The Evolution of Controllability in
Enzyme System Dynamics
Jonathan Vos Post
Mathematics Department
Woodbury University
7500 Glenoaks Blvd.
Burbank, CA 91510-7846
(818) 767-0888
Previously unpublished excerpt from 1977 Doctoral Dissertation
"Molecular Cybernetics"
[University of Massachusetts, Amherst, Massachusetts]
ABSTRACT:
A building block of all living organisms' metabolism is the "enzyme chain." A chemical
"substrate" diffuses into the (open) system. A first enzyme transforms it into a first intermediate
metabolite. A second enzyme transforms the first intermediate into a second intermediate
metabolite. Eventually, an Nth intermediate, the "product" diffuses out of the open system.
What we most often see in nature is that the behavior of the first enzyme is regulated by a
feedback loop sensitive to the concentration of product. This is accomplished by the first
enzyme in the chain being "allosteric", with one active site for binding with the substrate, and a
second active site for binding with the product. Normally, as the concentration of product
increases, the catalytic efficiency of the first enzyme is decreased (inhibited). To
anthropomorphize, when the enzyme chain is making too much product for the organism’s good,
the first enzyme in the chain is told: "whoa, slow down there." Such feedback can lead to
oscillation, or, as this author first pointed out, "nonperiodic oscillation" (for which, at the time,
the term "chaos" had not yet been introduced).
But why that single feedback loop, known as "endproduct inhibition" [Umbarger, 1956], and not
other possible control systems? What exactly is evolution doing, in adapting systems to do
complex things with control of flux (flux meaning the mass of chemicals flowing through the
open system in unit time)?
This publication emphasizes the results of Kacser and the results of Savageau, in the context of
this author’s theory. Other publications by this author [Post, 9 refs] explain the context and
literature on the dynamic behavior of enzyme system kinetics in living metabolisms; the use of
interactive computer simulations to analyze such behavior; the emergent behaviors "at the edge
of chaos"; the mathematical solution in the neighborhood of steady state of
previously unsolved systems of nonlinear Michaelis-Menton equations [Michaelis-Menten,
1913]; and a deep reason for those solutions in terms of Krohn-Rhodes Decomposition
1
of the Semigroup of Differential Operators of the systems of nonlinear Michaelis-Menton
equations.
Living organisms are not test tubes in which are chemical reactions have reached equilibrium.
They are made of cells, each cell of which is an "open system" in which energy, entropy, and
certain molecules can pass through cell membranes. Due to conservation of mass, the rate of
stuff going in (averaged over time) equals the rate of stuff going out. That rate is called "flux." If
what comes into the open system varies as a function of time, what is inside the system varies as
a function of time, and what leaves the system varies as a function of time. Post's related
publications provide a general solution to the relationship between the input function of time and
the output function of time, in the neighborhood of steady state. But the behavior of the open
system, in its complexity, can also be analyzed in terms of mathematical Control Theory. This
leads immediately to questions of "Control of Flux."
-----------------------------------"Control of Flux" has been analyzed [Kacser, 1973] in terms of sensitivity and controllability at
steady state (i.e. when the rate of change of the concentration of substrate, intermediates, and
product in the open system equals zero). Here, we follow this analysis of Kacser and Burns.
Their important study examines enzyme chains identical to those which this author analyzed,
except that Kacser's reactions were reversible (Post's were irreversible) and that Kacser
ignored transients (while the main thrust of Post’s work is the study of how those transients
propagate in chemical phase space). We concur: there is no general kinetic theory of enzyme
systems, but computer simulations (and Post’s solutions to Michaelis-Menton) help.
Each step of the enzyme chain has at least three parameters: (1) Vmax the enzyme velocity (how
many molecules of intermediate per second can one molecule of enzyme transform), i.e. the
catalytic efficiency; (2) Km, the enzyme’s Michaelis constant (to a quadratic fit, how rapidly
does the enzyme approach Vmax as the concentration of intermediate approaches infinity); and
(3) what is the concentration of the enzyme at time t=0? Post's solution shows how, for
transients ("enzyme waves") the three parameters can be lumped into a single "wave constant"
which is an eigenvalue of the corresponding eigensolution. The flux through the entire system
varies as the flux through each link of the enzyme chain varies.
