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Principles of Regulation and Control in Biochemistry: A Pragmatic, Flux‐Oriented Approach

B. Crabtree, E. A. Newsholme, N. B. Reppas

10.1002/cphy.cp140105

Source: Supplement 31: Handbook of Physiology, Cell Physiology

Originally published: 1997

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Flux‐Oriented Theory of Regulation
1.1 Flux‐Generating Steps
1.2 Branched Fluxes: Reversible Reactions
1.3 Other Types of Flux: Energy Fluxes and Equivalence
1.4 Internal and External Effectors
1.5 Control and Regulation of Metabolic Fluxes and Concentrations: Communication and Regulatory Sequences
1.6 Control Versus Regulation
1.7 Identification of Communication Sequences In Vivo
1.8 Need for Sensitive Regulation In Vivo
1.9 Metabolic Sensitivities
1.10 Analysis of Non‐Steady States
2 Two Other Important Approaches to Metabolic Control and Regulation
2.1 Biochemical Systems Theory (BST)
2.2 Metabolic Control Theory (MCT)
3 Summary and Future Prospects
Figure 1. Figure 1.

Simple linear flux. A‐D represent metabolic intermediates, E1‐E4 represent enzyme‐catalyzed reactions and symbol denotes saturation (here, with A).
Figure 2. Figure 2.

Metabolic carbohydrate flux, (a) Conventional pathway of glycolysis, originating with glucose entry into a cell, (b) whole‐body flux, of which (a) is a component. E1 = glycogen phosphorylase. In this, and other figures the cofactors are omitted, unless relevant to the discussions.
Figure 3. Figure 3.

Branched flux.
Figure 4. Figure 4.

Fluxes with reversible steps. (a) Hypothetical linear flux, (b) glycolysis: a = hexokinase (HK); b = phosphoglucose isomerase (PGI); c = phosphofructokinase (PFK); d = pyruvate kinase (PK); e = lactate dehydrogenase (LDH); f = pyruvate dehydrogenase (PDH).
Figure 5. Figure 5.

Energy fluxes, (a) Equivalent fluxes; JE=2xJl, (b) non‐equivalent fluxes; total energy flux = 2 × Jl + 38 × J0, which is not proportional to, and thus not equivalent to, the total carbon flux, Jo+Jl.
Figure 6. Figure 6.

Linear flux with external effectors. The broken lines denote regulatory (for example, allosteric) interactions.
Figure 7. Figure 7.

Partially external regulators. The concentrations of metabolites M and N are only partially determined by flux J. On the other hand, they are totally determined by the wider system, J + K. Regulator X (totally external for J) communicates with M, but M (partially external for J) does not communicate with X.
Figure 8. Figure 8.

Truncated flux. The reactions producing B are omitted, so that B is a partially external regulator of the truncated flux, Jt.
Figure 9. Figure 9.

Flux with feedback inhibition. (a) Simple linear system, (b) system with extended feedback (end product) inhibition. ⊖ denotes an inhibition, ⊕ denotes an activation.
Figure 10. Figure 10.

Outline of the control of glycolysis by ATP. E1 = phosphorylase b; E2 = phosphofructokinase (PFK). ⊕ denotes an activation.
Figure 11. Figure 11.

System with two potential control sites, only one of which is used. Here, as in subsequent figures, the greek letters denote relative‐change intrinsic or component sensitivities.
Figure 12. Figure 12.

Hill responses. The curves are calculated from equation 35, with K = 1 and Vm = 100.
Figure 13. Figure 13.

Reversible system spanning a membrane. XI and XO denote concentrations inside and outside a cell compartment, respectively. M is a metabolite present only on the inside.
Figure 14. Figure 14.

Substrate cycles. (a) Simple hypothetical cycle, (b) cycle between fructose‐6‐phosphate (F‐6‐P) and fructose‐1,6‐bisphosphate (FBP), (c) cycle between triacylglycerol and FFA; the 7 ATP molecules consumed per turn comprise two each for the conversion of the three molecules of FFA to their CoA esters, and one for the formation of glycerol phosphate.
Figure 15. Figure 15.

