J. Phys. Chem. B 2010, 114, 16105–16111
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Cyclic Conformational Modification of an Enzyme: Serial Engagement, Energy Relay, Hysteretic Enzyme, and Fischer’s Hypothesis† Hong Qian Department of Applied Mathematics, UniVersity of Washington Seattle, Washington 98195 ReceiVed: March 16, 2010; ReVised Manuscript ReceiVed: July 28, 2010
Reversible chemical modification of enzymes is one of the most important mechanisms in cellular signaling. We generalize this concept to include cyclic modification of enzyme conformations. The mechanism is fundamentally different from the ligand induced conformational change: It only requires a catalytic amount of ligand to activate an enzyme, but it does require an active chemical energy driving a “futile cycle” akin to the phosphorylation-dephosphorylation cycle. The mechanism covers several previously proposed models that include serial engagement for T-cell receptor activation, energy relay for proofreading in DNA replication and protein biosynthesis, the hysteretic enzyme, and Fischer’s hypothesis on protein tyrosine phosphatase action. While for small proteins operating under a funnel-shaped energy landscape, multiple conformations with sufficiently long dwell times are not common, recent experiments suggest that for larger, multidomain proteins, cyclic conformational modification (CCM) is much more likely and evolution presumably finds a way to capitalize on this mode of regulation. CCM can be difficult to identify in cells; however, it is potentially an important, and yet overlooked, regulatory mechanism in cellular signal transduction. We suggest the serial engagement mechanism in T-cell activation to be a possible testing case for the CCM mechanism. 1. Introduction Reversible chemical modifications of enzymes, first discovered in the phosphorylation and dephosphorylation of glycogen phosphorylase,1,2 is one of the most widely used signaling mechanisms in cellular biochemistry.3 The phosphorylated form of the substrate protein, or enzyme W*, usually has a structure, and thus biochemical activity, different from that of the unphosphorylated enzyme W.4,5 In the phosphorylationdephosphorylation cycle (PdPC) shown in Figure 1A, the phosphatase catalyzed dephosphorylation is not the reverse reaction of the kinase catalyzed phosphorylation reaction. Rather, a complete cycle accompanies one ATP hydrolysis. PdPC kinetics regulate cellular biochemical information with the expanse of cellular phosphorylation energy.6-9 Figure 1B shows something very similar in kinetics but very different in molecular nature: An enzyme has two conformations W1 and W2; after catalyzing the conversion of its substrate A to product B, the enzyme conformation W1 becomes W2. The enzyme remains in W2 for a significantly long time until it returns to W1. The returning of W2 f W1 can be, but not necessarily, assisted by a ligand or a protein L. We shall call this reaction scheme cyclic conformational modification (CCM). One important difference between parts A and B of Figure 1 is that W and W* are molecularly different, but W1 and W2 are isomers. Hence it is much more difficult, by chemical means, to identify the existence of W1 and W2 experimentally in a cell. CCM has been discussed repeatedly in the literature, under various different names by many authors: Frieden introduced the concept of hysteretic enzyme and was the first to realize important roles it possibly plays in cellular metabolic regulations.10,11 Hopfield suggested energy relay and showed its possibility for proofreading in connection to DNA replication and protein biosynthesis.12 Fischer et al. used such a model in explaining how protein †
Part of the “Robert A. Alberty Festschrift”.