The question of system response becomes: "what is the quantitative influence of one parameter
on … one particular flux?" By definition:
R = (dF/F) / (dP/P) = (P/F)(Partial derivative of F with respect to P)
Where R is a "response coefficient", F the flux in question, and P any parameter. This can only
be calculated of the flux is known as a function of the parameter, which is precisely what Post's
solutions describe. Any particular R has two parts: one for an enzyme in isolation, and one for
the whole system.
When an enzyme velocity can be controlled from outside the system, P is an “effector.” How
well does the parameter P control? By definition:
2
Kappa = (dV/V) / (dP/P) = (P/V)(Partial derivative of V with respect to P)
Where Kappa is a "controllability coefficient" which measures only the local response of the
enzyme in isolation to input (changes in the effector parameter P).
We can measure system response in principle by perturbing the concentration of a single enzyme
and observing a change in flux. By definition:
Z = (dF/F) / (dE/E) = (E/F)(Partial derivative of F with respect to E)
Where Z is a "sensitivity coefficient" of system flux, F, with respect to enzyme concentration E.
This, too, is given by Post's solution. As Z approaches 1, we call the enzyme "fully controlling",
"a pacemaker", or a "bottleneck."
The relationship between the three coefficients, defined above, justifies the partitioning of
system response into local (changes on enzyme) and global (enzyme changes flux). Response is
partitioned multiplicitively:
R = Z Kappa
Just as Post's transient analysis shows how enzyme system response depends collectively on all
parameters, Kacser and Burns argue formally that sensitivity is distributed. With N enzymes in
the chain:
(Sum from I=1 to N) Zi = 1
All enzymes share, perhaps unequally, but there is NOT a single rate-limiting "bottleneck"
enzyme. An evolutionary implication, which we examine later, is that most enzymes exist "in
excess." A typical enzyme could have its concentration, or its catalytic efficiency, somewhat
decreased without a catastrophic decrease in system performance, where performance, is
narrowly identified with product flux. This notion is also a rejection of the naïve "bottleneck
hypothesis" wherein only enzyme protein mutations in a single rate-limiting enzyme will affect
the organism's fitness by changing system performance.
In summary, Post's analysis dovetails with that of Kacser and Burns. The question of control of
flux, F, in an enzyme chain as a function of change in a parameter, P, is grossly described by:
dF/F = Z Kappa (dP/P)
or, equivalently,
(Partial derivative of F with respect to P) = (F/P) (Z Kappa)
This is seen in greater detail when Z (sensitivity) and Kappa (controllability) are calculated from
Post's solution.
3
Kacser and Burns' analysis touches on endproduct inhibition, when the last enzyme in the chain
has Z =1, but specifically ignores transients from one steady state to another (there being
multiple solutions for different values of flux, after transients have damped out and all
derivatives of parameters are zero).
Since the derivative of Post's solutions with respect to any of the parameters is rather
complicated (whether Post's solution is expressed in terms of matrix exponents or in terms of
Laplace transforms and Transfer Function), we limit our discussion of flux control in this paper.
We have indicated the necessity of a closed-form mathematical solution of input-output
relationships in the calculation of R, Kappa, and Z, and we examine elsewhere the qualitative
nature of Post's solutions to the Michaelis-Menton equations.
Is it reasonable to limit our evaluation of enzyme system performance to control of flux?
Sensitivity and controllability are standard coefficients in Control Theory, but are not the only
characteristics of system behavior which may be involved in optimization. Another important
paper [Savageau, 1974] concentrates on evolutionary reasons for the frequent occurrence of
enzyme system with our model’s structure: endproduct inhibition [Umbarger, 1959]: an
enzyme chain with a single feedback loop from the last step to the first step.
