INTRODUCTION

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One of the requirements of life at the single-cell or multicellular level is the ability to detect external stimuli and convert them into biologically meaningful intracellular signals. Such events underlie unicellular chemotaxis, sensory reception by specialized cells, and the intercellular communications that are necessary for the development and functioning of multicellular animals. In the majority of these cases, the external signal is a molecular ligand that, by binding to a specific membrane receptor protein, triggers a cascade of enzymatic reactions that ultimately lead to the activation of an effector. The resulting cellular response is capable of adapting to the level of the external signal and may be contingent on the presence or absence of other signals (Koshland, 1980; Gerhart and Kirschner, 1997). Enzymatic signal transduction pathways are characteristically heavily regulated through feedback and multiple modulators. The control of gene expression, for example, typically involves the integration of many signals and employs complex enzymatic networks, which effectively implement logical functions (Bray, 1995; Wray, 1998; Ptashne, 1992). In contrast, olfaction and photoreception involve simpler enzymatic cascades which may be thought of as adaptive amplifiers or transducers (Reed, 1990; Stryer, 1991; Koshland, 1980) that detect an extracellular stimulus and convert it into an intracellular signal that can effectively control the information content of the cellular output signal, i.e., neurotransmitter release. Photoreceptorsthe rod and cone cells of the retinaare unique because instead of molecular ligands they transduce a particularly potent input, the visible light quanta, which carry ~50 kcal of energy. However, downstream from the specially adapted 7-helix transmembrane receptor protein, rhodopsin, the enzymatic cascade responsible for phototransduction employs molecular elements that are ubiquitous and standard components in biological signaling pathways. These include a heterotrimeric G-protein (transducin) (Alberts et al., 1994; Stryer, 1995; Simon et al., 1991), an effector enzyme (phosphodiesterase, PDE), and intracellular signals that are carried by changes in a cyclic nucleotide (cGMP) and Ca. In addition, the components of this cascade are organized in a way that is similar to many other chemical signal transduction pathways. Below we shall take advantage of the great deal of knowledge of the electrophysiology and biochemistry of rods and cones and use photoreceptors as a case study in our discussion of the general engineering principles of signal transduction.

Modern genetic and biochemistry methods have led to the discovery of a multitude of signaling cascades. The identification of the molecular elements and their interactions has provided detailed information about "how" a wide variety of specific pathways work. But little attention has been given to considering broader questions about the general system-level properties of signaling cascades and "why" they are designed the way they are. Such an approach might start with the formulation of the engineering requirements and the physical constraints on signal transduction and, by identifying common biochemical modules and their regulatory motifs, demonstrate how these functional requirements are met. The ultimate goal is to provide a unified view of the comparative physiology and biochemistry of signal transduction in the context of evolution (Gerhart and Kirschner, 1997) and to give quantitative insight into the regulatory mechanisms and the system-level consequences of the modulation/modification of the components.

Below, with the example of vertebrate phototransduction in mind, we shall examine an enzymatic cascade in general engineering terms. Following Stadman (Stadman and Chock, 1977; Chock and Stadman, 1977) and Koshland (Koshland et al., 1978), we identify the amplifier modules of the cascade as irreversible messenger activation loops with "push-pull" control by opposing enzymes. We then characterize this enzymatic amplifier in terms of the engineering parameters such as gain, bandwidth (i.e., inverse characteristic time of the response), and noise and demonstrate how these parameters may be "tuned" by adjusting enzyme concentrations. We will make clear the competition between the gain and bandwidth and the relation between the noise (due to fluctuations in reactions), bandwidth, and dissipated power. These are the engineering characteristics that determine the rate of information transfer in the signal transduction channel, and we provide the information theoretic considerations which govern the optimization of the design of the cascade; e.g., we determine the gain requirements and the optimal input/output relation. We shall also discuss the role of adaptation, which corresponds to a slow change in the optimal input/output mapping in response to a change in the statistical properties of the input.

