13 Flux Modulation: Resistance

Reaction rate depends on the thermodynamic driving force that pushes reactants toward products and on the resistance that opposes the reaction. Equation 11.9 expresses reaction flux by an approximate analogy with Ohm’s law as flux = force / resistance.

Chapter 12 focused on the driving force of reactions, which is the increase in entropy between reactants and products (eqn 11.4). This chapter considers resistance, which impedes reactions.

The first section reviews how resistance alters chemical reaction flux. The second section describes mechanisms that modulate resistance. Constraints limit the control of resistance and flux.

The third section raises genetic drift as a fundamental constraining force on design. In small populations, stochasticity in reproduction can overwhelm any fitness differences between alternative traits. Drift may be particularly important when analyzing the design of metabolic control for individual biochemical reactions.

The fourth section highlights general challenges in the study of metabolic control. The fifth section lists specific problems of reaction flux. The final section notes gaps in current understanding and prospects for further work.

The design of regulatory control provides a natural extension to this book, setting a primary task for the future.

13.1 Resistance Impedes Flux

ACTIVATION ENERGY BARRIER

Organisms inevitably decompose into water, carbon dioxide, nitrogen, and other simple inorganic compounds. Those compounds have much lower free energy than the organic compounds of life. The driving force toward decay is strong. Yet decay happens slowly enough to allow life.

In other words, thermodynamic driving force tells us where things end up in the long-run equilibrium. But the thermodynamics of decay does not by itself explain chemical kinetics on the timescales that matter for life.

We must also consider the driving force of negative entropy from food, flowing from the sun or from geochemical disequilibria. And we must consider the resistance that opposes reactions.

With regard to resistance, consider that both diamond and graphite are composed of carbon atoms bound to one another. Graphite has lower free energy, so the thermodynamic driving force favors diamond to decompose into graphite. However, that happens so slowly in the conditions in which we live that essentially it does not happen. The resistance is very high.

Resistance occurs because the transition requires diamond to break its strong carbon-carbon bonds before reforming a different pattern of carbon bonding in graphite. The intermediate stage with broken carbon bonds has much higher free energy than the diamond crystal. So the intermediate transition almost never forms spontaneously, and the transformation to graphite almost never occurs.

In general, an intermediate reaction state with higher free energy than the initial reactants is called an activation energy barrier. Figure 13.1 shows an example. The heights of the molecular forms correspond to relative free energy levels.

The change from reactants to products, R → P, decreases free energy and has a strong thermodynamic driving force. However, the reaction happens very slowly. The reaction must go through an intermediate, R → C → P, in which thermodynamic driving force works against the formation of the intermediate complex, C. Thus, C imposes high resistance, impeding the driving force that favors R → P flux.

Mechanisms that alter driving force and resistance include changes in the concentrations of reactants and products, changes in conditions such as pH or the surrounding solvation environment, barriers to diffusion that resist encounters between reactants, catalysts that alter the intermediate complex, and so on.

The variety of biophysical mechanisms can only roughly be divided into abstract driving force and resistance categories. For example, reduced diffusion alters molecular concentrations and the thermodynamic driving force. Alternatively, one may consider limited diffusion as a physical resistance mechanism that impedes molecular motion and reaction. Nonetheless, the abstractions of force and resistance provide insight.

Figure 13.1 The activation energy barrier impedes a reaction with strong thermodynamic driving force. (a) The initial reactant, R, must change into an intermediate reaction complex, C, before completing the reaction by forming the product, P. Greater height of a molecular form signifies higher free energy. (b) Symbolic expression of reaction fluxes between states. We may denote each rate of transition as k_ij, the transition from i to j. From the left panel, we can derive the relative rates as $k_{p c} < k_{r c} < k_{c r} < k_{c p}$ . (c) When the intermediate complex, C, decays rapidly and is not observed, one instead observes forward and backward transitions between the reactants and products at rates k_pr < k_rp. Those rates depend on the free energy driving force between the reactants and products and on the resistance imposed by the increased free energy of the intermediate complex.

SHORT-TERM KINETIC VERSUS LONG-TERM THERMODYNAMIC CONTROL

Useful organic molecules must be sufficiently stable so that they do not decay too rapidly. And those molecules must be sufficiently reactive so that they can be changed into other forms or be destroyed as needed.

Figure 13.2 shows the tension between thermodynamically driven decay and kinetically driven transformation between molecular forms. We start with some reactant molecules, R. Very strong driving force favors decay to inorganic products, P. But strong resistance greatly slows decay because the high free energy intermediate complex, C, rarely forms.