tyrosine phosphatase might act synergistically with tyrosine kinase to elicit a full physiological response from tyrosine phosphorylations.13 And the same idea is behind the serial engagement mechanism proposed by Valitutti et al.14 to account for the puzzling observations in T-cell receptor activation. Cyclic conformational modification is a more general, but much more subtle, concept than reversible chemical modification. In this paper, we provide a quantitative theory as well as discuss possible applications. In particular, we shall discuss the fundamental difference between the CCM and the ligand induced reversible, allosteric conformational change. The key distinction is that the CCM requires a chemical driving force
Figure 1. (A) Canonical PdPC with kinase (not shown) catalyzed phosphorylation of a substrate enzyme, W + ATP f W* + ADP, and phosphatase (not shown) catalyzed dephosphorylation of the enzyme, W* f W + Pi. The entire cycle is accompanied with one ATP hydrolysis ATP f ADP + Pi. (B) Essentially the same chemical kinetics is the cyclic conformational modification (CCM) of an enzyme W: the conformational change W1 f W2 is accompanied by the conversion of substrate A to product B. The W2 then returns to W1 with the assistance of a “catalyst” L. The entire cycle is accompanied with one A f B turnover. Both reactions, of course, could also have their respective catalyzing enzymes, not shown in the figure. Just as kinase and phosphatase regulate the interconversion between W and W* in (A), the balance between W1 and W2 in (B) can be regulated by the catalyzing enzymes as well as by L.
10.1021/jp102400u 2010 American Chemical Society Published on Web 09/24/2010
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in analogous to the ATP hydrolysis in PdPC. Hence it is a phenomenon akin to living cells. 2. Basic Theory of CCM Kinetics and Thermodynamics of CCM. To have a solid understanding of the CCM mechanism, we start with its thermodynamics. To do so, one has to consider reversibility of each and every reactions in Figure 1B:3,6 k1
W1 + A y\z W2 + B
(1a)
k-1
k2
W2 + L y\z W1 + L
(1b)
Figure 2. Enzyme activation by ligand binding (A) and ligand triggered enzyme activation via CCM (B). Activation by binding is an equilibrium process, and it requires a stoichiometric amount of ligand. On the contrary, activation by CCM needs only a trace amount of ligand, but it functions in an open chemical system and requires a sustained ∆GAB * 0. Kinetic module shown in (B) is widely known as a futile cycle.8
k-2
For simplicity, we first assume that the reactions follow massaction kinetics rather than Michaelis-Menten kinetics, which will be considered later. This means all the enzymes involved are operating in their linear regimes. The conclusions we obtain below, however, do not change if the Michaelis-Menten kinetics is carried out. The steady state of the CCM system has the fractions of W in states 1 and 2:
f1 )
[W1] k-1[B] + k2[L] ) WT k1[A] + k-1[B] + (k2 + k-2)[L]
(2a) f2 )
[W2] k1[A] + k-2[L] ) WT k1[A] + k-1[B] + (k2 + k-2)[L]
(2b) and the steady state cycle flux
Jss )
(k1k2[A] - k-1k-2[B])[L] W k1[A] + k-1[B] + (k2 + k-2)[L] T
(2c)
WT is the total amount of [W1] + [W2]. k1k2/k-1k-2 ) Keq is Keq
the equilibrium constant for the reaction A y\z B. In a cellular environment, [A] and [B] are assumed to be constant. We see that if [B]/[A] ) Keq, Jss ) 0, and the entire reaction system is in a chemical equilibrium. If γ ) Keq[A]/[B] > 1, then Jss > 0; and if γ < 1, then Jss < 0. The system then is in a nonequilibrium steady state (NESS).6 Note
[A] RT ln γ ) ∆G°AB + RT ln ) ∆GAB [B]
that a given ligand can act either as an activator or as an inhibitor to a same target was first discussed in ref 8 and has been independently discovered and systematically investigated by Jia et al.15 What determines this “cellular context”? We note that f2(0) > f2(∞), f2(0) ) f2(∞), and f2(0) < f2(∞) correspond precisely to ∆GAB ) k1k2[A]/k-1k-2[B] > 0, ∆GAB ) 0, and ∆GAB < 0, respectively! Therefore, whether L is an activator or an inhibitor for W2 depends upon whether ∆GAB > 0 or 0. Therefore, we need to identify the W1 as the “activated” W in Figure 2B and call it Wa. Then the f1 in eq 2a can be rewritten as
1 + σ[L] µγ fa ) 1 1+ + (1 + µ)σ[L] µγ
(3)
is the free energy difference of the reaction; i.e., the chemical driving force for the kinetic cycle. γ ) 1 is equivalent to ∆GAB ) 0. Increasing the Amount of L Activates W1 f W2. Equation 2b shows that f2([L]) changes monotonically from f2(0) )k1[A]/ (k1[A] + k-1[B]) to f2(∞) ) k-2/(k2 + k-2) when [L] increases from 0 to ∞. Hence, if k1[A] , k-1[B] and k-2 . k2, f2 increases from zero to unity, exhibiting a switching-on like behavior. However, if k1[A] . k-1[B] and k-2 , k2, then f2 exhibits a switching-off behavior with increasing [L]. The ligand L can be either an activator or an inhibitor of W2, depending upon the cellular context (i.e., the amount of A and B). The possibility
(4)
in which σ)
k2 k1[A]
µ)
k-2 k2
γ)
k1k2[A] ) e∆GAB/RT k-1k-2[B]
(5) We see that when µ is very small and γµ is very large, fa ) σ[L]/(1 + σ[L]). However, if ∆GAB ) 0, i.e., γ ) 1, then fa ) {1/µ + σ[L]}/{1 + 1/µ + (1 + µ)σ[L]} ) {1/µ + σ[L]}/{(1/µ + σ[L])(1 + µ)} ) 1/(1 + µ), which is independent of [L].
