurland and Ehrenberg1987). Considering the typical lengths (~100–1000) and total abundance of proteins (10⁸) in eukaryotic cells, one can expect 10¹⁰–10¹¹ protein-incorporated amino acids to exist at any time. Without error correction this could imply the constant existence of 10⁶–10⁸ erroneous amino acids in a typical eukaryote cell. This would make translation-error induced proteome variation of similar importance as typical, mostly heterozygote, natural sequence variation in a population. This of course raises the question how much of the actual observed proteome variation is due to genetic inheritance, somatic mutations, and translation errors. To be sure, one needs to sequence each gene and protein many times for several cells. Regardless of this complication, it is clear that the proteome varies much more in composition than implied by genetic variance alone.

Considering this, because the typical non-native residue destabilizes by ~5 kJ mol⁻¹ (Tokuriki et al., Reference Tokuriki, Stricher, Schymkowitz, Serrano and Tawfik2007), as much as 10% of a proteome could be less stable than commonly assumed purely from wild-type sequence. For a cell with 10⁸ proteins, this implies that 10⁷ protein copies are randomly destabilized and subject to higher turnover that expected from their wild-type sequence. Post-translational modifications and specific degrons further diversify the proteome and complicate turnover further. Considering this, the additional destabilization from new arising mutations will aggravate costs only if the affected protein is quite abundant or subject to high turnover.

If the misfolded protein is selected against, regardless of the reason, highly expressed proteins are under stronger selection pressure because the copy number of misfolded proteins U_i scales with the total abundance of the protein A_i. Drummond and co-workers suggested a fitness function Φ depending exponentially on the total copy number of all misfolded proteins U = ∑U_i, with an unknown scaling constant c (Drummond and Wilke, Reference Drummond and Wilke2008):

(1)$${\it \Phi} \propto \exp \lpar {-cU} \rpar $$

The constant c can be derived from fundamental and simple assumptions and related directly to the cost of protein turnover, as discussed below.

The theory of proteome cost minimization

Darwin's theory of selection and the theory of nearly neutral evolution (Kimura, Reference Kimura1962, Reference Kimura1991; Ohta, Reference Ohta1992) together explain evolution as a process of selection and drift, whereas structural biology explains the molecular language of evolution via the central dogma. However, a complete theory of evolution requires us to also know the properties of the evolving protein that contributes to the organism phenotype, why it contributes, to what extent it contributes, and how this affects the wider evolution of the population in its ecological and historical context. As discussed extensively in the literature, it is increasingly clear that the functional traits selected for in classical positive Darwinian evolution have relatively little importance in many cases relative to other, partly hidden and perhaps universal properties of the proteins (Hurst and Smith, Reference Hurst and Smith1999; Bloom and Adami, Reference Bloom and Adami2003, Reference Bloom and Adami2004; Drummond et al., Reference Drummond, Bloom, Adami, Wilke and Arnold2005; Lobkovsky et al., Reference Lobkovsky, Wolf and Koonin2010; Wylie and Shakhnovich, Reference Wylie and Shakhnovich2011).

The most obvious universal property subject to selection pressure is arguably the cellular energy state. Before the era of structural biology and proteomics, Boltzmann (Reference Boltzmann1886) and Schrödinger (Reference Schrödinger1944) already speculated that life characteristically represents a well-defined organized (low-entropy) structure that maintains a thermodynamic non-equilibrium state relative to its high-entropy surroundings by constant energy turnover and associated heat dispersion. By this definition, expansion of life (fitness) implies expansion of this energy turnover. Lotka applied these ideas to Darwin's selection theory via his maximum power principle, arguing that evolution occurs by selection of the most energy-efficient organisms (Lotka, Reference Lotka1922). These ideas were then expanded into a much broader ecological view by Odum (Reference Odum1988). Thermodynamically, the system most capable of maintaining its structure by energy dissipation and with the ability to grow and reproduce these structures will prevail over other similar systems, and thus, be most fit.

The theory of proteome cost minimization (PCM) presented below was inspired by these views and further supported by the observations of consistent cost-bias in amino acid use across all kingdoms of life first discovered by Akashi and Gojobori (Reference Akashi and Gojobori2002). These findings were confirmed by Swire (Reference Swire2007) and explained in a fitness model by Wagner who showed, among other things, that gene duplications are highly selected against in terms of cellular energy costs (Wagner, Reference Wagner2005). The theory builds substantially on Wagner's seminal quantitative considerations (Wagner, Reference Wagner2005) and the important considerations of Brown et al. (Reference Brown, Marquet and Taper1993) who used Lotka's ansatz to explain mass and size optima of biological taxa in terms of evolutionary fitness caused by the different scaling of metabolic rates and reproductive rates with mass. The theory's central ansatz, inspired by these minds, is as follows: ‘Fitness is proportional to the energy per time unit available for reproduction after subtracting (proteome) maintenance costs’. Because fitness always has to be measured relative to a wild type after an instant of time, the energy of interest becomes a power (measured in watts or J s⁻¹) as in Lotka's original thinking, and as such directly relates to the respiration rate of the organism, as discussed below.

The mechanistic basis for the theory is that (i) protein degradation increases many-fold with the lack of structure and partial unfolding in protein copies (Gsponer et al., Reference Gsponer, Futschik, Teichmann and Babu2008) and (ii) the cost of protein turnover is more than half of total metabolic costs in growing microorganisms (Harold, Reference Harold1987), and at least 20% in humans (Waterlow, Reference Waterlow1995). Accordingly, any increase in these costs reduces the energy available for other energy-demanding processes, notably reproduction (fitness) of microorganisms (Dasmeh and Kepp, Reference Dasmeh and Kepp2017) and cell signaling (cognition) in higher organisms (Kepp, Reference Kepp2019). One of many implications of the theory is that selection against misfolded proteins and toxicity of misfolding proteins measured in cell viability assays is not due to a specific toxic molecular mode of action as widely assumed, but to the generic adenosine triphosphate (ATP) burden of turning over the misfolded proteins within the cell.

In its simplest form, which is easily expanded, we assume a life cycle of a protein i as:

(2)$${\rm mRN}{\rm A}_i\buildrel {k_{{\rm s}_i}} \over \longrightarrow F_i\mathrel{\mathop{\kern0pt\rightleftharpoons}\limits_{{k_{{\rm 2}_i }}}^{k_{{\rm 1}{_i }}}} U_i\buildrel {k_{{\rm d}_i}} \over \longrightarrow D_i$$

F_i represents the folded proteins, U_i represents misfolded proteins, and D_i represent the degradation products, many of which are recycled for use in other proteins; the rate constant of each process is specific to the protein in question. Because the ultimate selection pressure acts only on U_i, one can easily relax the assumption of one-step unfolding to account for complex situations.