The initial reactants, R, can also change to an alternative molecular state, S, that may be a useful variant form. The alternative state, S, has only slightly lower free energy and relatively low driving force toward formation. However, the resistance for R → S is relatively low, causing that transformation to happen relatively quickly.

Figure 13.2 Kinetic versus thermodynamic control. Initial reactants, R, may decay to P or be modified to an alternative molecular form, S. In the short term, kinetics favors transformation to S because the lower activation barrier for B makes R → S relatively rapid. In the long term, thermodynamics favors the transformation of molecules to P because molecular forms eventually move to the lowest free energy state. (b) If the intermediates B and C transform into other molecules very quickly, one only observes the fluxes between the more stable molecular forms, represented by this diagram. See Mallory et al.260 for flux analysis in relation to reaction barriers and thermodynamic driving force.

Figure 13.3 shows an example of the reaction dynamics. Initially, all molecules are in state R. The alternative S form arises quickly (light curve) because of the low resistance for the R → S transformation. Over a long period of time, all molecules move toward the lowest free energy state, P, because thermodynamic driving force eventually dominates (dark curve).

For cells to function, their high free energy organic molecules must not decay rapidly to low free energy inorganic components. In other words, the building, maintenance, error correction, and recycling of organic molecules requires sufficient resistance to impede the long-term driving force toward decay.

Given sufficient resistance to long-term thermodynamic decay in the R → P transition of Fig. 13.2, cells still have the short-term problem of controlling the kinetics of the R → S reaction. Altering short-term resistance provides one mechanism to modulate flux.

Figure 13.3 Kinetics dominates in the short term and thermodynamics dominates in the long term. The plot shows an example of the dynamics implied by Fig. 13.2. Initially, all molecules are in state R. The light and dark curves show the fraction of molecules in states S and P, respectively. The dynamics are based on linear transition rates *k_rs* = 60, *k_sr* = 0.5, *k_rp* = 0.05, and *k_pr* = 0.00001.

13.2 Mechanisms to Alter Resistance and Flux

ENZYME CONCENTRATION

A protein catalyst (enzyme) may reduce the free energy of the intermediate complex. Reduced activation energy lowers the resistance barrier between reactants and products, increasing the reaction flux.

Changing the enzyme concentration provides a mechanism to control flux. The concentration of a particular enzyme depends on four rate processes.174 Transcription creates mRNA. Transcript decay removes mRNA. Translation creates proteins in proportion to mRNA abundance. Protein decay removes proteins.

Suppose decay rates are constant. Then enzyme concentration varies with the transcription and translation rates. Cells can make the same concentration by raising one of the rates and lowering the other. In theory, fast transcription and slow translation produce approximately the same concentration as slow transcription and fast translation.

In a comprehensive dataset, the combination of high transcription and low translation rates rarely occurred.¹⁷⁴

What might favor relatively low transcription and high translation rates? Perhaps the benefits of limiting expensive transcription outweigh the costs of greater noise with low mRNA transcript numbers.

On the cost side, lower transcription rate increases noise. Noise increases because the number of mRNA transcripts may drop to the point at which stochastic production of one more or one less transcript strongly influences output.^271,293

On the benefit side, lower transcription rate increases observed growth. Additional mRNA may be inefficient and costly for growth because each transcript is used less often to make a given amount of protein.^145,200,271

The rarity of high transcription and low translation suggests that the loss in growth rate from excess transcription typically outweighs the gain from suppressing noise.¹⁷⁴

COMPARATIVE PREDICTIONS

The tentative conclusions about the biophysical constraining forces of production cost and noise pose a puzzle about the control of enzyme concentration. How do the forces of design, which weight noise and growth components of fitness differently in different environments, affect the noise-growth tradeoff in transcription and translation? Comparatively, can we predict how changes in the environment alter expression?

Comparative tests could be applied to whole genomes, analyzing the overall tradeoff in metabolism between noise and efficiency. Or the tests could be applied to particular reactions, analyzing the noise-efficiency tradeoff in the control of particular metabolic steps.

The latter tests link flux control for particular aspects of metabolism to the ways in which changed environments alter fitness components. Natural history shapes biochemistry, the primary theme of this book.