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With sufficiently large ∆GAB in a cell, the CCM behaves essentially the same as the ligand-binding induced activation. The “effective association constant” σ, however, can be regulated by the amount of [A]. Furthermore, in a chemical equilibrium in a test tube, fa is independent of [L] as if σ ) 0. If one measures the transient kinetics of A f B in vitro, one will observe a time lag; hence W is a hysteretic enzyme.16,17 Systems involving CCM are much more dynamic with multiple regulatory capabilities than systems with simple ligand-induced allosteric regulation. 3. Applications Serial Engagement. Cellular response of T lymphocytes is initiated at the immunological synapse by the interaction of T-cell surface receptors (TCR) and a very small number of MHC (major histocompatibility complex)-peptide complex on the surface of antigen-presenting cells (APCs).18 For a long time, it had been puzzling how this system works since the affinity between the TCR and the peptide is low with a rapid dissociation rate constant: the half-life is on the order of 1-20 s. The number of MHC-peptide molecules on APCs is often less than 100; but over a period of hours several thousand TCR are internalized. If all the internalized receptors are activated, as widely believed to be the case, then most activated TCR must be in their activated state even when the peptide is dissociated from the receptors. In other words, the TCR activation in the immunological synapse is likely to employ the CCM mechanism. Taking the association of the ligand and enzyme in Figure 2B explicitly, we have the kinetic scheme: a2
b2
d2
q2
L + W y\z WL y\z Wa + L
k1
Wa + A y\z W + B
(6a)
(6b)
k-1
Even when LT , WT, there will be still significant fraction of activated Wa, fa ) [Wa]/WT. In fact, if we assume q2 ) 0 and WT . Km2, then (see Methods)
1 + κ(LT /L*) T if LT e L*T 1+κ fa ) 1 if LT g L*T
fa )
and
(7)
where κ ) k1[A]/(k-1[B]), L*T ) k1Km2[A]/b2. Note that b2/Km2 is the apparent bimolecular rate constants, ∼k2. Thus, L *T ) 1/σ. Whether the number of MHC-peptide complexes on an APC is sufficient to activate a T-cell is not determined by the relative numbers to the TCR, but to the L*T . One prediction from the CCM model is that there must be a driven chemical reaction, yet to be identified in cells, that is tightly coupled to the TCR activation. If there is not a coupled driven reaction A f B, then peptide binding will not affect the fa. It is also significant to note that if the coupled reaction 6b is catalyzed by another enzyme which operates in its saturated regime, then the reaction system in (6) is equivalent to the Goldbeter-Koshland ultrasensitive PdPC.9,19 Then two different peptides with a small difference in their Km2 will have a large difference in corresponding fa. The ligand discrimination is no longer determined by the equilibrium affinities alone, but rather a “proofreading” mechanism is able to amplify the specificity.