$k_{{\rm d}_i}$ is the rate constant (in units of protein molecules per s) for degrading the misfolded protein copies. The in vivo rate constants reflect the half-life (t _½) of the fully folded protein, and can thus be written at steady state as:

(3)$${k}^{\prime}_{{\rm d}_i} = \displaystyle{{k_{{\rm d}_i}k_{1_i}} \over {k_{2_i}}} = \displaystyle{{k_{{\rm d}_i}} \over {K_{{\rm f}_i}}} = \displaystyle{{{\rm ln}\,2} \over {t_{1/2}}}\;$$

which varies substantially with the protein i, giving half-lives from minutes to days (Hargrove and Schmidt, Reference Hargrove and Schmidt1989). The model assumes that unfolded protein copies are always kept at a very small number in the cell, compared to folded copies, such that $k_{2_i}$ is much larger than $k_{{\rm d}_i}$ and $k_{1_i}$. This is generally a good approximation, because ${k}^{\prime}_{{\rm d}_i}$ is typically of the order of 10⁻⁴ s⁻¹ but with order-of-magnitude variations. In contrast, $k_{{\rm d}_i}$ acts directly on already misfolded protein and represents the rate of protein degradation if the chemical activation barrier to unfolding has been removed. Thus, $k_{{\rm d}_i}$ is limited by the number of active proteases, the diffusion and proper orientation of the exposed peptide bond, and the actual k _cat/K _M of the proteases, with an upper limit of perhaps 10⁶ to 10⁸ M⁻¹ s⁻¹ per peptide bond hydrolysis (Wolfenden and Snider, Reference Wolfenden and Snider2001; Bar-Even et al., Reference Bar-Even, Noor, Savir, Liebermeister, Davidi, Tawfik and Milo2011). In terms of steady-state turnover, misfolded proteins are immediately targeted for degradation (Gsponer et al., Reference Gsponer, Futschik, Teichmann and Babu2008) and recruited by the ubiquitin–proteasome pathway that takes the protein out of the pool, and thus this process is not rate-limiting the overall protein flux but arguably operates near the diffusion limit.

Assuming one-step misfolding, U_i is related to the folding free energy of the protein ΔG_i = −RT ln($K_{{\rm f}_i}$) via the equilibrium constant $K_{{\rm f}_i}$ = F_i/U_i:

(4)$$U_i = A_i\left({\displaystyle{1 \over {1 + \exp \lpar {-\Delta G_i/RT} \rpar }}} \right)\approx A_i\exp \left({\displaystyle{{\Delta G_i} \over {RT}}} \right)$$

The last expression follows if there are many more folded than unfolded copies of the protein, which is almost always the case. Because folding equilibrium constants easily reach 10¹¹ for a protein of typical stability (65 kJ mol⁻¹ at 37 °C), the number of misfolded proteins at any given time is typically negligible, as they are immediately subject to turnover. Reasonable experimental values of $k_{{\rm d}_i}$ = 10⁷ s⁻¹, $K_{{\rm f}_i}$ = 10¹¹, and ${k}^{\prime}_{{\rm d}_i}$ = 10⁻⁴ s⁻¹ satisfy the relationship in Eq. (3) and thus justify the use of Eq. (2).

Equation (4) is well established and was first used in a fitness function by Bloom et al. (Reference Bloom, Wilke, Arnold and Adami2004) and has been specifically used to explain some of the E-R anticorrelation (Serohijos et al., Reference Serohijos, Rimas and Shakhnovich2012) and additional variations in evolutionary rates (Dasmeh et al., Reference Dasmeh, Serohijos, Kepp and Shakhnovich2014a). The advantage of this expression is that we can relate the number of misfolded protein copies, which is the property selected upon, directly to the total copy number A_i of the protein within the cell and to its thermodynamic stability, via the free energy of folding ΔG_i (a negative number in kJ mol⁻¹). RT is the thermal energy of the cell, and thus temperature enters directly as a fundamental physical parameter determining proteome U_i and ultimately cellular proteome costs and fitness, as discussed further below.

The critical step is now to write the fraction of the total respiration rate (in watts, or J s⁻¹) of the cell due to the maintenance of a single protein i:

(5)$$\displaystyle{{dE_{{\rm m}\comma i}} \over {dt}} = A_i\exp \left({\displaystyle{{\Delta G_i} \over {RT}}} \right)k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar $$

In this equation, in addition to the parameters already described above, $N_{{\rm a}{\rm a}_i}$ represents the number of amino acids in the protein i, and the cost constants $C_{{\rm s}_i}$ and $C_{{\rm d}_i}$ describe the average synthetic and degradation cost per amino acid in protein i in units of J (Kepp and Dasmeh, Reference Kepp and Dasmeh2014).

For the whole proteome of the cell, we can write the total cost per time unit as the sum of the costs of maintaining steady-state folded protein copy numbers within the cell:

(6)$$\displaystyle{{dE_{\rm m}} \over {dt}} = \alpha \mathop \sum \limits_i \displaystyle{{dE_{{\rm m}\comma i}} \over {dt}} = \alpha \mathop \sum \limits_i A_i\exp \left({\displaystyle{{\Delta G_i} \over {RT}}} \right)k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar $$

Importantly, we see that the total energy costs scale with A_i. Because A_i varies substantially for different proteins, e.g. from zero to a million, some proteins are much more important to the cell's energy budget than others. The scaling constant α represents the activity of the proteasome, which may be controlled with proteasome inhibitors, but a slight expansion of this expression can be done to (α + β + ⋯) taking into account the contributions of various degradation pathways (lysosome, proteasome, effects of N-end rule, etc.) to the overall turnover. Figure 1 summarizes some typical values for the parameters of the model applicable to eukaryote cells.

Fig. 1.

Schematic overview of order-of-magnitude terms of the PCM model. Typical values for yeast used as example. All values are subject to the well-known variations in copy numbers of individual proteins, degradation constants, length of proteins, and total number of proteins copies in a cell.

Selection dynamics of PCM

To understand how protein turnover costs affect evolution, we now use the central ansatz that fitness scales with the energy available for reproduction dE _r/dt after subtracting the proteome costs of Eq. (6) from the total energy available to the cell either by production or supply, dE _t/dt, divided by the respiration rate needed to run an individual, also taken to dE _t/dt:

(7)$${\it \Phi} = \displaystyle{{dE_{\rm r}/dt} \over {dE_{\rm t}/dt}} = \displaystyle{{dE_{\rm t}/dt-dE_{\rm m}/dt} \over {dE_{\rm t}/dt}} = 1-\displaystyle{{dE_{\rm m}/dt} \over {dE_{\rm t}/dt}}$$

The division by dE _t/dt formally ensures a dimensionless fitness function. For simplicity, we ignore the non-proteome energy costs because the purpose is to show that the cost of the proteome exerts a major effect on evolution by itself. Assuming that the total energy production is constant for all competing cells, minimization of dE _m/dt maximizes fitness. When a new mutation arises in protein i, the selection coefficient is:

(8)$$s_i\lpar M \rpar = \displaystyle{{{\it \Phi} _i\lpar M \rpar } \over {{\it \Phi} _i\lpar {WT} \rpar }}-1 = \displaystyle{{{\it \Phi} _i\lpar M \rpar -{\it \Phi} _i\lpar {WT} \rpar } \over {{\it \Phi} _i\lpar {WT} \rpar }} = \displaystyle{{\;\;dE_{\rm m}/dt\lpar {WT} \rpar -dE_{\rm m}/dt\lpar M \rpar } \over {dE_{\rm t}/dt\lpar {WT} \rpar -dE_{\rm m}/dt\lpar {WT} \rpar }}$$