Interconversion cycle. Enzyme E exists in two forms, a and b, which are interconverted by E1 and E2. The energy consumption here is represented by a net hydrolysis of ATP to ADP and Pi. In the steady state the rates of E1 and E2 are equal, and there is no net flux across the cycle.
Figure 16. Figure 16.

Regulation of pyruvate dehydrogenase (PDH) by pyruvate. Pyruvate has a direct effect (rm + rc) and an indirect one, via inhibition of the phosphatase component of the interconversion cycle.
Figure 17. Figure 17.

Reversible reactions in situ. (a) Simple hypothetical system, (b) flux of serine biosynthesis in vivo. E1 = phosphoglycerate dehydrogenase plus phosphoserine aminotransferase; E2 = phosphoserine phosphatase. The symbol, denotes saturation.
Figure 18. Figure 18.

Basic control of glycolysis by ATP. This diagram illustrates the general principles: the presence of auxiliary regulators and other mechanisms makes the actual system much more complicated.
Figure 19. Figure 19.

Linear flux with external regulator of E2.
Figure 20. Figure 20.

Linear flux with reversible reaction at E2.
Figure 21. Figure 21.

Linear flux with partially external regulator at E2.
Figure 22. Figure 22.

Opening the system in Figure 21.
Figure 23. Figure 23.

Branched flux with two external regulators.
Figure 24. Figure 24.

The pathway of glutaminolysis.
Figure 25. Figure 25.

Relationship of serine biosynthesis to glycolysis.
Figure 26. Figure 26.

Linear flux with a regulator interacting at two sites.
Figure 27. Figure 27.

Linear non‐steady state system. Here v1, v2 and v3 are not mutually equal, so that S and P vary with time and there is no defined system flux; α‐γ denote absolute‐change sensitivities.
Figure 28. Figure 28.

Effect of the speed of response on a pulsed system. The pulsed stimulus lasts for time τ. Curves 1–4 show the effect of a coordinated increase of the initial rate of cycling between the b and a forms of an interconversion cycle. In curve 4 a steady state has been reached within time τ and any further increase in the rate of initial cycling has no further effect.
Figure 29. Figure 29.

Metabolic buffers. (a) Equilibrium. Q = Keq.S and the resulting larger pool of S is shown in parentheses. Fluxes I and O are equal in the steady state. (b) Receptor recycling. The hormone binds tightly to its receptor, but the complex is destroyed when it is internalized (via F); the free receptor is then recycled (via C). Here, fluxes I and O represent the delivery and removal of the hormone, respectively. (c) Substrate cycle. Here, in the steady state there is no net flux across the cycle, that fluxes I and O are equal. S and P may denote (1) glucose and glycogen, (2) amino acids and protein, (3) fatty acids and triacylglycerol, in which case the cycles may be regarded as storage cycles. E1 and E2 are assumed to have first‐order responses to S and P, respectively.
Figure 30. Figure 30.

Oscillations. Xs is the steady state value, or balance point around which X oscillates. The broken line shows a fading, or damped, oscillation.
Figure 31. Figure 31.

System with indirect removal of an intermediate. The indirect communication, S∼M∼E2, introduces a delay between the production of S (by E1) and its removal (by E2).
Figure 32. Figure 32.

System with a long feedback loop.
Figure 33. Figure 33.

A branched system in BST formulation.
Figure 34. Figure 34.

Linear flux in MCT formulation. A comparison with Figure 11 shows that ε 1/s = α; ε 2/s = β.
Figure 35. Figure 35.

Interconversion cycle with a single molecule of protein. a, b, and c refer to the same molecule and hence cannot coexist. k1 and k2 are classical rate constants that would give the observed rates as a function of the fractional times spent as a or b. The conversion of a into c takes a time π.
Figure 36. Figure 36.

Flux‐generating step with a particulate system. G is a glycogen particle, to which phosphorylase and other enzymes (E) are tightly bound. E does not dissociate as glucose‐1‐phosphate is released.
Figure 37. Figure 37.