Savageau describes endproduct inhibition with generalized Michaelis-Menton kinetics. He
reviews the complications stemming from the non-linearity of the equations. He makes a powerlaw approximation (interpolating between 1st and 2nd order reactions, i.e. linear and quadratic
kinetics) for example, and a logarithmic reformulation. This is said to respect the non-linearities
more than does linearization. But his reformulation is most valid near steady state
(when rate of change of flux is zero). These approximations have been tested in vivo [Kohen
1972] in the glycolytic sequence of enzymes.
The steady state is calculated by matrix techniques. The first order response of the system, in
terms of logarithmic gain factors, is calculated by partial differentiation. Similarly, a sensitivity
of the system to perturbations of internal parameters is derived.
These gains and sensitivities can be calculated for other patterns of feedback control by
inhibition. We limit our discussion to "unbranched" systems (neglecting those branched systems
where the product from one open system serves simultaneously as the substrate for more than
one downstream open system). There can be several feedback loops, in series. There
can be several feedback loops, nested inside each other. There can be several overlapping
feedback loops. In what sense is our prototypical endproduct inhibition single feedback loop
optimal?
If gain is taken as a measure of sensitivity of endproduct availability to substrate perturbation, we
guess that an optimal gain is a high one: the pathway (enzyme chain) can manufacture extra
product faster when extra substrate is on hand, with possibly greater Darwinian fitness and
selective advantage.
4
If sensitivity to internal parameters is seen as a measure of insensitivity of maximal production
rate (flux) to perturbations (mutation, transcription/translation error, temperature, pH...)
then we guess that an optimal gain is a low one: the pathway adaptively damps out potentially
disrupting biases. This gives superior stability, smoothness, and reliability of metabolic
functioning.
Savageau’s work allows precise comparison of all alternative control systems. The striking
result is abstracted here.
The single-feedback case (our model) is optimal with respect to ability to do each of these things:
(1) meet demand for above-normal endproduct production (flux);
(2) limit the accumulation of intermediate metabolites;
(3) respond to above-normal substrate availability;
(4) maximize the gain/sensitivity ratio (a crucial "figure of merit" of the system).
The first three of these conclusions show, cybernetically, the extreme flexibility of the most
common endproduct inhibition system – a formal measure of adaptability to various
physiological changes. The 4th relates to the balance of conflicting demands put
on enzyme chains by natural selection in rapidly changing environments.
More complex feedback networks do, in fact, occur. One product sometimes inhibits several
enzymes. One enzyme may be inhibited by the concentrations of several intermediates. Enzyme
chains sometimes branch, with each branch inhibiting the other. The preceding analysis presents
these cases as paradoxical: evolved but not optimized.
While that analysis is not a causal demonstration of the evolution by natural selection of enzyme
chain control systems, it does include:
(1) a formalism for such hypotheses;
(2) a hint of the transformations needed to parameterize selective advantage (fitness) in
metabolisms under complexity;
(3) boundary conditions which must be met by any evolutionary adaptation of enzyme pathways
in metabolisms.
The strictly evolutionary arguments in this paper are incomplete, and perhaps sometimes
misleading. It is not true, a priori, that natural selection optimizes systems in the Control Theory
sense.
The process by which non-optimal cases are eliminated are taken for granted: "Other things
being equal, cases such as these would ... be easily lost by mutation ... since they have no
selective advantage based on the criteria of minimum sensitivity."
[Savageau].
5
This ignores the subtleties of multi-gene molecular population genetics, the effect of complexity
of regulatory enzyme function on rate of evolution as exemplified by the careful studies of the
enzyme Cytochrome C [Dickerson, 1972], and the role of gene duplication on enzyme evolution
[Koch, 1972] [Leidigh, 1973]. That analysis also fails to consider the possible role of transients
on fitness as more than noise to be dealt with by insensitivity. Post considers the transients
(enzyme waves) the be perhaps the basic mechanism of information transmission within
metabolic systems, and capable of reverse-engineering and modification to build informationprocessing and communication devices inside of single living cells -- a type of protein
"nanotechnology" described by Post before the modern use of that term was introduced by
K. Eric Drexler, with whom Post had corresponded before Drexler’s breakthrough papers on the
subject.