For clarity, we have assumed that the mutation only affects maintenance turnover costs and not energy production, and thus the total energy produced is the same before and after mutation and cancels in Eq. (8). If we further neglect epistasis, selection only acts on the mutated protein i:

(9)$$s_i\lpar M \rpar = \displaystyle{\matrix{{A_i\exp \lpar {\Delta G_i/RT} \rpar k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar \lpar {WT} \rpar}\cr{ -A_i\exp \lpar {\Delta G_i/RT} \rpar k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar \lpar M \rpar}} \over {dE_{\rm t}/dt\lpar {WT} \rpar -A_i\exp \lpar {\Delta G_i/RT} \rpar k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar \lpar {WT} \rpar }}$$

This selection coefficient is a function only of protein properties, scaled by the general energy spent for reproduction of the organism, dE _r/dt(WT), which can be taken as a constant of the order of 10⁻¹¹ J s⁻¹ (Harold, Reference Harold1987). It is perhaps more convenient to write Eq. (9) in terms of copy numbers and half-lives (t _½) which can be measured in live cells:

(10)$$s_i\lpar M \rpar = \displaystyle{\matrix{{A_iN_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar {\rm ln}\,2/t_{1/2}\lpar {WT} \rpar}\cr{-A_iN_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar {\rm ln}\,2/t_{1/2}\lpar M \rpar}} \over {dE_{\rm r}/dt\lpar {WT} \rpar }}$$

where we have used the relationship:

(11)$${k}^{\prime}_{{\rm d}_i} = \displaystyle{{{\rm ln}\,2} \over {t_{1/2}}} = \exp \left( {\displaystyle{{\Delta G_i} \over {RT}}} \right)k_{{\rm d}_i}$$

For a haploid organism, the probability of its fixation P _fix is approximately (Kimura, Reference Kimura1962; Ohta Reference Ohta1992):

(12)$$P_{{\rm fix}} = \displaystyle{{1-{\rm e}^{{-}s_i}} \over {1-{\rm e}^{{-}s_iN}}}\cong \displaystyle{{s_i} \over {1-{\rm e}^{{-}s_iN}}}$$

where N is the effective population size, and the last term comes from expanding the exponential of the small s_i. For neutral evolution, as s_i → 0, P _fix → 1/N, and does not depend on any properties of the protein. At significant positive selection, s_i N is large, s_i is positive, and P _fix → s_i. Very similar behavior applies to diploid organisms with slightly different factors of 2 and 4 (Kimura, Reference Kimura1962).

The simple kinetic scheme assumed for the PCM model is highlighted in Fig. 2a. From Eq. (10), considering the variations in the parameters, most of the proteome cost selection occurs by affecting the ratio A_i/t _½. Mutations that reduce the half-life of abundant proteins are thus particularly selected against. The typical behavior of P _fix with N and s_i is shown in Fig. 2b. The absolute rate of evolution ω scales with the mutation rate and the probability of fixating new arising mutations:

(13)$$\omega = uNP_{{\rm fix}} = u\displaystyle{{s_i} \over {1-{\rm e}^{{-}s_iN}}}$$

where u is the absolute mutation rate; this expression can be expanded by life history variables such as generation time (Martin and Palumbi, Reference Martin and Palumbi1993), but this is beyond the scope here, as the proportionality of Eq. (13) generally applies, and P _fix thus measures evolution rate. For an optimized evolutionary system, a typical arising mutation has a negative selection coefficient; if small relative to 1/N, it is subject to random fixation drift. From Eq. (13), such mutations will reduce the probability of fixation (and evolution rate) in proportion to the size of the negative selection coefficient. Figure 2b also illustrates why the molecular clock is generally successful at dating phylogenies, because 90% of randomly occurring mutations in the relevant selection-fixation space are subject to neutral evolution.

Fig. 2.

(a) Schematic overview of the processes of protein turnover, with the central dogma to the left and the proteome maintenance, the concern of the present paper, to the right. (b) Probability of fixation (P _fix) plotted against selection coefficient s and log N (effective population size). Beneficial mutations with s > 0.001 have relevant P _fix of more than 1% for most populations. Only in very small populations (<100) do other mutations get fixated (P _fix ~ 1%), and neutral and slightly deleterious mutations become fixated to a similar extent until s approaches −1/N, whence P _fix rapidly decreases.

To understand the slow evolution of abundant proteins discussed in the literature (Drummond et al., Reference Drummond, Bloom, Adami, Wilke and Arnold2005; Bloom et al., Reference Bloom, Drummond, Arnold and Wilke2006a; Drummond and Wilke, Reference Drummond and Wilke2008), we should identify low values of P _fix in the evolution rate space of Fig. 2b. Most arising mutations (Fig. 2b) remain subject to nearly neutral evolution. However, more extreme selection coefficients will occur for highly abundant proteins, because the selection coefficient of a new arising mutation in a protein scales with the abundance and turnover rate of the affected protein. In contrast, less abundant proteins will typically have numerically smaller selection coefficients at any given effective population size. The next section gives a quantitative estimate of the fixation probabilities.

Typical PCM selection pressures and fixation probabilities for yeast

Table 2 summarizes some typical selection scenarios in yeast cells. A typical yeast cell respires at ~1 J s⁻¹ g⁻¹ and has a mass of 3 × 10⁻¹¹ g, giving dE _t/dt ≈ 3 × 10⁻¹¹ J s⁻¹. $C_{d_i}$is perhaps 1 ATP per peptide bond or 30 kJ mol⁻¹ (Benaroudj et al., Reference Benaroudj, Zwickl, Seemüller, Baumeister and Goldberg2003). The biosynthetic costs of the amino acids vary from 10 to 80 ATP (Wagner, Reference Wagner2005), the average amino acid composition of the yeast proteome gives ~25 ATP, or 750 kJ mol⁻¹ as typical. If half of the amino acids are recycled, neglecting amino acid transport cost (Waterlow, Reference Waterlow1995), this reduces to 375 kJ mol⁻¹. Additional costs of the polypeptide chain synthesis, neglecting chaperones, is ~11–19 ATP, or 330–660 kJ mol⁻¹ (De Visser et al., Reference De Visser, Spitters and Bouma1992). Amino acid transport and chaperones (which need to be synthesized independently) increase costs further. Under growth conditions where most selection probably occurred historically, very few amino acids are recycled, and thus the specific turnover costs per amino acid in a protein molecule $\lpar C_{{\rm s}_i} + C_{{\rm d}_i}\rpar$ may easily reach 1500 kJ mol⁻¹. However, the amino acid-specific values vary little compared to the protein-specific ${k}^{\prime}_{d_i}$ and A _i, and thus we use a value of 1500 kJ mol⁻¹ in Table 2. With a typical protein of 400 amino acids, this implies 10⁻¹⁵ J s⁻¹ of turnover cost per protein molecule, which varies perhaps by 3–4 orders of magnitude, mostly due to $N_{{\rm a}{\rm a}_i}$ (protein length) and $C_{{\rm s}_i}$ (the biosynthetic cost of the amino acids) consistent with the empirically known sequence biases (Akashi and Gojobori, Reference Akashi and Gojobori2002; Wagner, Reference Wagner2005; Swire, Reference Swire2007).

Table 2.

Effect of arising mutants in a haploid organism ($N_{{\rm a}{\rm a}_i}$ = 400; dE _t/dt  = 3 × 10⁻¹¹ J s⁻¹)

WT, wild-type value of property; M, Mutant value of property.