Figure 3. Energy relay: a proofreading scheme based on the generic CCM mechanism in which L is the same as A, and the W2A complex can either dissociate and regenerate W1 or be converted into W1A. The proofreading is achieved at this branching point. The two parallel pathways are A + W2 f W2A f W1A f B + W2 and A + W2 f W 2A f A + W 1 f W 1A f B + W 2 .
There is no chemical difference between W and Wa; it is purely conformational. Hence, in a cell biology experiment W and Wa are likely to be indistinguishable. In addition, in immunological synapse, the molecular resolution is not sufficient to distinguish W and WL. Hence, for a long time the kinetic scheme in eq 6a could have been easily mistaken as a ligand binding leading to activation of W, shown in Figure 2A. In this case, the peptideassisted turnover of TCR, W f Wa, could be interpreted as reversible binding and unbinding, and fast turnover interpreted as rapid off-rate. And it would be indeed puzzling how a very few ligands L, with weak affinity, were able to activate a large quantity of TCR. The “puzzling” facts have lead Valitutti et al. to propose the serial engagement mechanism.14,20 What has not been explicitly made clear is the necessity of the second half of eq 6a to be a driven chemical reaction. If the kinetic cycle in eq 6a is not driven, i.e., [B]/[A] ) Keq and γ ) 1, then an increase in the concentration (or activity) of the L will not be able to elicit an increase in the population of Wa (see Methods). In a closed reaction system at chemical equilibrium, the forward and backward reactions of W + L h Wa + L have an equal probability following the principle of detailed balance.6 Amplification of Substrate Specificity. We have already seen how the CCM can amplify specificity for its ligand, L, with the presence of a driven reaction A f B. This is the essential idea of Hopfield-Ninio’s kinetic proofreading,6,21,22 and it shares a mechanism with the zeroth-order ultrasensitivity.3,19 We now show how the reaction system in eq 1b can perform specificity amplification on its substrate A, while using the energy from the same reaction A f B. Note, in DNA replication, there is not a separate energy source besides dNTP hydrolysis, while in protein biosynthesis there is a separate GTP hydrolysis. Hopfield’s energy relay kinetics is a variation on the theme in Figure 1B in which the ligand L is itself the substrate A. Consider both reactions in eq 1a as Michaelis-Mentenian, we have the kinetic scheme for energy relay shown in Figure 3.12 Let us now consider two possible substrates: A and A′, with ′ for W1, and their respective Michaelis constants Km1 and Km1 ′ for W2. We assume that the intrinsic substrate Km2 and Km2 ′ /Km1 ) Km2 ′ /Km2 ) b2′/b2 ) θ > 1; i.e., enzyme specificity Km1 A′ is the substrate with less affinity. For standard Michaelis-Menten kinetics, when an enzyme is operating in the linear regime, its substrate specificity is determined by θ. If the enzyme is operating in the saturated regime, then its specificity is less. However, Hopfield showed that the kinetic scheme in Figure 3 is capable of proofreading
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Figure 4. From reversible chemical modification perspective to cyclic conformational modification perspective. (A) The target enzyme has three states: an inactive dephosphorylated state W, an active phosphorylated state W*, and an active dephosphorylated state Wa. The a phosphorylation turns out not to correlate strictly with its function. Rather, a conformational modification is correlated with the activity of the target enzyme, shown in (B). Still, the kinetic cycle is driven by ATP hydrolysis, where the phosphate group is taken off, while biochemically important, is not functionally relevant.