With the framework of the engineering description in place, one can inquire how a cell controls the basic parameters of its transduction pathway. In the past, much of the discussion of enzymatic cascades has focused on the often remarkable properties of its molecular components (e.g., the impressive catalytic efficiency of phosphodiesterase; Stryer, 1995). Yet the time scale for protein evolution is slow, and the relevant engineering parameters of the transduction system would be more readily modified through the adjustment of molecular concentrations rather than their kinetic constants. We shall explicitly identify the parametric dependence of the engineering characteristics of the phototransduction cascade on the concentration of its key protein elements. This has allowed us to obtain bounds on the minimal amount of enzymes required to achieve the observed functional performance of rods which are consistent with prior measurements and to identify different means of controlling and regulating their performance characteristics.

In the next section, a generic enzymatic amplifier unit is described and analyzed in terms of gain, bandwidth, and power dissipation. The following section presents a general treatment of noise in the enzymatic amplifier. The section, Enzymatic Cascade, deals with general properties of amplifier cascades, and the following section discusses and parameterizes the effect of feedback. The section, Minimal Required Gain and Minimal Messenger Concentration, shows how the signal-to-noise considerations lead to a minimal gain requirement and to a bound on the necessary amount of transduction messenger molecules. The section, Optimization of Input/Output Relation and Adaptation, outlines the information theoretic considerations governing the design of the signal transduction system and discusses optimization and adaptation. Enzymatic Amplifier Cascade in Phototransduction analyzes the organization of the vertebrate rod phototransduction cascade from the engineering point of view, identifies the way in which all of the relevant engineering parameters are controlled by enzyme concentrations, and gives bounds on the required numbers of enzymes. The final section summarizes the lessons of the analysis and suggests avenues for future work. The summary of the chemical kinetics equations describing the phototransduction cascade may be found in Appendix A; Appendix B discusses the Ca feedback loop of rod phototransduction; Appendix C lists relevant biochemical parameters; and Appendix D provides details of the information theoretic arguments.

	BASIC ENZYMATIC AMPLIFIER

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In this section we consider the basic module of an enzymatic amplifier where the input is converted into a change in the number of messenger molecules. We show that this amplifier can be characterized by a static number gain and the response time. Signal transduction cascades can be understood as a series of such modules coupled to each other.

Let us begin by considering the basic step of chemical signal transduction: the mechanism by which the input signala change in the concentration of some messenger moleculemodulates the activity of the effector enzyme. The detection of weak signals requires that the input signal be converted into a significant and macroscopic change in the output level. Therefore, amplification and not just faithful transduction is necessary. No stoichiometric equilibrium mechanism can, by itself, provide such amplification.

To show this, let us consider the simplest example of a generic receptor (Lauffenburger and Linderman, 1993), R, which upon binding the messenger ligand L undergoes conformational change to an enzymatically active state R^*_L. Consider how a small change in the total number of ligand molecules may be detected. In response to a small change in the total number of ligands, dL_tot, no more than that many additional activated receptors, dR^*_L, can be produced. Thus, the number gain, g₀ = dR^*_L/dL_tot, is necessarily smaller than 1. (Changing the stoichiometry and going to cooperative binding of n ligands (i.e., high Hill coefficient) may offer higher sensitivity to the fractional changes of the input d ln[R^*_L]/d ln[L] = n. However, in this case the gain is even lower: dR^*_L/dL < 1/n.)

The amplification can be achieved in an enzymatic push-pull loop (Stadman and Chock, 1977; Koshland et al., 1978) where a messenger X is activated to X* in a nearly irreversible reaction (e.g., phosphorylation or GDP/GTP exchange) catalyzed by an activating enzyme E_a and X* is deactivated back to X via another nearly irreversible reaction (e.g., hydrolysis) catalyzed by a deactivating enzyme E_d (see Fig. 1). The push-pull module is described by

(1)

where _a,d are the rates of activation and deactivation which depend on the concentrations of the enzymes and substrates. We shall arbitrarily take the activating enzyme E_a (which controls the activation rate _a) to be the input signal. We will show that the response of X* to small modulations in the input, E_a, are characterized by the static gain g₀ and the time constant . In this case, g₀ can be made as large as one wants because a single signal molecule can excite many messenger molecules. However, as we will see, this comes at the cost of increasing  and the energy consumption.