The exponential of Eq. (1) can be expanded as 1 − cU because the values of cU are much smaller than 1. Accordingly, the empirically proposed (Drummond and Wilke, Reference Drummond and Wilke2008) fitness cost constant c can be expressed in terms of fundamental protein turnover parameters, and we argue that c is protein-specific. The PCM fitness function, Eq. (7), can be written as:

(14)$$\Phi = \displaystyle{{dE_{\rm t}/dt-\mathop \sum \nolimits_i A_i\exp \lpar {\Delta G_i/RT} \rpar k_{d_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar } \over {dE_{\rm t}/dt}} = 1-\displaystyle{{\mathop \sum \nolimits_i U_ik_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar } \over {dE_{\rm t}/dt}}$$

Comparing the exponential-expanded fitness functions 1 − cU proposed by Drummond and Wilke (Reference Drummond and Wilke2008) and Eq. (14), the dimensionless protein-specific and effective total cost constants are:

(15)$$c_i = \displaystyle{{k_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar } \over {dE_{\rm t}/dt}}\semicolon \;\quad c = \displaystyle{{\mathop \sum \nolimits_i U_ik_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar } \over {dE_{\rm t}/dt\;U}}$$

Separation of U_i from its cost constant c_i does not apply in general, as each type of unfolded protein has specific costs, and thus c represents an average cost of handling all misfolded proteins regardless of type. Using the typical values of $k_{{\rm d}_i}$ = 10⁷ s⁻¹ and $N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar$ = 10⁻¹⁵ J s⁻¹ (Fig. 1, Table 2) gives 10⁻⁸ J s⁻¹ for one molecule of protein i. When dividing by dE _t/dt ~ 10⁻¹¹ J s⁻¹, this gives a cost constant c _i ~ 1000. Summing over all misfolded copies (U ~ 10⁻³) gives a correction to the fitness function of the order of unity, in agreement with energy allocated to reproduction and proteome turnover being of the similar magnitudes as total respiration rates of growing cells (Harold, Reference Harold1987).

A single protein's contribution to fitness is proportional to its relative abundance, all else being equal. If A_i = 1000, then U_i = 10⁻⁸ misfolded copies of this particular protein exist at any time, using the typical parameters given in Fig. 1 and Table 2, giving a total contribution to fitness of 10⁻⁵. Typically arising, slightly deleterious mutations in typical proteins will affect evolution rates in small populations of the order of N ~ 10⁴, which probably played a major role in evolution in the wild (Gillespie, Reference Gillespie2001; Piganeau and Eyre-Walker, Reference Piganeau and Eyre-Walker2009), mainly because historic population bottlenecks dominate the apparent effective population size (Willis and Orr, Reference Willis and Orr1993; Hawks et al., Reference Hawks, Hunley, Lee and Wolpoff2000; Bouzat, Reference Bouzat2010). The calculation example in Table 2 gives a fixation probability of 7.9 × 10⁻⁵ for such typical mutations.

However, some proteins are much more systemically important than such a typical protein. The most important contributor to c_i is the degradation rate constant $k_{{\rm d}_i}$, which varies by many orders of magnitude for different proteins, and to obtain the fitness we need to multiply this constant by A _i, or equally, the fold-stability weighted U_i. Abundance can span 5–7 orders of magnitude (Jansen and Gerstein, Reference Jansen and Gerstein2000; Ghaemmaghami et al., Reference Ghaemmaghami, Huh, Bower, Howson, Belle, Dephoure, O'Shea and Weissman2003; Beck et al., Reference Beck, Schmidt, Malmstroem, Claassen, Ori, Szymborska, Herzog, Rinner, Ellenberg and Aebersold2011; Milo Reference Milo2013), whereas protein length $N_{{\rm a}{\rm a}_i}$ spans about three orders of magnitude, up to ~30 000 amino acids (e.g. titin), with a reasonably small variance of gamma-distributed protein sizes (Zhang, Reference Zhang2000). PCM theory thus suggests that selection acts both on expression level and protein length, as indeed seen experimentally (Bloom et al., Reference Bloom, Drummond, Arnold and Wilke2006a). In small populations (N = 10⁴), a typical slightly deleterious mutation (less stable by 5 kJ mol⁻¹, or a 10-fold higher turnover rate) in a highly expressed protein (10⁵ copies) will have essentially no probability of fixation (<10⁻²⁰, middle right, Table 2). Cost selection in such moderate-sized populations can thus explain the relatively slower evolution of abundant proteins.

Large effective populations can also contribute to the E-R anti-correlation: random mutation-selection dynamics resulting from purifying or compensatory selection of new residues after accepting slightly deleterious mutations occur more frequently in less abundant proteins that have more neutral selection coefficients. In contrast, these dynamics are less important near the steeper fitness optimum of the more optimized, abundant proteins that pose larger costs to the proteome. The relative importance of these two mechanisms depends on the historic effective population size and the population bottlenecks on long evolutionary timescales. One can model such effects by explicit evolution simulations but this is beyond the scope of the current study.

For comparison to experiment, it is more convenient to use the fitness function:

(16)$$\Phi = 1-\displaystyle{{\mathop \sum \nolimits_i A_iN_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar {\rm ln}\,2/t_{1/2i}} \over {dE_{\rm t}/dt}}$$

where t _½i is the experimental in vivo half-live of the protein i, which accounts for real cellular life-times distinct from biophysical protein stability, e.g. effects of the N-end rule (Varshavsky, Reference Varshavsky1997; Mogk et al., Reference Mogk, Schmidt and Bukau2007; Gibbs et al., Reference Gibbs, Bacardit, Bachmair and Holdsworth2014). All the properties in Eq. (16) are either observable or deducible from the protein's sequence.

Scaling relations of proteome costs: mass, metabolism, and eukaryote evolution

The examples given have centered on yeast as model cell, with ∑A_i = 10⁸. Eukaryote cells vary greatly in size, the total copy number of proteins, and metabolic respiration rates, and prokaryotes typically feature smaller volumes, protein copy numbers and lower metabolic total respiration rates by 2–3 orders of magnitude (Milo, Reference Milo2013). The question then emerges how these orders-of-magnitude differences affect the proteome turnover and the associated effects described above. Proteins are degraded differently due to specific degrons of their sequences, but the overall rate of protein turnover typically scales with the general activity of the proteasome (except for those proteins that are not degraded by the proteasome). Accordingly, a scale factor of proteasome activity α (Eq. (6)), as modulated by proteasome inhibitors, will be an important control parameter in experimental tests of the theory as well as in efforts to understand protein turnover in relation to cellular energy costs, cell viability, and fitness. Although long-term proteasome inhibition is toxic, mild instantaneous proteasome inhibition should prove a useful tool in testing some of the mechanisms described here.