We have found that our model fits closely with other studies of related interest. These other
studies have isolated and inter-related various coefficients of enzyme system response. They
agree with the conclusions here as to the adequacy of non-systemic models, and of certain naïve
classical hypotheses.
Post’s solution to the Michaelis-Menton equations, which completely describes system behavior
as a function of many parameters (and a matrix of the effect of every parameter on every step
"downstream" of the enzyme with that parameter; and the description of more complex systems
with multiple branches and feedback by manipulation of the Transfer Functions of each of
several interconnected enyme chains), may be directly coupled with the mathematical analysis as
given by Kacser and by Savageau.
All three approaches (Kacser, Savageau, Post) reveal the difficulty of drawing evolutionary
conclusions from enzyme system behavior. Post's approach alone is not limited to steady state
behavior. Post's general solution embodies complexity, and makes answering any
particular question of enzyme protein behavior rather difficult. Hopefully, such a unified model
allows such questions to be more precisely framed, and to be ultimately answered in a consistent
and quantitative way.
Although most of the references given here are from the 1970s, these studies have recently
gained new importance, due to:
(1) billion-fold increase in computing power, now allowing (in principle) complete simulation of
the dynamic behavior of a typical cell's metabolism, with on the order of 20,000 different
substrates, products, and intermediate metabolites all changing concentration over time in a
complex network of branches and feedback loops;
(2) improvements in the analytical capabilities of measuring enzyme systems in vivo;
(3) the rise of Complexity as a discipline, with deeper understanding of chaos and other
phenomena;
(4) the rise of nanotechnology, now funded at several billion dollars per year, in which reverseengineering and modification of protein systems is now seen as a plausible technology, and not
the science fiction it was accused of being when Richard Feynman [Feynman, 1959; 1960] as the
great-grandfather of nanotechnology, Post and over a dozen other researchers were grandfathers
6
of nanotechnology, and Drexler (with some early PR assistance by Post in popular magazines
such as Omni and Analog) became acknowledged as the father of nanotechnology.
The questions are being widely asked now, which Kacser, Savageau, Post and others wrestled
with in obscurity three decades ago. The goal of complete description of complex living
systems in computer simulation is a goal to many research institutions. The goals of modifying
those living systems to perform their functions better, and to perform completely new
functions, is widely seen as important to the progress of the 21st century.
This publication is part of a reappraisal of earlier research, which came “before its time” had
come. That time, in the discipline of Complexity, is now.
ControlOfFlux.doc [spellchecked, some refs added draft of 29 Dec 2003]
7
REFERENCES:
Richard E. Dickerson, “The Structure and History of an
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Kacser, H. & Burns, J.A., “The control of flux”
Symp. Soc. Exp. Biol. 27 (1973), 65-104.
Arthur L. Koch, [1972] ENZYME EVOLUTION: I. THE IMPORTANCE OF
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Jonathan V. Post, "Analysis of Enzyme Waves: Success through Simulation", Proceedings of
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Jonathan V. Post, "Alternating Current Chemistry, Enzyme Waves, and Metabolic Chaos",
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Jonathan V. Post, "Is Functional Identity of Products a Necessary Condition for the Selective
Neutrality of Structural Gene Allele?", Population Biologists of New England (PBONE),
Brown University, Providence, RI, June 1976
Jonathan V. Post, "Enzyme Kinetics and Selection of Structural Gene Products -- A Theoretical
Consideration", Society for the Study of Evolution, Ithaca, NY, June 1977
Savageau, M. A. [1974]: expands on his earlier
M. A. Savageau (1969) J. Theor. Biol. 25: 365-379
8
possibly: Savageau, M.A. (1976) Biochemical Systems Analysis – A Study of Function
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Program, pp. 1-379.
Umbarger, H.E. (1956) Evidence for a Negative-Feedback
Mechanism in the Biosynthesis of Leucine, Science, 123, 848 [1959]
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9
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10
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11
Experimental
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Related topics
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systems: experiments and models. Adv. Mol. Cell Biol. 11, 1-19.
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Chock, P.B. & Stadtman, E.R. (1980) Covalently interconvertible enzyme cascade systems.