with increased specificity (i.e., greater than θ).12 The steady state of the kinetics is readily obtained and the steady state production rates (fluxes) for B and B′, (JB)ss and (JB′)ss are computed (see Methods). It can be shown that the ratio (JB)ss/(JB′)ss e θ[A]/ [A′] for both very large and very small R. However, for the optimal R ) (b2b2′)1/2, one has
{
( (
) )
d1 b1 + + θKm1 a a 2 1 [A] √ ) θ d1 [A'] b1 (JB')ss √θ + + Km1 a2 a1 (JB)ss
√θ
}
(8)
The conditions to obtain this optimal situation are (i) q1, q2, and β ≈ 0, the standard Michaelis-Menten assumptions; (ii) [A] , Kmi, both W1 and W2 are operating in their linear regimes with respect to their substrate A, thus giving maximal specificity; and (iii) b2/Km2 . b1/Km1, the dominant substrate enzyme association is A + W2 f W2A rather than A + W1 f W1A. Equation 8 indicates that if d1 , b1 , a2d1/a1, then both the numerator and denominator are dominated by the Km1 terms. Hence with equal amount of A and A′ the specificity is related to θ3/2. If θ ) 100, then there will be an additional factor of 10 increase in the specificity due to the energy relay mechanism. We note that these conditions imply that the reaction cycles are strongly driven clockwise, i.e., γ . 1. Kinetic Model for Protein Tyrosine Phosphatases. The tyrosine phosphorylation and dephosphorylation system is an important component of transmembrane signaling in cytoskeletal dynamics and the cell cycle. It is widely accepted that the protein tyrosine phosphatases (PTP), together with protein tyrosine kinase (PTK),23,24 effectively control the level of phosphorylation of the target enzymes in cells. However, there are some puzzling features of tyrosine phosphorylation-dephosphorylation cycle (PdPC), as opposed to that of serine-threonine PdPC. First, tyrosine phosphorylation is often very transient and substoichiometric: That is, the physiological activity is not simply proportional to the level of phosphorylation. In fact, it has been suggested that the target proteins can have a full physiological response irrespective of the state of their phosphorylation.13,25 To reconcile this noncanonical behavior, Fischer et al.13 have proposed a kinetic model shown in Figure 4. Here, as conceptually very similar to the serial engagement model, the active Wa is no longer strictly tied to its phosphorylation: Figure 4 is another variation on the theme of CCM.
A detailed kinetic model is given in the Methods. The steady state fractions of phosphorylated W, f *a ) [W*]/W a T, and activated W, fa ) [W*] a + [Wa]/WT, can be computed (see eqs 20 and 21). Fischer et al.13 suggested that in the PTP system, increasing the phosphatase activity, φ, can decrease the level of phosphorylation and at the same time increase the level of activity. In terms of the kinetic model, this means increasing fa while decreasing f *a due to increasing φ. This behavior is predicted by the model. One can show that f *a is always a decreasing function of φ; however, with a moderate kinase activity (i.e., κ < k2γ/k-2 in eq 21), fa can be a decreasing function of φ. Similar to what we have discussed above, the same phosphatase can be either an activator or an inhibitor depending on the magnitude of the chemical driving force. In fact, there is another critical condition κ ) k2γ/k-2 under which the fa is independent of the phosphatase activity! The other critical condition is γ ) 1. Again, one can show that if γ ) 1, then neither f *a nor fa will be changing with φ. In this case, the equilibrium population distributions among W, W*, a and Wa are independent of the phosphatase, as expected for an enzyme. Hysteretic Enzyme. Both the serial engagement model and Fischer’s hypothesis articulate a memory effect in protein conformations: In the former, the MHC-peptide complex induces a conformational change in a TCR that remains active even after the peptide is dissociated; and in the latter, tyrosine phosphorylation activates a target protein that retains its activity even after PTP catalyzed