Additional scaling relations are relevant to discuss. Notably, from Eq. (8), any scaling of the metabolic rate by a number a characteristic of the organism will not affect the selection coefficient, if the fraction of energy devoted to reproduction is constant, commonly between 0.1 and 0.7 of total respiration costs (Harold, Reference Harold1987; Hawkins, Reference Hawkins1991), because the advantage of the mutation with lowered maintenance costs can be considered a perturbation:

(17)$$s_i (scaled)\lpar M \rpar = \displaystyle{{\;\;adE_{\rm m}/dt\lpar {WT} \rpar -adE_{\rm m}/dt\lpar M \rpar } \over {adE_{\rm t}/dt\lpar {WT} \rpar -adE_{\rm m}/dt\lpar {WT} \rpar }} = s_i\lpar M \rpar \;$$

This relation requires comparison of the mutant and wild-type proteins under the same growth conditions.

Based on cell volume and protein copy measurements and associated calculations (Milo, Reference Milo2013), and using the assumption that a typical protein volume is 10 000 Å³, proteins take up 1–4% of the cell volume of any cell and more importantly, regardless of the cell type, across prokaryotes and eukaryotes, including human cells. From this, we conclude that the total protein copy number A_i scales approximately linearly with cell volume. In contrast, the basal specific metabolic rate of both cells and whole organisms tends to scale with M ^3/4, rather than M (Kleiber's law) (Kleiber, Reference Kleiber1932, Reference Kleiber1947; Savage et al., Reference Savage, Allen, Brown, Gillooly, Herman, Woodruff and West2007). Size, all-else-being equal, lowers the specific surface area of the organism and thereby increases metabolic efficiency by reducing the mass-weighted thermodynamic force required to maintain the non-equilibrium boundary (reduced heat dispersion per unit of biomass). Size also potentially minimizes average, mass-specific chemical and electric signaling distances within the organism. Such scaling laws of mass and volume and their implication for bioenergetic costs were discussed by Lynch and Marinov (Reference Lynch and Marinov2015).

For these reasons, the specific resting metabolism decreases with volume or mass, and equally, with total protein copy number of the organism. Accordingly, size carries an evolutionary advantage of the order of the mass-specific metabolic rate, as explained in detail by Brown and co-worker who developed the framework relating mass to fitness (Brown et al., Reference Brown, Marquet and Taper1993). The advantage is of the order of:

(18)$$\eqalign{s\lpar M \rpar & = \displaystyle{{{\it \Phi} \lpar M \rpar } \over {{\it \Phi} \lpar {WT} \rpar }}-1 = \displaystyle{{\;dE_{\rm r}/dt\lpar M \rpar } \over {\;dE_{\rm r}/dt\lpar {WT} \rpar }}-1 \cr & = \displaystyle{{-aM^{3/4}\lpar M \rpar } \over {-aM^{3/4}\lpar {WT} \rpar }}-1\sim \left({\displaystyle{{M\lpar M \rpar } \over {M\lpar {WT} \rpar }}} \right)^{3/4}-1\;}$$

However, as pointed out by Brown et al. (Reference Brown, Marquet and Taper1993) whereas ecological life-history variables (e.g. foraging efficiency) favor large organisms, the reproduction rate favors smaller organisms and scales with M ^−1/4. Thus, organism size has an evolutionary optimum with respect to both energy and time, which is distinct for different taxa due to the different life-history variables and associated scaling parameters (Brown et al., Reference Brown, Marquet and Taper1993). A yeast mutant with a larger size of 1%, all-else being equal, would thus be predicted by PCM theory to have a selective advantage of (1.01/1)^3/4 -1 = 0.007 if all the saved energy is spent on reproduction. This energy is clearly enough to enforce positive selection at all relevant population sizes from 10² to 10⁷, including early population bottlenecks (Fig. 2).

Combining the ansatz of PCM theory (that fitness scales with the energy left for reproduction per time unit after subtracting maintenance costs) with Kleiber's law leads to several potentially important explanations for size advantage relevant to emergence of life in general and eukaryotes in particular. A central weakness of endosymbiont theory, not mentioned by the otherwise important reviews on this topic (Gray et al., Reference Gray, Burger and Lang1999; Lane Reference Lane2011), is the problem of evolutionary advantage immediately after the symbiosis event. The argument goes as follows: at the very beginning, the actual process of symbiosis must have had immediate costs of intrusion and aligning the cellular machineries, and must thus also have provided immediate selective advantages in competition will non-symbiotic cells. According to PCM theory, fitness scales with energy left for reproduction, and thus the immediate total maintenance costs must have reduced.

Imagine a simple doubling of the cell size by a unification event. All else being equal, the new organism would carry the double amount of proteins, the double volume, the double mass, and would require the double amount of energy to reproduce these cell constituents, giving the same fitness as the competing non-symbiont cells, but then reduced by the costs of the endosymbiosis event itself. However, the immediate advantage offered by reducing the specific surface area of the ancestral eukaryote cell would reduce the basal metabolic maintenance rate. The saved energy could then be immediately converted into a larger fraction of the total energy budget being devoted to the proteome of larger cells and organisms, thus compensating the cost of the actual symbiosis event. If this is correct, endosymbiosis will be successful only when and if the mass-specific metabolic rate saved by mass increase outweighs the energy costs of the symbiosis event itself.

Evidence for PCM during evolution

Some support for the theory of proteome cost minimization is summarized in Table 3. The following section discusses some of these facts briefly.

Table 3.

Events and facts supporting the PCM theory

Trading function for cost

Classical Darwinian evolution considers the struggle and selection for optimal function the primary mode of evolution (Richmond, Reference Richmond1970; Hurst, Reference Hurst2009). This aspect of Darwinism has dominated biochemical views of enzymes as perfectly optimized proficient catalysts that accelerate chemical reactions by orders of magnitude, implying that evolution strives toward optimal function per se, including maximal substrate turnover of enzymes (Radzicka and Wolfenden, Reference Radzicka and Wolfenden1995; Cannon et al., Reference Cannon, Singleton and Benkovic1996; Zhang and Houk, Reference Zhang and Houk2005). However, proteins are also subject to non-function selection pressures that are distinct from, and sometimes in conflict with, optimality of function (Hurst and Smith, Reference Hurst and Smith1999; Bloom and Adami, Reference Bloom and Adami2003; Reference Bloom and Adami2004; Drummond et al., Reference Drummond, Bloom, Adami, Wilke and Arnold2005; Lobkovsky et al., Reference Lobkovsky, Wolf and Koonin2010; Wylie and Shakhnovich, Reference Wylie and Shakhnovich2011). Indeed, actual comparison of enzyme kinetic parameters shows that many enzymes are distinctly suboptimal, most likely because of evolutionary and biophysical constraints (Bar-Even et al., Reference Bar-Even, Noor, Savir, Liebermeister, Davidi, Tawfik and Milo2011).