View larger version (8K): [in this window] [in a new window]   FIGURE 1   Schematic representation of a push-pull amplifier loop. The messenger molecule X is activated to the by enzyme Ea and deactivated by enzyme Ed. The activation-deactivation cycle is driven by ATP as a metabolic energy source.

In the simplest case of Michaelis-Menten kinetics one would have

(2)

with K_a being the Michaelis constant and k_aK_a being the catalytic velocity. The dependence of the reaction on the concentration of energy-supplying molecules (e.g., ATP/ADP or GTP/GDP) is subsumed into the effective reaction rate k_a. We assume the reaction to be far from equilibrium and proceed unidirectionally. Here and in Eq. 1 [A] denotes the concentration of molecule A, while A refers to the total number: A = [A] * Volume. The deactivation rate _d is taken to be of the same form as Eq. 2 but dependent on [E_d] and [X*] with different parameters, k_d and K_d. Finally, Eq. 1 is supplemented by a constraint on the total number of the messenger molecules X + X* = X_tot.

In the steady state,

(3)

where the quantities with bars represent their steady-state values. The approximate expression holds for low substrate concentrations, when the saturation effects are negligible. Below, for the sake of simplicity, we shall restrict ourselves to this regime (unless stated otherwise).

Note that the ratio of active and inactive messenger concentrations in the steady state depends on the ratio of enzyme concentrations: [*]/[] = k_a[_a]/k_d[_d]. This steady state must be contrasted with the thermodynamic equilibrium, where this ratio would be fixed by the free energy difference and thus would be independent of [_a,d]. Thus the signal transduction capability of this enzymatic circuit is entirely due to its nonequilibrium, dissipative nature. Each activation-deactivation event dissipates G_cycle worth of energy; our neglect of reverse reactions is consistent only to the extent that this energy is large compared to k_BT (where k_B is the Boltzmann constant and T is temperature). The total power dissipation in the steady state is

(4)

Let us now consider the enzymatic circuit set in a certain steady statei.e., at a certain "operating point"and consider the behavior of small deviations about it: X*  X*  * in response to small fluctuations of the "input," E_a  E_a  a. Linearizing Eq. 1 yields

(5)

where

(6)

is the time constant of the response which controls how fast the perturbation decays back to the steady state and

(7)

is the differential static gain, defined as the change in the steady state * in response to a small increment in _a. Equation 5 can be solved explicitly by Fourier transform. The response to input modulation at frequency : E_a(t) =  de^itÊ_a() defines frequency-dependent gain:

(8)

(Note that g() is defined as a complex number, the phase of which fixes the time lag between the input and output oscillations.) The amplitude of the frequency-dependent gain decreases rapidly at frequencies higher than ¹, so that high amplification is limited to the frequencies within a bandwidth  = ¹. The maximal gain g₀ is achieved at  = 0. Note that both g₀ and  depend on the operating point of this enzymatic amplifier, which is characterized by (, _a, _d). From Eq. 8, we see that small, time-dependent variations in E_a produce changes in X* according to

(9)

The beauty and the presumed evolutionary advantage of the push-pull scheme are in its tunability. Assuming k_a,d and K_a,d to be fixed molecular "hardware" parameters leaves the concentrations [_a,d], [_tot] available for tuning. For example, the ratio [_a]/[_d] controls the fraction of activated messenger [*]/[X_tot] in the steady state, while the response time constant may be tuned independently by scaling both E_a,d concentrations up or down. The gain may be increased in two ways: 1) by decreasing enzyme concentrations [_a,d] and thereby increasing the time constant, or 2) by increasing the total messenger concentration [X_tot] and hence []. Increasing  corresponds to the longer lifetime of the active messenger, resulting in larger cumulative changes in X* in response to a change in E_a. However, long  also means that the X* cannot follow rapid changes in E_a: high gain comes at the expense of sluggish behavior. The compromise between high gain and fast response is quantified by the gain-bandwidth product, g₀¹, which is bounded because

(10)

The product K_ak_a is just the catalytic velocity of the enzyme. Increasing [X_tot] and therefore [] regulates the gain directly but ineffectively once the saturation regime [] > K_a is reached. Also from (4) it is clear that scaling up the total messenger concentration increases the rate of dissipation.