Switch mechanism at an interconversion cycle. (a) Cycle, (b) response of E1 to b: substrate inhibition, (c) response of active, a form to regulator X. At X2 there is a large increase of a, which is not reversed until X is lowered to X1, a value much lower than X2. This system before shows hysteresis and enables a response to be maintained in the absence of the original stimulus 36.
Figure 38. Figure 38.

A fully reversible system. The net flux J (=A‐B; =F‐C; =D‐E) can reverse and hence change sign.
Figure 39. Figure 39.

Evaluating core sensitivities by graph theory, (a) Digraph for the system in Figure 11 (omitting Z). Variables J and S are nodes, (b) directed circuits and connections for the digraph, (c) directed paths and one‐connections for each variable. For farther details see text.


Figure 1.

Simple linear flux. A‐D represent metabolic intermediates, E1‐E4 represent enzyme‐catalyzed reactions and symbol denotes saturation (here, with A).



Figure 2.

Metabolic carbohydrate flux, (a) Conventional pathway of glycolysis, originating with glucose entry into a cell, (b) whole‐body flux, of which (a) is a component. E1 = glycogen phosphorylase. In this, and other figures the cofactors are omitted, unless relevant to the discussions.



Figure 3.

Branched flux.



Figure 4.

Fluxes with reversible steps. (a) Hypothetical linear flux, (b) glycolysis: a = hexokinase (HK); b = phosphoglucose isomerase (PGI); c = phosphofructokinase (PFK); d = pyruvate kinase (PK); e = lactate dehydrogenase (LDH); f = pyruvate dehydrogenase (PDH).



Figure 5.

Energy fluxes, (a) Equivalent fluxes; JE=2xJl, (b) non‐equivalent fluxes; total energy flux = 2 × Jl + 38 × J0, which is not proportional to, and thus not equivalent to, the total carbon flux, Jo+Jl.



Figure 6.

Linear flux with external effectors. The broken lines denote regulatory (for example, allosteric) interactions.



Figure 7.

Partially external regulators. The concentrations of metabolites M and N are only partially determined by flux J. On the other hand, they are totally determined by the wider system, J + K. Regulator X (totally external for J) communicates with M, but M (partially external for J) does not communicate with X.



Figure 8.

Truncated flux. The reactions producing B are omitted, so that B is a partially external regulator of the truncated flux, Jt.



Figure 9.

Flux with feedback inhibition. (a) Simple linear system, (b) system with extended feedback (end product) inhibition. ⊖ denotes an inhibition, ⊕ denotes an activation.



Figure 10.

Outline of the control of glycolysis by ATP. E1 = phosphorylase b; E2 = phosphofructokinase (PFK). ⊕ denotes an activation.



Figure 11.

System with two potential control sites, only one of which is used. Here, as in subsequent figures, the greek letters denote relative‐change intrinsic or component sensitivities.



Figure 12.

Hill responses. The curves are calculated from equation 35, with K = 1 and Vm = 100.



Figure 13.

Reversible system spanning a membrane. XI and XO denote concentrations inside and outside a cell compartment, respectively. M is a metabolite present only on the inside.



Figure 14.

Substrate cycles. (a) Simple hypothetical cycle, (b) cycle between fructose‐6‐phosphate (F‐6‐P) and fructose‐1,6‐bisphosphate (FBP), (c) cycle between triacylglycerol and FFA; the 7 ATP molecules consumed per turn comprise two each for the conversion of the three molecules of FFA to their CoA esters, and one for the formation of glycerol phosphate.



Figure 15.

Interconversion cycle. Enzyme E exists in two forms, a and b, which are interconverted by E1 and E2. The energy consumption here is represented by a net hydrolysis of ATP to ADP and Pi. In the steady state the rates of E1 and E2 are equal, and there is no net flux across the cycle.



Figure 16.

Regulation of pyruvate dehydrogenase (PDH) by pyruvate. Pyruvate has a direct effect (rm + rc) and an indirect one, via inhibition of the phosphatase component of the interconversion cycle.



Figure 17.

Reversible reactions in situ. (a) Simple hypothetical system, (b) flux of serine biosynthesis in vivo. E1 = phosphoglycerate dehydrogenase plus phosphoserine aminotransferase; E2 = phosphoserine phosphatase. The symbol, denotes saturation.