A standard view is that proteins have evolved to use their excess fold-free energy to optimize the active sites for function, the most notable example being pre-organized active sites with electrostatic fields favoring the free energy of the transition states, to increase k _cat/K _M (Cannon et al., Reference Cannon, Singleton and Benkovic1996; Warshel, Reference Warshel1998; Adamczyk et al., Reference Adamczyk, Cao, Kamerlin and Warshel2011; Morgenstern et al., Reference Morgenstern, Jaszai, Eberhart and Alexandrova2017; Fuller et al., Reference Fuller, Wilson, Eberhart and Alexandrova2019). Although not directly pointed out by Warshel and co-workers, this mechanism contributes to making proteins marginally stable because, all-else-being equal, any potential excess fold-free energy has been diverted into optimizing the electrostatic field of the folded structure to reduce the transition state's free energy and thereby increase catalytic proficiency. The mechanism also largely explains the widely observed stability-function trade-offs in protein engineering (Tokuriki et al., Reference Tokuriki, Stricher, Serrano and Tawfik2008). Correspondingly, in the laboratory, without many biological constraints, function of a high-stability starting protein may be optimized beyond the level seen in the wild (Bloom et al., Reference Bloom, Labthavikul, Otey and Arnold2006b; Tokuriki and Tawfik, Reference Tokuriki and Tawfik2009). This is particularly relevant in the context of ‘directed evolution’, i.e. the intended human evolution of new improved protein mutants employing yeast cells with short generation times in static environments where selection pressure can be effectively controlled (Francis and Hansche, Reference Francis and Hansche1972; Hall Reference Hall1981).

PCM theory argues that even functional proficiency often evolved conditionally on cost. To appreciate this, we consider the requirement of a certain total substrate turnover of each enzyme per time unit to maintain homeostasis. The proficiency of function is for enzymes typically defined by k _cat, measuring how many substrate molecules convert into product per time unit per enzyme molecule. At steady-state, both the maximum turnover (V _max) and the turnover at low substrate concentration are proportional to the total enzyme concentration [E] and k _cat (Northrop, Reference Northrop1998; English et al., Reference English, Min, van Oijen, Lee, Luo, Sun, Cherayil, Kou and Sunney2005).

Now consider a typical arising mutation in an enzyme i required to make a product at a certain rate, i.e. dP_i/dt. Because the protein is evolutionarily optimized (but not necessarily optimal), mutations will tend on average to be hypomorphic and reduce the turnover constant k _cat,i but with a broad scatter and many nearly neutral effects with a random chance of fixation. If the mutation reduces k _cat,i substantially, e.g. by modifying the active site, the substrate turnover will be greatly reduced, and the organism will need to increase the local enzyme concentration [E] by expressing more enzyme per time unit to maintain a comparable substrate turnover (compensatory expression), thereby increasing A_i. More specifically, the rate of product formed by enzyme i under Michaelis–Menten kinetics is (Cannon et al., Reference Cannon, Singleton and Benkovic1996; Northrop, Reference Northrop1998)

(19)$$\displaystyle{{dP_i} \over {dt}} = A_i\;k_{{\rm cat}\comma i}\displaystyle{{\lsqb S \rsqb } \over {K_{{\rm M}\comma i} + \lsqb S \rsqb }}$$

Equation (19) represents the standard equation multiplied on both sides by the cell volume to convert from concentrations to absolute copy numbers. For simplicity, we can ignore the last term and assume zero-order kinetics in [S], which represent selection of the enzyme for maximum rate at saturated substrate concentration when [S] is much larger than the Michaelis constant K _M,i. The cost of maintaining the enzyme is

(20)$$\displaystyle{{dE_{{\rm m}\comma i}} \over {dt}} = A_i{k}^{\prime}_{{\rm d}_i}N_{aa_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar $$

Accordingly, the specific cell-wide cost of maintaining steady state produced concentration of P_i is

(21)$$\displaystyle{{dE_{{\rm m}\comma i}} \over {dP_i}} = \displaystyle{{A_i{{k}^{\prime}}_{{\rm d}_i}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar } \over {A_ik_{{\rm cat}\comma i}}} = \displaystyle{{{{k}^{\prime}}_{{\rm d}_i}} \over {k_{{\rm cat}\comma i}}}N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar $$

If measured in concentrations instead, the cost scales with the volume of the cell V _cell to which the steady state applies. We have ignored the costs associated with producing the substrate and transporting the substrate and products, which can easily be included into the model.

Equation (21) predicts that the ratio of the two time constants for turnover of the enzyme and turnover of the substrate together define the cost of producing P_i at steady state. The two time constants are in units of s⁻¹, and $N_{{\rm a}{\rm a}_i}\lpar {C_{{\rm s}_i} + C_{{\rm d}_i}} \rpar$ is of the order of 10⁻¹⁵ J for a typical protein. Considering again a typical arising mutation, even if ${k}^{\prime}_{{\rm d}_i}$ is not increased (which it typically is), a reduction in k _cat,i of a typical hypomorphic mutation will require compensatory expression of the enzyme, increasing A_i to maintain the rate of production of P_i, Eq. (19). This increase in A_i will then increase the total cost of obtaining the product with the same factor (Eq. (20)). Equation (21) summarizes this cost–function relationship because k _cat,i and A_i are inversely related if homeostasis in P_i is required. If compensatory expression is 100%, a ten-fold reduction in the enzyme's k _cat,i requires a 10-fold increase in the enzyme's expression, and the specific and total costs of producing P_i increases 10-fold.

Accordingly, even mutations that only impair function also increase the proteome costs: a 10-fold increase in ${k}^{\prime}_{{\rm d}_i}$ (loss of kinetic stability, misfolding) or decrease in k _cat,i will have approximately the same 10-fold increase in cellular costs, according to Eq. (21), ignoring the mutation-induced changes in the amino-acid synthesis and degradation costs. If required, the assumption of 100% compensatory expression can easily be modified by a scale factor between 0 and 1 in the above equations. Evidence for compensatory expression is well-known, a dramatic example being homozygous sickle cell disease (Table 3), where dysfunctional, instable hemoglobin mutants cause a doubling of protein turnover and degradation in patients and a 20% increase in total resting metabolism (Badaloo et al., Reference Badaloo, Jackson and Jahoor1989). Considerations of loss and gain of function mutations associated with other diseases may be viewed in this light (Kepp, Reference Kepp2015, Reference Kepp2019).

Because of the above considerations, we expect a function–cost trade-off acting during evolution of many proteins. We obtain the important possibility that the main advantage of a mutant may not be a functional improvement of the protein per se, but a reduction of its cost per unit of function, in the simplest case the ratio ${k}^{\prime}_{{\rm d}_i}/k_{{\rm cat}\comma i}$. Co-optimization of cost versus function is fundamental to many optimization processes and follows the basic principle that if several inputs are available at different functionality and price, the optimal system uses the input whose cost per unit of function is lowest. Such systems will tend to use less functional input if its cheaper price outweighs the loss of function. This suggests that at least some of the widely observed inverse relationships between function and stability (Tokuriki et al., Reference Tokuriki, Stricher, Serrano and Tawfik2008; Bonet et al., Reference Bonet, Wehrle, Schriever, Yang, Billet, Sesterhenn, Scheck, Sverrisson, Veselkova, Vollers, Lourman, Villard, Rosset, Krey and Correia2018; Du et al., Reference Du, Zielinski, Monk and Palsson2018) in reality reflect a cost–function trade-off as summarized by Eq. (21). The laboratory can change selection pressures drastically away from those in the wild, notably in the form of ‘directed evolution’ (Francis and Hansche, Reference Francis and Hansche1972; Hall Reference Hall1981). In nature however, the situation is more complicated, because the stability affects the proteome costs and thus fitness. Newly arising mutations may impair both stability and function, but both have a direct negative fitness effect in terms of cost.

The theory thus predicts that highly abundant proteins, because they are more cost-selected, are more likely to display suboptimal functionality, all else being equal (after adjusting for other correlating variables such as size). The trade-offs will be habitat- and strategy-dependent, and the preferential use of very functional but expensive input may be restricted to high-nutrient habitats and growth media.