ENZYME MODIFICATION

Modifying an enzyme can significantly change its catalytic properties. For example, adding a phosphate group to a single amino acid can alter how the enzyme binds to its substrates. Phosphorylation commonly occurs by taking a phosphate group from ATP and adding that phosphate group to an enzyme,

$ATP + E \to_{}^{kinase} ADP + E - Pi,$

in which ATP gives up a phosphate group to produce ADP, and the phosphate group is attached to the enzyme as E − Pi. A kinase enzyme catalyzes the phosphorylation reaction.⁴²⁸

A phosphatase enzyme removes the phosphate group, reversing the phosphorylation modification,

$H_{2} O + E - Pi \to_{}^{phosphatase} Pi + E .$

An enzyme can have many hundreds of amino acids. Phosphorylation or similar covalent modification typically changes only one small molecular group on one amino acid. Altering flux by enzyme modification is much faster and less costly than altering enzyme concentration through protein production and degradation pathways.

Eukaryotes rapidly adjust enzymes by kinase-phosphatase pairs or by other modifications.¹⁸⁹ Several studies suggest that prokaryotes also regulate flux via enzyme modification.^56,158,232

Enzyme modification gains the benefits of speed and relatively low cost. However, small modifications typically cannot provide the level of enzymatic specificity and efficiency achieved by a custom enzyme for a specific task. Different challenges acting over different timescales likely favor different mechanisms for controlling reaction flux.¹⁰⁰

ALLOSTERIC CONTROL

Small molecules can bind to enzymes, changing enzyme structure. Such allosteric change in structure often modifies the catalytic activity of the bound enzyme.^287,412

The aggregate catalytic activity of a target enzyme can be modulated by altering the concentrations of small allosteric effectors. Allostery provides another relatively fast and inexpensive way to control flux.³¹⁹

Aspartate transcarbamoylase (ACTase) provides a classic example of allostery.⁴²⁸ This enzyme catalyzes a key step in pyrimidine synthesis and the production of nucleotides.

A later step in the pyrimidine synthesis pathway makes cytidine triphosphate (CTP). Binding of CTP to ACTase reduces enzyme activity. This allosteric binding of CTP to ACTase creates a negative feedback loop that prevents overproduction.

A separate nucleotide synthesis pathway makes purines. The cell requires a balance between purines and pyrimidines. Balance arises by an excess of purines stimulating pyrimidine production.

Cells use the purine ATP to achieve positive feedback from purines to pyrimidines. Binding of ATP to ACTase stimulates enzyme activity, creating a positive feedback that balances purines and pyrimidines.

These two allosteric modifications of ACTase create a positive and negative feedback pair that regulates nucleotide synthesis.

TRANSCRIPTION FACTORS AND PHYSIOLOGICAL RESPONSE

A typical transcription factor is a protein that binds to DNA, altering the rate of transcription and protein production of nearby genes.^24,180 A single transcription factor may bind to multiple DNA sites, altering expression for a set of genes.

Transcription factors regulate the production of enzymes that modulate the resistance and flux of reactions. Cells often initiate wide changes in their biochemical flux pathways by altering the abundance of particular transcription factors.

Shifts in enzyme concentrations typically happen on a slower timescale than covalent modification or allosteric binding of enzymes.

Space and production costs may limit the total amount of protein, including transcription factors. Suboptimal control of gene expression may occur widely in prokaryotes because of limitations on the abundance of transcription factors.³²⁹

Much research focuses on the biophysical structure of transcription factors and the mechanisms by which they control the production of proteins.³¹⁵ Broader design puzzles have received relatively little attention. For example, how do the life history forces of design alter the characteristics and expression of particular transcription factors?

PROTEOME LIMITATION

Space and resource constraints limit total protein production. Proteome limitation implies tradeoffs in flux because cells cannot make enough enzyme to control the flux of all reactions.^28,165,454

Fast growing cells may reduce proteome limitation by increasing ribosome count and protein production.³⁴³ Fast growing cells sometimes have larger cell size,^417,445 which may reduce proteome space constraints.

The potential for cells to modify constraints raises questions about design. What environmental conditions favor cells to increase proteome size? How do demographic factors influence the relative weighting of growth and other fitness components, setting the costs and benefits of cell and proteome size?

SPATIAL SEPARATION

Within the cellular cytosol, many reactions appear to be limited by diffusion.^355,359 In diffusion-limited reactions, chemical transformation from the collision of reactants in one spatial location happens faster than the time it takes for other potential reactants to re-equilibrate into spatially homogeneous concentrations.

Uneven spatial distribution of reactants impedes reaction flux, increasing resistance against the potential driving force. Creating localized reaction centers and modulating diffusion within the cytosol provide mechanisms to alter the resistance that impedes reactions.