Figure 18.

Basic control of glycolysis by ATP. This diagram illustrates the general principles: the presence of auxiliary regulators and other mechanisms makes the actual system much more complicated.



Figure 19.

Linear flux with external regulator of E2.



Figure 20.

Linear flux with reversible reaction at E2.



Figure 21.

Linear flux with partially external regulator at E2.



Figure 22.

Opening the system in Figure 21.



Figure 23.

Branched flux with two external regulators.



Figure 24.

The pathway of glutaminolysis.



Figure 25.

Relationship of serine biosynthesis to glycolysis.



Figure 26.

Linear flux with a regulator interacting at two sites.



Figure 27.

Linear non‐steady state system. Here v1, v2 and v3 are not mutually equal, so that S and P vary with time and there is no defined system flux; α‐γ denote absolute‐change sensitivities.



Figure 28.

Effect of the speed of response on a pulsed system. The pulsed stimulus lasts for time τ. Curves 1–4 show the effect of a coordinated increase of the initial rate of cycling between the b and a forms of an interconversion cycle. In curve 4 a steady state has been reached within time τ and any further increase in the rate of initial cycling has no further effect.



Figure 29.

Metabolic buffers. (a) Equilibrium. Q = Keq.S and the resulting larger pool of S is shown in parentheses. Fluxes I and O are equal in the steady state. (b) Receptor recycling. The hormone binds tightly to its receptor, but the complex is destroyed when it is internalized (via F); the free receptor is then recycled (via C). Here, fluxes I and O represent the delivery and removal of the hormone, respectively. (c) Substrate cycle. Here, in the steady state there is no net flux across the cycle, that fluxes I and O are equal. S and P may denote (1) glucose and glycogen, (2) amino acids and protein, (3) fatty acids and triacylglycerol, in which case the cycles may be regarded as storage cycles. E1 and E2 are assumed to have first‐order responses to S and P, respectively.



Figure 30.

Oscillations. Xs is the steady state value, or balance point around which X oscillates. The broken line shows a fading, or damped, oscillation.



Figure 31.

System with indirect removal of an intermediate. The indirect communication, S∼M∼E2, introduces a delay between the production of S (by E1) and its removal (by E2).



Figure 32.

System with a long feedback loop.



Figure 33.

A branched system in BST formulation.



Figure 34.

Linear flux in MCT formulation. A comparison with Figure 11 shows that ε 1/s = α; ε 2/s = β.



Figure 35.

Interconversion cycle with a single molecule of protein. a, b, and c refer to the same molecule and hence cannot coexist. k1 and k2 are classical rate constants that would give the observed rates as a function of the fractional times spent as a or b. The conversion of a into c takes a time π.



Figure 36.

Flux‐generating step with a particulate system. G is a glycogen particle, to which phosphorylase and other enzymes (E) are tightly bound. E does not dissociate as glucose‐1‐phosphate is released.



Figure 37.

Switch mechanism at an interconversion cycle. (a) Cycle, (b) response of E1 to b: substrate inhibition, (c) response of active, a form to regulator X. At X2 there is a large increase of a, which is not reversed until X is lowered to X1, a value much lower than X2. This system before shows hysteresis and enables a response to be maintained in the absence of the original stimulus 36.



Figure 38.

A fully reversible system. The net flux J (=A‐B; =F‐C; =D‐E) can reverse and hence change sign.



Figure 39.

Evaluating core sensitivities by graph theory, (a) Digraph for the system in Figure 11 (omitting Z). Variables J and S are nodes, (b) directed circuits and connections for the digraph, (c) directed paths and one‐connections for each variable. For farther details see text.

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Article Title: Principles of Regulation and Control in Biochemistry: A Pragmatic, Flux‐Oriented Approach
 

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B. Crabtree, E. A. Newsholme, N. B. Reppas. Principles of Regulation and Control in Biochemistry: A Pragmatic, Flux‐Oriented Approach. Compr Physiol 2011, Supplement 31: Handbook of Physiology, Cell Physiology: 117-180. First published in print 1997. doi: 10.1002/cphy.cp140105