Methods Enzymol. 64, 297-325.
MORE REFERENCES
[not to be included in conference Proceedings;
here only as supporting notes for oral presentation]:
Introduction to Biochemical Regulation
Prepared by Franklin R. Leach
May 24, 2002
For BiochemI
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Biochemical Regulation became a separate area of biochemical research in the 1950's. The first
researches expounded the concept of feed-back inhibition, which has received the most
emphasis. But there are many other means of regulating that have now been recognized.
The Origin – 1950-1959
Umbarger, H.E. (1956) Evidence for a Negative-Feedback Mechanism in the Biosynthesis of
Leucine, Science, 123, 848.
Yates, R.A., and Pardee, A.B. (1956) Control of Pyrimidine Biosynthesis in Escherichia coli by
a Feed-Back Mechanism, J. Biol. Chem., 221, 757-770.
Pardee, A.B. (1959) The Control of Enzyme Activity, The Enzymes, 2nd Edition, 1, 681-716.
Expansion and Inventory – 1960-1969
Hathaway, J.A., and Atkinson, D.E. (1963) The Effect of Adenylic Acid on Yeast Nicotinamide
Adenine Dinucleotide Isocitrate Dehydrogenase, a Possible Metabolic Control Mechanism, J.
Biol. Chem., 238, 2875-2881.
Atkinson, D.E. (1968) Biological Feedback Control at the Molecular Level, Science, 150, 851857.
Atkinson, D.E. (1966) Regulation of Enzyme Activity, Ann. Rev. Biochem., 35, 85-124.
Codification of the Concepts Enzyme Regulation in Metabolism, The Enzymes, 3rd edition, 1,
397-459.
Atkinson, D.E. (1970) Enzymes as Control Elements in Metabolic Regulation, The Enzymes,
3r d edition, 1, 461-489.
Savageau, M.A. (1972) The Behavior of Intact Biochemical Control Systems, Curr. Top. Cell.
Regul., 6, 63-130.
Savageau, M.A. (1976) Biochemical Systems Analysis – A Study of Function and Design in
Molecular Biology, Addison-Wesley Publishing Co., Advanced Book Program, pp. 1-379.
Monographs and Meeting Proceedings on Regulation
Larner, J. (1971) Intermediary Metabolism and Its Regulation, Prentice-Hall, Inc., Englewood
Cliffs, NJ, 308 pp.
Newsholme, E.A., and Start, C. (1973) Regulation in Metabolism, John Wiley & Sons, London,
349 pp.
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Bacchus, H. (1976) Essentials of Metabolic Diseases and Endrocrinology, University Park Press,
Baltimore, 511 pp.
Denton, R.M., and Pogson, C.I. (1976) Outline Studies in Biology: Metabolic Regulation,
Halsted Press, London, 64 pp.
Cohen, P. (1983) Outline Studies in Biology: Control of Enzyme Activity, 2nd Edition, Chapman
and Hall, London, 96 pp.
Herman, R.H., Cohn, R.M., and McNamara, P.D. (Eds.) (1980) Principles of Metabolic Control
in Mammalian Systems, Plenum Press, NY, 669 pp.
Ochs, R.S., Hanson, R.W., Hall, J. (Eds.) (1985) Metabolic Regulation, Elsevier,Amsterdam,
246 pp.
Brown, R.F. (1985) Biomedical Systems Analysis via Compartmental Concept, Abacus Press,
Tunbridge Wells Kent, 684 pp.
Goodridge, A.G., and Hanson, R.W. (Eds.) (1986) Metabolic Regulation: Application of
Recombinant DNA Techniques, Ann. N. Y. Acad Sci., 478, 320+ pp.
Martin, B.R. (1987) Metabolic Regulation: A Molecular Approach, Blackwell Scientific
Publications, 299 pp.
Torriani-Gorini, A., Rothman, F.G., Silver, S., Wright, A., and Yagil, E. (Eds.) (1987) Phosphate
Metabolism and Cellular Regulation in Microorganisms, ASM, Washington, 316 pp.