Eukaryotes, with their internal membranes and phase-separated partitions, have greater intracellular barriers than do prokaryotes.³⁷⁷ Within prokaryotes, much diversity likely occurs in the mechanisms by which reactions are localized, reactants are separated, and gradients are modulated and exploited.^109,355,398 The study of separation mechanisms is currently an active and controversial topic.²³³

MEMBRANES AND DISEQUILIBRIUM

Membranes create a primary physical barrier that impedes reactions. Cells modulate spatial gradients across membranes by altering diffusion or transport.^35,86

Changing the concentrations across membranes modifies the resistance against reactions. Dissipating disequilibrium across membranes can drive coupled reactions that would otherwise be unfavorable. Exploiting flux across membranes to drive other processes provides a primary force for much of life.^86,467

Chemical gradients between cells also create resistance and disequilibria. Groups of cells may exploit intercellular resistance and flux to create physiological processes across a social network.³⁵

TEMPERATURE AND THERMODYNAMIC INHIBITION IN METABOLISM

Temperature affects reactions.¹⁰ Heat speeds things up, alters entropy changes and driving force, and modifies resistance via diffusion. Temperature also influences the enzymes and regulatory mechanisms that control flux.³²⁷

When temperature changes, flux may be perturbed, potentially leading to product inhibition or otherwise creating bottlenecks. Temperature may more strongly perturb reactions with low net free energy change because small perturbations significantly alter the flux of those intrinsically slow reactions.

Organisms that live in stable temperatures may be tuned differently from organisms that face fluctuating temperatures. With fluctuation, organisms may require special designs to cope flexibly with altered rates. They may also need special functions to clear product inhibition or other bottlenecks that arise from concentration mismatches between flux pathways.

Reactions typically occur as parts of pathways. Thus, temperature effects must be considered in the context of pathway flux rather than as a single reaction step.³⁶⁵

Multicellular eukaryotes often control their temperature, which influences metabolic flux. Do microbes alter temperature to change flux?¹⁴¹

13.3 Genetic Drift

This section notes a common constraint on design forces. Evolutionary processes are inherently stochastic. As population size declines, stochasticity in reproduction between alternative traits may overwhelm any fitness differences between those traits.71

Put another way, stochastic genetic drift imposes a constraint on the potential for weak forces of design to shape traits. As always, comparative predictions give the most insight. For example, Lynch²⁵⁴ showed that a reduction in population size and the associated increase in genetic drift have shaped many aspects of genomes.

Lynch considers genetic drift as a nonadaptive force. I agree. In this book, I typically label nonadaptive forces as forces of constraint.

For most problems discussed in this book, the forces of design are likely to overwhelm the weak constraining force of genetic drift. However, for the flux control of individual metabolic steps, drift may sometimes be important.

The challenge is to formulate comparative predictions. How do changes in population size and genetic drift alter expectations for the control of enzyme abundance levels? How do changes in robustness that protect against perturbations alter the relative strength of design forces and genetic drift (p. 107)?^125,134

I leave those important questions and return to my main goal, clarifying design forces and the best ways to study those forces.

13.4 Challenges in Control Design

Metabolic components must adjust to each other. The overall metabolic system forms a major part of the environment for each component.

Functions of the metabolic system include sensing information, filtering out false signals, correcting errors, speeding adjustment, and enhancing stability. It can be difficult to match the particular biochemical mechanisms of metabolic components with their functional attributes at the system level.227

Sometimes we can think directly about component design. For example, the ATP–ADP disequilibrium provides a focal point for contrasting the efficiency of free energy capture with the use of disequilibria to drive growth. We can study how different environmental challenges alter the balance between efficiency and growth.

However, it can be difficult to match abstract system aspects of metabolic control to observable biochemical components. For example, error-correcting feedback can maintain overall system homeostasis, balancing allocations to stress resistance, maintenance, and growth. How do we relate those system-level controls and functions to the biochemical component traits that we can measure?

With those difficulties in mind, the next section lists a few problems of flux control that arose in prior sections. The last section considers the broader challenges and prospects for studying the design of control systems. The key is to focus on functional aspects, such as sensing, filtering, correcting, speed, and stability (Chapter 7).

13.5 Problems of Flux Modulation

DRIVING FORCE VERSUS RESISTANCE

The prior sections raised several problems.