dephosphorylation. The memory has to be sufficiently long to be biologically relevant. This means kinetically there is a “slow” transition for the activated protein/ receptor to return to their resting states. Enzymes with slow kinetics in response to a rapid change in ligand/effector concentration have been extensively studied in enzyme kinetics in test tubes. Kinetic characteristics of such reactions will have a sigmoidal “lag” as a function of time. This is the concept of hysteretic behavior.10,16 Hysteresis has been discussed in connection with enzyme memory in the past.26 We have shown, however, that for an enzyme to have a memory, the slow kinetic step has to be coupled to irreversibility.3,6 This distinguishes the present idea of CCM and the hysteretic enzyme. Nevertheless, hysteresis is likely to be the kinetic signature, in a test tube, of an enzyme that involves CCM in living cells. 4. Discussion Can Conformational Modification Be Relevant to a Cell without Chemical Modification? We have stated that CCM is a very subtle regulatory mechanism, and it will be difficult to identify without sensitive spectroscopic methods. In living cells, it is equally difficult to rule out the possibility that a CCM is coupled to another chemical modification step; thus, Wa is in fact stabilized by further phosphorylation. Nevertheless, such examples are indeed known in src family kinase,4 and has been observed by NMR spectroscopy.5 How likely is the existence of different conformations of a protein with sufficiently long dwell time without chemical modification? In the past, among the many well studied small proteins, one has not observed many cases of such behavior. However, we shall point out that the difference between the Monod-Wyman-Changeux (MCW) model and the KoshlandNemethy-Filmer (KNF) model for hemoglobin cooperativity was precisely the existence of multiple conformational states (i.e., T and R in the MCW model) in the absence of ligand
Cyclic Conformational Modification
Figure 5. Difference between the MCW model and the KNF model for cooperative oxygen binding by hemoglobin (Hb). The energies of the various states are shown in (A) for MCW and (B) for KNF. The square and circle represent two different conformations of the Hb subunits; the orange and turquoise subunits represent unbound and oxygen-bound. One sees that both orange squares and orange circles are present in the MCW model, indicating the coexistence of two conformations in the absence of the oxygen binding. However, this is not the case for the KNF model in which the orange color and circular shape are correlated. (C) In the MCW model, oxygen binding changes the relative population of two existing conformations, represented by the color change from orange to turquoise. (D) In the KNF model, oxygen binding shifts the conformational structure of the Hb: There is only one conformational population at any given oxygen concentration.
binding (see Figure 5). In recent years, single-molecule studies have provided a sizable body of literature that supports the existence of such multiple conformational states with slow fluctuations between them.27,28 The more precise definition of allostery is “protein activity switches via regulation by communication between two sites in the proteinsthe active site and the site of modification or binding”.29 This is clearly the dominant mechanism for small proteins and enzymes whose native structures follow the principle of “funnel-shaped energy landscape”.30,31 However, for larger proteins with multisubunits, one expects the situation to be quite different. Our current understanding of the sequence/ structure/dynamics/function relationships of large proteins is still rather rudimentary. The possibility of a CCM regulatory mechanism points to the importance for further progress in protein science. Nonequilibrium Steady State and Transient Kinetics. Equilibrium thermodynamics is the theoretical foundation of allosteric conformational change.3,32 A biochemical system in