Ottaway, J.H. (1988) In Focus: Regulation of Enzyme Activity, IRL Press, Oxford, 88 pp.
Morgan, N.G. (1989) Cell Signalling, Guilford Press, New York, 203 pp.
Cornish-Bowden, A., and Cardenas, M.L. (Eds.) (1990) Control of Metabolic Processes, NATO
ASI Series, 190, Plenum Press, NY, 454 pp.
Srere, P.A., Jones, M.E., and Mathews, C.K. (Eds.) (1990) Structural and Organizational Aspects
of Metabolic Regulation, UCLA Symposia on Molecular and Cellular Biology, New Series, 133,
Wiley-Liss, NY, 416 pp.
Beitner, A. (Ed.) (1990) Regulation of Carbohydrate Metabolism, CRC Press, Boca Raton, FL,
Vol. 1, 176 pp. and Vol. 2, 224 pp.
The Academic Press Series, Current Topics in Cellular Regulation, Preceding volumes
beginning in the 1970's with Vol. 33 of 435 pp. to be published April 1992.
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Retrospective Review
Richard, J., and Cornish-Bowden, A. (1987) Co-Operative and Allosteric Enzymes: 20 Years
On, Eur. J. Biochem., 166, 257-272.
1990s Monograph
Frayn, K.N. (1996) Metabolic Regulation: A Human Perspective, Portland Press, London, pp.
265.
Fell, D. (1997) Understand the Control of Metabolism, Portland Press, London, pp. 301.
The Quintessence of Regulation (Franklin R. Leach)
1. Determining the current status of metabolic compounds in the cell because regulation
functions to maintain homeostasis and balance.
2. The status is communicated via chemical or electrical signals to organs, cells, or other
functional units which detect the signal and respond.
3. The adjustments are made by modulating the activity and/or quantity of key enzymes in a
pathway in response to effectors taking into account the substrate available and the amount of
end-product. After a response has been made the system is still monitored and readjusted if
required.
4. The pathways are connected into a network and are regulated together by several
mechanisms which allow the proper distribution to be made of resources.
5. Regulation functions to maintain order at the price of cellular energy and to separate the
system from the detrimental effects present in the environment.
The whole system must be dampened so that wild responses don’t tear it apart and yet they
must be sensitive enough to respond to the small inputs which occur in a physiologically
functional system. These exquisitely balanced responses are often achieved by opposing
controls, reciprocal functions, cascades, oppositely directed reactions, or multiple controllers.
But control of controls makes a more complicated system for us to understand - a network of
interactions.
The divisions and classifications that we impose as aids to learning may only
circumstantially resemble a portion of the real order. We look for steps and breaks when there is
a continuum. We try to make the complex simple and if we ever find anything that is simple,
then we try to make it more complex.
Step back and look at the whole of metabolism and regulation - reflect on their organization.
Realize that we can only give a floor plan and a building directory for much of what we study.
We don’t have time or the information to map all of the electrical and computer connections.
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Models - beautiful models - nice to observe and they function to give us a means of testing
our predictions and to determine if the theory adequately explains the real world. When we can
quantitate and generate expected responses, then we can design experiments and compare
resulting observations to test and evaluate our perception.
Biochemical regulation controls the dynamic networks of reactions and interactions
responsible for the balanced output of products from minimal and stable concentrations of
intermediates and the orderly accomplishment of processes both in spit of environmental
fluctuations and with response to precursor availability. This regulation requires assessment of
the situation, communication of the status, reception of these signals, their amplification, if
necessary, and making corrective actions to turn on, turn off, modulate, or otherwise control the
information or functional units (mainly enzymes) with continual monitoring of the responses.
The network of pathways and interrelated signals that exist in a stable biochemical system,
require a biochemical systems analysis approach to their quantitative understanding. While the
concepts and mathematical techniques of a biochemical systems analysis are foreign to many,
Pardee said “eventually, we biochemists would hope to describe many of the essentials of
biochemistry - the network of life - in mathematical terms”. While many don’t like the
mathematic approach, that approach is essential to the quantitation necessary. I will provide
some additional references and materials on biochemical systems analysis.
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