Time or energy?

We expect that variations in the habitat's selection pressure should affect the proteome function–cost trade-offs. This should be evident when comparing organisms adapted to different environments. The most obvious biophysical properties of the habitat are time, energy, space, and temperature, which all enter directly in the model, Eq. (12). Selection for time, i.e. ‘survival of the fastest’, can be considered the default mode, and enters via the central ansatz of the theory, that ‘fitness is proportional to the energy per time unit available for reproduction after subtracting maintenance costs’, i.e. Eq. (6): Φ = dE _r/dt = dE_t/dt − dE _m/dt. Fitness scales inversely with the time step dt required for directing a unit of surplus energy sufficient to complete a reproductive event. Temperature enters as a modifier of the protein stability's role in the turnover ΔG_i/RT. We also note that a model of protein minimization driven mainly by considering space as a limiting parameter leads to some of the same consequences as protein cost minimization (Brown, Reference Brown1991). Accordingly, all these biophysical properties may potentially act as selection pressures.

Reasonably, cells have been optimized to maximize growth rates by enabling their proteomes to be produced as fast as possible within the necessary function and stability restrictions. Translational speed and accuracy imply selection for smaller and more streamlined genomes, and accuracy mainly reflects the time–cost trade-off of correcting errors during protein synthesis rather than correcting them later in e.g. a misfolded protein (Kurland and Ehrenberg, Reference Kurland, Ehrenberg, Cohn and Moldave1984; Drummond et al., Reference Drummond, Bloom, Adami, Wilke and Arnold2005). We can thus reasonably view time as the ‘default mode’ of selection when energy is plentiful (i.e. survival of the ‘fastest’). If the growth rate is proportional to the synthesis rate of the proteome, then large, highly expressed, and slowly folding proteins will be growth-limiting either at the ribosome or during subsequent folding by chaperones of the rate-limiting proteins.

To account for both time and energy together, for simplicity we only consider two processes, one that is energy-limited and one that is time-limited:

1. (1)

   (22)$$\hskip-6pc{\rm ATP} + {\rm cell}\to {\rm Budding}\;{\rm cell}\;\lpar {{\rm G1\semicolon \;}\,{\rm S}} \rpar $$

2. (2)

   (23)$$\hskip-7.5pc{\rm Budding}\;{\rm cell}\to 2\;{\rm cells}\;\lpar {{\rm G2}\comma \;{\rm M}} \rpar $$

In this simple model, if energy is limited, the cell will enter a dormant state and growth rates are controlled by energy efficiency of the proteome according to PCM theory. If energy is plentiful, growth rates are limited by the rate of producing the new cell, restricted by the speed of synthesizing the proteome rather than its cost. Other models of proteome optimization have emphasized translational speed and accuracy and minimization of protein size (Ehrenberg and Kurland, Reference Ehrenberg and Kurland1984; Brown, Reference Brown1991) due to space restrictions on flux control. One can also consider analogous microkinetic models such as:

1. (1)

   (24)$$\hskip-7pc{\rm ATP} + {\rm R}\hbox{-}{\rm chain}\to {\rm R}\hbox{-}{\rm chain}\hbox{-}{\rm ATP}$$

2. (2)

   (25)$$\hskip-5pc{\rm R}\hbox{-}{\rm chain}\hbox{-}{\rm ATP} + {\rm aa}\to {\rm R}\hbox{-}{\rm chain}\hbox{-}{\rm ATP}\hbox{-}{\rm aa}$$

Here, the ATP needed for the ribosome (R) to catalyze chain elongation by an amino acid (aa) must be available to the ribosome, and if the concentration of ATP is low, then this step is rate-limiting the protein synthesis. If energy is plenty, then step 2, the chain elongation (and subsequent protein folding by e.g. chaperones) is limiting growth and subject to selection pressure.

It should be clear that both the cell-cycle and microkinetic model imply that both energy and time can be relevant selection modes, i.e. survival of the ‘fastest’ (scenario 2) survival of the ‘cheapest’ (scenario 1). One can consider r- and k-strategies as resulting from specialization toward these regimes. Experimentally, one may test the two cases via competitive growth assays with variable space and energy restrictions. Importantly, the two selection modes (time and energy) lead to several of the same implications, notably with a selective advantage for streamlining and particular selection on highly expressed proteins as they may limit both time and energy costs of growth (Wang et al., Reference Wang, Kurland and Caetano-Anollés2011).

One recent study that casts light on this is a study of pathways choices among different sequenced organisms (Du et al., Reference Du, Zielinski, Monk and Palsson2018). The study found that different organisms select specific choices of precursor pathways based on both metabolic cost and synthetic efficiency. Cost selection occurs in energy-poor habitats, whereas in energy-rich habitats, the default selection mode is time. There are correlations between time and energy advantages. Notably, the synthesis time of expensive amino acids is all-else-being-equal long as more phosphate bonds must be recruited during synthesis. The cost of handling misfolded proteins can limit growth substantially, as seen in a case of ~3% growth rate reduction in yeast upon folding-stability-impaired mutants of only one protein (YFP) (Geiler-Samerotte et al., Reference Geiler-Samerotte, Dion, Budnik, Wang, Hartl and Drummond2011).

The shift in selection pressure from time to energy can also explain the important phenomenon of overflow metabolism, the tendency of using more expensive, but faster fermentation rather than respiration during growth (Basan et al., Reference Basan, Hui, Okano, Zhang, Shen, Williamson and Hwa2015). PCM theory implies that microorganisms shift to fermentation in rich habitats and growth media, because time is the main selection pressure, whereas in poorer habitats, respiration becomes favored and selected upon because energy is restrictive, although combinations of strategies will probably be common. The choice between these options depending on energy availability could be relevant to many growth assays, but perhaps also to the Warburg effect of cancer cells (Basan et al., Reference Basan, Hui, Okano, Zhang, Shen, Williamson and Hwa2015). Cancer cells are remarkable by being under selection both for time and space in competition with each other against the selection pressure of the body's immune system. Cancer cells tend to use cheaper amino acids (Zhang et al., Reference Zhang, Wang, Li, Chen, He, Zhang, Liang and Lu2018), in accordance with PCM theory, but when energy is widely available, growth-limiting space and time restrictions would favor the Warburg effect over oxidative phosphorylation, although other contributing effects such as mutation impacts and oxygen availability are relevant as well.

Temperature, thermostable proteins, and thermophilic organisms

As mentioned above, the habitat temperature also imposes a selection pressure on evolution according to the PCM theory, because it directly modifies protein stability ΔG_i/RT and thereby, the fitness function, Eq. (11). To appreciate this, we used a sign convention of negative ΔG_i for a stable protein, and the ΔG_i is the optimal stability of the protein at its temperature of operation (sometimes called T*), typically reflecting to some extent the organism's experienced extrema temperatures in the relevant habitat (Robertson and Murphy, Reference Robertson and Murphy1997). The protein has been optimized to display its maximal stability at this T*, with ΔG_i typically harmonic in the temperature, and increasing or decreasing the temperature away from T* will thus increase the number of misfolded proteins U_i and increase the associated turnover costs, thereby reducing fitness, Eq. (11) (Robertson and Murphy, Reference Robertson and Murphy1997).