Changes in reactant concentrations alter driving force. Changes in enzymes and spatial barriers alter resistance. What conditions favor controlling flux by modulating force or modulating resistance?
Low resistance caused by excess enzymes or other causes typically leads to near-equilibrium flux. Flux sensitivity to small changes in driving force increases as resistance drops.
On the benefit side, near-equilibrium flux reduces the loss of free energy and raises sensitivity to force-altering changes in reactant and product concentrations, providing a fast and simple way to modulate flux.
On the cost side, flux slows near equilibrium, product inhibition stops or reverses flux, and low resistance may require costly production of enzymes. How do differing conditions alter the weighting of these costs and benefits for near-equilibrium flux?
Far from equilibrium, flux becomes sensitive to small changes in enzyme activity, or spatial barriers, or whatever is resisting flux. Small changes in reactant concentrations have little effect on flux. Reactions that are far from equilibrium dissipate a lot of free energy.
When is it advantageous to regulate reactions near or far from equilibrium? How do the biophysical mechanisms that influence metabolic concentrations and resistance properties affect the design of flux control?

These problems of biochemical control form a broad area of research. Most of the work focuses on reaction dynamics and biophysical mechanisms that alter flux.9,100,307,371,429

CELL SIZE AND PROTEOME SIZE

Microbial cell and genome sizes vary widely.^{194,206,237,325,373,391,415} Size correlates with environmental attributes and with many cellular traits, including lifespan, growth rate, and abundance. Size may influence metabolic flux.

Size alters the ratio between membrane surface area and cell volume, which affects opportunities for membrane-based reactant gradients and for internal diffusion barriers.
Cell volume may limit proteome size, which influences tradeoffs between the abundances of transcription factors, enzymes, and modifiers of enzyme activity.
Fast cellular growth correlates with larger cell size. How does increased cell size alter the modulation of metabolic flux? Does a rise in proteome size and the opportunity for more proteins shift control in predictable ways?
Larger cells for the same genome have more room for proteins. Does varying cell size reveal the relative importance of limited proteome space versus the limited genomic coding capacity?
Advancing technology will improve measurements of kinases, transcription factors, and other proteome components. How do the forces of design shape allocations to these various classes?
For example, do particular environmental changes favor greater response speed, relatively more kinases, or enhanced allosteric modification?

13.6 Limitations and Prospects

We can often measure how things change within cells. The problem of design concerns understanding why they change.

SCALE

The scale at which we measure often does not match the scale at which function arises. Function has to do with how cellular traits affect components of fitness. To understand design, we must match how a change in a cellular trait alters the various components of fitness.

Consider an example. We can often work out how one molecule affects another. An increase in A may enhance or repress B. Similarly, B may enhance or repress A and also affect C. The six paths between the three molecules form a little network.

If we code each path as plus, minus, or no effect, then there are 13 possible network motifs between three molecules.9 It turns out that cells use some motifs much more often than others.

We can think of each motif as a little input-output machine. An input alters the dynamics of one molecule, such as its production or decay rate. That input-induced change shifts the molecular abundances of the network nodes, creating the output consequences of the input.

Each motif has different input-output properties. The properties include such things as how a motif amplifies an input signal, filters noisy inputs when producing outputs, or keeps the abundances of the three molecules in relative balance by feedbacks.

These facts tell us how cells deploy biochemistry to make component input-output modules. It is a bit like how computer components process electron flow to create particular logical operations. Cells use those small-scale biochemical modules to build larger networks that execute cellular control programs.

LIMITATIONS

Large-scale computer programs depend on their small-scale logical components. But we cannot understand the design of computer programs to achieve particular real-world functions by knowing only how the underlying components manipulate electrons to create logical operations.

Similarly, we cannot understand the design of cellular control to influence fitness components by knowing only how small biochemical motifs process chemical input-output operations.

Many people have understood the need to match cellular control traits to fitness components.¹⁹⁰ A few studies have linked cellular traits to growth rate or yield efficiency. Focus on those two fitness components arises because common methods can measure them in the laboratory.

However, one cannot understand design by those common laboratory methods of measurement. Studies must extend to the broader array of environmental challenges and fitness components that matter.

PROSPECTS

The solution is always the same. Make comparative predictions about how changing environmental challenges alter fitness components and design. Figure out how to test those predictions.

For example, how does an increase in the genetic variation between competitors alter flux control? How much does the understanding of metabolic flux and biochemistry depend on such links to natural history?

This book does not develop comparative predictions for metabolic and cellular control. That development requires synthesizing observations and extending the formulation of the key control concepts in Chapter 7 to create an applicable framework. The study of cellular control design remains an important open challenge for future work.