equilibrium has constant concentrations of all its species, and zero fluxes of all its reactions.6 For a nonequilibrium system, however, either it can be in the relaxation process approaching its equilibrium or it can be in a steady state with sustained chemical sources and sinks. The former is more relevant to the kinetic study of enzymatic systems in test tubes,32 and the latter is more appropriate for living cells in a homeostasis.33 Since the 1970s, the concept of hysteretic enzyme has been widely accepted. It turns out, such enzymes with slow conformational dynamic changes exhibit interesting behaviors in transient kinetics, in terms of a kinetic lag, and in its steady state turnover, sometimes called mnemonic enzyme.26 These are two different manifestations of the same dynamic origin.17,34 However, to observe the former, one needs a fast kinetic measurement, and to observe the latter, one needs to have a
J. Phys. Chem. B, Vol. 114, No. 49, 2010 16109 sustained nonequilibrium steady state: the substrate and product of the enzymatic reaction has to be approximately at constant levels but not at equilibrium. While this condition is naturally satisfied inside a living cell, it requires a regenerating system for in vitro studies. Beyond Structure: Correlation between Fluctuating Enzyme Dynamics and Functions. The discovery in section 2 that the same ligand L can be an activator and also an inhibitor for the enzyme activity of W, modulated by the concentrations of A and B, illustrates that the function of a biomolecule may not be completely determined by its molecular structure but can be reaction-system dependent. To a large extent, mean protein structures are still playing a dominant role in current molecular biology. It is a widely held belief that the function of an enzyme can be sufficiently understood from knowing its 3-dimensional structure at the atomic level. Borrowing a statement from Karplus and McCammon, “The intrinsic beauty and remarkable details of the protein structures obtained from X-ray crystallography resulted in the view that proteins are rigid. This created the misconception that atoms in a protein are fixed in position ...”.35 While the functions of some biological macromolecules can be understood almost completely from their respective 3-dimensional structures, cf. the DNA double helix, there is growing evidence that dynamics and fluctuations of an enzyme play an important role in its function.36 Recently, we have shown that, at least in theory, the range (i.e., amplitude) and rate of the conformational fluctuations of an enzyme, information that is precisely lacking in the mean 3-dimensional structure, could play a significant role in the specificity of an enzyme at steady state.37 5. Methods and Mathematical Analyses Serial Engagement. The kinetic system in eq 6a has its steady state concentrations satisfy
a2[W][L] - (d2 + b2)[WL] + q2[Wa][L] ) 0
(9a)
k1[A][Wa] - k-1[B][W] - a2[W][L] + d2[WL] ) 0 (9b) [W] + [WL] + [Wa] ) WT
(9c)
[L] + [WL] ) LT
(9d)
If LT , WT, then fa ) [Wa]/WT and 1 - fa ) [W]/WT. fa can be solved from the equation
[ µγ1 - (1 + µγ1 )f ][1 + K
WT
a
LT )
(1 - fa) +
m2
σ[µfa - (1 - fa)]
]
WT f Qm2 a
(10) where
d2 + b2 a2 b2 σ) k1Km2[A]
Km2 )
d2 + b1 q2 d2Km2 µ) b2Qm2
Qm2 )
γ)
k1b2Qm2[A] k-1d2Km2[B] (11)
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If we further have Km2, Qm2 . WT, then the eq 10 becomes
LT )
k-1 - k1fa b2 d2 fa (1 - fa) Qm2 Km2
constants b2 and b2′, and all the other rate constants the same. ′ /Kmi ) b′2/b2 ) θ > 1. When R/b2 is very We shall denote Kmi small, we have
(12)
{ }
′ ′ Km1 Km2 + b' b J 2 1 [A] ) ss K [A'] K (J) m2 m1 + b2 b1 ss
Identifying d2/Qm2 ) k-2 and b2/Km2 ) k2, we obtain eq 4 as expected. Also, if γ ) 1, then fa ) 1/(1 + µ), again as expected. On the other hand, the more interesting case is when q2 ) 0 and WT . Km2, then eq 10 becomes
{ }
and
d1 1 + ′ a a J 1′b1 [A] 2 ) d1 [A'] 1 (J)ss + a2 a1b1
(13)
ss