Using the theory, we can better understand adaptation of proteomes to hot or cold environments (thermophiles and psychrophiles, respectively) (Li et al., Reference Li, Zhou and Lu2005; Mozo-Villiarías and Querol, Reference Mozo-Villiarías and Querol2006; Luke et al., Reference Luke, Higgins and Wittung-Stafshede2007; Fu et al., Reference Fu, Grimsley, Scholtz and Pace2010). Adaptations to a warmer habitat is largely expected to be a question of optimizing the proteome's copy-number-weighted median protein T* (the most representative T* of the proteome of the cell) toward the T of the habitat, to minimize the average copy number of misfolded protein copies in the cell at any given time, again to minimize proteome costs and maximize energy available for reproduction. Many studies of thermophilic proteins and thermophilic adaptation may be seen in this light, without going into further details, as this is a large and complex topic (Tekaia et al., Reference Tekaia, Yeramian and Dujon2002; Sawle and Ghosh, Reference Sawle and Ghosh2011; Venev and Zeldovich, Reference Venev and Zeldovich2018), but the essential implications should be clear. In particular, thermophilic organisms are predicted to adjust protein thermostability mainly for the most abundant and quickly turned-over proteins that pose the largest economical cost to the proteome.

PCM, aging, and neurodegenerative diseases

Proteome cost minimization has been argued to explain a substantial part of the evolution on longer evolutionary timescales, producing clear biases in the use of amino acids and explaining the E-R anti-correlation by slowing the probability of fixating new mutations in abundant, expensive proteins, and giving rise to important cost–function trade-offs. The evolution that shaped these relations mainly occurred in single-cell organisms, and it is thus of interest to consider whether the theory has implications also for evolution of higher organisms and in particular the evolution of aging.

A note is required first on intrinsically disordered proteins (IDPs), which make up a substantial fraction of all proteins in a typical cell. IDPs are disordered as part of their natural function, which can be expected to require structural plasticity or specific conformational changes as the local environment changes, or upon interaction with binding partners (Uversky et al., Reference Uversky, Oldfield and Dunker2008). The required disorder may lead to particular sensitivity and potential elevated cost of turnover. The common involvement of IDPs in protein misfolding diseases hints to the importance of proteome maintenance, which we argue should be counted in bioenergy units (Kepp, Reference Kepp2019).

All higher organisms use oxidative phosphorylation as the most effective energy-producing process, using the O₂ of the planet's atmosphere produced by the photosynthetic organisms as primary electron acceptor. The free radical theory of aging argues that aging arises from the incurred damage due to the activity of reducing O₂ to water, as the radical side products of the respiratory chain leads to a consistent mutagenic pressure that needs to be countered by DNA repair and antioxidant defenses (Speakman et al., Reference Speakman, Selman, McLaren and Harper2002; Harman, Reference Harman2003).

Different higher organisms have evolved different trade-offs between life history variables relating mainly to the generation time (Kirkwood and Rose, Reference Kirkwood and Rose1991; Shanley and Kirkwood, Reference Shanley and Kirkwood2000; Kirkwood, Reference Kirkwood2011). Shorter lifespan implies specialization toward shorter generation time, which again implies less energy invested in maintenance of the proteome. Based on the discussion above, this specialization emphasizes time over energy. Each strategy probably involves an aging program to ‘dispose the soma’ after reproduction to make space for the next generation, although this remains debated (Westendorp and Kirkwood, Reference Westendorp and Kirkwood1998; Speakman et al., Reference Speakman, Selman, McLaren and Harper2002). Aging may thus be a direct consequence of the reproductive strategy. Some organisms specializing toward long lifespan (i.e. r- versus k-strategists) also diversify toward complex lifestyles with capacity for technology transfer, e.g. cetaceans and apes. Compared to primates, rodents on average have shorter generation times, lifespans, larger litter size, and have traded lifespan for fecundity (Speakman et al., Reference Speakman, Selman, McLaren and Harper2002; Wensink et al., Reference Wensink, van Heemst, Rozing and Westendorp2012). In long-living organisms, proteome misfolding may cause death, perhaps because PCM can no longer be afforded beyond what was evolutionarily beneficial. It is reasonable to argue that the aging program of long-living mammals largely reflect the (active or passive) giving up of the maintenance of the proteostatic machinery to enable the rise of the next generation (Taylor and Dillin, Reference Taylor and Dillin2011; Hipkiss, Reference Hipkiss2017).

This discussion is well illustrated by superoxide dismutase 1 (SOD1). SOD1 is one of the most abundant proteins in primates and A_i can reach 100 000 copies per cell (Dasmeh and Kepp, Reference Dasmeh and Kepp2017), it is the central antioxidant defense protein of the mitochondria thus directly linking energy and aging (Perry et al., Reference Perry, Shin, Getzoff and Tainer2010), it is one of the few proteins known to directly extend lifespan upon induction (Tolmasoff et al., Reference Tolmasoff, Ono and Cutler1980; Landis and Tower, Reference Landis and Tower2005), and one of the few genes of great apes known to have undergone non-synonymous positive selection (Fukuhara et al., Reference Fukuhara, Tezuka and Kageyama2002; Dasmeh and Kepp, Reference Dasmeh and Kepp2017). Deposits of misfolded SOD1 is a hallmark of age-triggered amyotrophic lateral sclerosis (Valentine et al., Reference Valentine, Doucette and Potter2005). The tendency toward aggregation and misfolding of natural human SOD1 variants correlates with their pathogenicity (Lindberg et al., Reference Lindberg, Byström, Boknäs, Andersen and Oliveberg2005; Wang et al., Reference Wang, Johnson, Agar and Agar2008; Kepp Reference Kepp2015), and wild-type overexpression by itself is enough to trigger disease (Wang et al., Reference Wang, Deng, Grisotti, Zhai, Siddique and Roos2009). Recent amino acid substitutions in SOD1 of great apes correlate with longer life span and tend to increase the net charge and stability of SOD1, thus increasing the thermodynamic and kinetic stability of the protein (k _d and ΔG_i) (Dasmeh and Kepp, Reference Dasmeh and Kepp2017). Via its abundance and functional importance, any impairment of SOD1 either in terms of function or stability will produce comparatively very large PCM costs. The combination of the features summarized above strongly argues for a relationship between PCM, evolution of aging, and age-triggered neurodegenerative diseases.

According to the PCM theory, neurodegenerative diseases are caused by the increased energy spent on maintaining the proteome of old humans, which leaves less energy available for neuron and motor neuron function. Protein turnover and neuron signaling costs perhaps 20–25% and 50% of the brains energy budget (Hawkins, Reference Hawkins1991; Attwell and Laughlin, Reference Attwell and Laughlin2001; Raichle and Gusnard, Reference Raichle and Gusnard2002), respectively, and as age advances, the supply of energy may no longer satisfy the increasing maintenance costs of the proteome (Kepp, Reference Kepp2019). Familial inherited mutations that tend to produce more aggregation-prone protein will increase turnover costs per time units according to PCM theory and will accordingly also accelerate the time at which available energy no longer satisfies the needs of synaptic transmission, leading to earlier clinical age of onset of disease (Kepp, Reference Kepp2019).