Hopfield’s Energy Relay. Strictly speaking, the rate constants in Figure 3 have to satisfy the detailed balance condition Rd1q2/ βb2a1 ) 1 and equilibrium equation a2b1R/q1d2β ) Keq. We, however, treat the kinetics in Figure 3 by the standard Michaelis-Menten assumptions q1, q2, and β ≈ 0. Then the steady state concentrations
[W2] )
a1a2[A]2(R + b2) ∆
a1a2[A] b1 ∆
in which
∆ ) a1a2[A]2(R + b2) + a1b1[A](R + b2 + d2) + a1a2[A] b1 + a2[A](b1b2 + b2d1 + Rd1) The steady state production rate of B:
a1a2b1[A]2(R + b2) ∆
(14)
The specificity of an enzyme is maximized when it is operating in its linear regime, i.e., [A] , Kmi ) di + bi/ai, i ) 1, 2. Therefore, the [A]2 terms in ∆ are negligible:
Jss )
(
d1 R 1 + b2 a2 a1b1
)
)
R b2 Km1 Km2 + + b2 b1
[A] 1 +
{
( (
√θ
) )
k1[K] k3 k2[P] W y\z W*a y\ z Wa y \z W
2
(
(17)
}
(18)
The expression in brackets is g1 but e θ. Fischer’s Model for Protein Tyrosine Phosphatases. Assuming the kinase and the phosphatase are both operating in their linear regimes, the kinetic equations for the signaling mechanism in Figure 4 is
a2[A](b1b2 + b2d1 + Rd1) ∆
Jss )
[A] [A']
d1 b1 + + θKm1 a2 a1 J [A] ) √θ d1 [A'] b1 (Jss) √θ + + Km1 a2 a1 ss
2
[W1] )
eθ
The energy relay works when R ) (b2b2′)1/2,12 and the majority of the substrate bind enzyme via W2: b2/Km2 . b1/Km1, where bi/Kmi are apparent bimolecular rate constants. They determine the specificities of enzyme substrate interactions.32 Therefore, we have
a1b1[A](R + b2 + d2) ∆
[W2A] )
(16)
Similarly, when R/b2 is very large, we have
1 + µγσLT fa ) if σLT e 1 1 + µγ fa ) 1 if σLT g 1
[W1A] )
[A] [A']
eθ
(15)
Now if there are substrate A and its analogue A′, with corresponding Michaelis constants Kmi and Kmi ′ , dissociation
k-1[K]
k-2[P]
(19)
k-3
k-1 ) k°-1[ADP], and k-2 ) k°-2[Pi]. [K] where k1 ) k°[ATP], 1 and [P] are the concentrations, or activities, of the kinase and the phosphatase. We introduce the parameters φ ) k-2[P]/k3 and κ ) k1[K]/k-3 to represent the activities of the phosphatase and the kinase. γ ) k1k2k3/k-1k-2k3, and kBT ln γ is the free energy of ATP hydrolysis. Then the concentration of target protein in its phosphorylated state is [W*], a and the concentration of target protein in its active state is [W*] a + [Wa]. Let f * a be the former and fa be the latter in fractions of the total,
f *a )
k2
( φκ + κ + 1) + k
-2
( φκ + κ + 1) kk ( γφκ + κ + 1) + k k ( γφκ + γκ + 1) 2 3
-2 -3
(20)
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fa )
(
k2 κ κ +κ+1 + +κ+1 φ k-2 γφ k2 κ k2k3 κ κ κ +κ+1 + +κ+1 + + +1 φ k-2 γφ k-2k-3 γφ γ (21)
( )
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Both eqs 20 and 21 can be written as [(c1/φ) + c2]/[(c3/φ) + c4] with ci g 0. It increases with φ if c2/c4 > c1/c3 and decreases with φ if c2/c4 < c1/c3. Applying this rule, we can show that f *a is always a decreasing function of φ as long as γ > 1. However, applying the same rule to eq 21, we have fa decreasing with increasing phosphatase activity φ if the kinase activity κ < k-2/ k2γ. Acknowledgment. I thank W. M. Atkins, J. A. Cooper, E. H. Fischer, C. Frieden, K. E. Neet, H. Sauro and Z. J. Zhao for helpful comments and discussions. I also am indebted to J. A. Cooper and an anonymous reviewer for carefully reading the manuscript. This work was partly supported by NSF grant No. EF0827592. References and Notes (1) Fischer, E. H.; Heilmeyer, L. M. G.; Haschke, R. H. Curr. Top. Cell. Regul. 1971, 4, 211–251. (2) Kreb, E. G. Curr. Top. Cell. Regul. 1981, 18, 401–419. (3) Beard, D. A.; Qian, H. Chemical Biophysics: QuantitatiVe Analysis of Cellular Systems; Cambridge Texts in Biomedical Engineering; Cambridge University Press: New York, 2008. (4) Cooper, J. A.; Howell, B. Cell 1993, 73, 1051–1054. (5) Volkman, B. F.; Lipson, D.; Wemmer, D. E.; Kern, D. Science 2001, 291, 2429–2433. (6) Qian, H. Annu. ReV. Phys. Chem. 2007, 58, 113–142. (7) Qian, H. Biophys. Chem. 2003, 105, 585–593. (8) Qian, H.; Beard, D. A. IET Proc. Syst. Biol. 2006, 153, 192–200. (9) Qian, H.; Cooper, J. A. Biochemistry 2008, 47, 2211–2220.
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