Phototransduction cascade provides two examples of enzymatic amplifier loop (Stryer, 1995) (see Figs. 2 and 3). In its first (membrane) stage, the activated rhodopsin (Rh*)the photoreceptor proteincatalyzes GDP/GTP exchange and the consequent activation of -transducin, G^* (a member of the G-protein family). The deactivation of G^* via GTP hydrolysis is catalyzed by the inhibitory subunit of the phosphodiesterase PDE to which G^* binds. Thus G plays the role of X, Rh* plays the role of E_a, and PDE plays the role of E_d. Of course, viewed in full detail, the G-protein mechanism is more complex than the push-pull cartoon: in particular, the loop involves the release and recovery of G subunits. This complication, however, is inessential (which does not mean that G, itself in many cases (Stryer, 1995) acting as a messenger, is irrelevant!), while the presence of the activation/deactivation loop powered by the out-of-equilibrium GTP/GDP ratio is fundamental.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   Phototransduction cascade. The incident photon activates rhodopsin, which in turn activated G-protein to form G*GTP. The activated G-protein binds to PDE and activates it, PDE*. These reactions take place on the surface of a disc. Activated PDE hydrolyzes cGMP in the cytoplasm. The drop in cGMP concentration causes some of the cyclic nucleotide-gated channels in the surface membrane of the outer rod segment to close, reducing the current into the cell and repolarizing it. Synthesis of cGMP by GC restores its concentration.

View larger version (10K): [in this window] [in a new window]   FIGURE 3   The two amplifying modules in phototransduction. (a) Activated rhodopsin catalyzes activation of transducin (the G-protein). This loop is powered by the GTP-GDP hydrolysis. (b) Active phosphodiesterase hydrolyzes cyclic GMP to 5'-GMP; cGMP is synthesized by GC; the loop is closed by the metabolic process which maintains GTP concentration.

The "readout" of the G-protein stage (see Fig. 3) is provided via the activation of the catalytic subunit of PDE through G^*-PDE binding. Active PDE* enzymatically hydrolyzes cGMPthe active messenger of the second (cytoplasmic) stagedown to GMP. cGMP is resynthesized by a guanylyl cyclase (GC) from the constant supply of GTP and plays the role of X* in Eq. 1. Even though in this case the full messenger activation/deactivation loop, GTP  cGMP  GMP, is only closed via a metabolic pathway, the quantitative analogy with the "push-pull" scheme is unmistakable. The quantitative description of the two-stage phototransduction cascade may be found in Appendix A; we shall discuss its engineering aspects in detail in the section Enzymatic Amplifier Cascade in Phototransduction.

	FLUCTUATIONS AND NOISE

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Chemical reactions are stochastic processes, and hence there are random fluctuations in the number of excited messenger molecules. The noise caused by these fluctuations determines how small a signal can be transduced faithfully. The design of any signal transduction system cannot be understood without considering its noise characteristics.

To that end, let us describe the fluctuations in the push-pull amplifier loop illustrated in Fig. 1. This amplifier loop consists of a forward reaction, exciting the messenger X at a rate r₊, and a backward reaction involving the deexcitation of X* at a rate r. In Eq. 1, r₊ is just _aX and r is _bX*. The fluctuations in the number of X and X* molecules are due to the Poisson nature of chemical reactions. Let us say that during a time interval t, the forward reaction produces n₊ more molecules, while the backward reaction leads to a loss of n molecules of X*. The net production of X* is n₊  n. The forward and backward reactions are independent statistical processes. The average number of X* produced is just n₊  n = (r₊  r)t. Because these processes are Poissonian, the variance of n₊ is (n₊  n₊)² = n₊ = r₊t, and, similarly, that of n is (n  n)² = n = rt. The variance of the total increment of X* is the sum of the two variances, i.e., (r₊ + r)t. This statistical behavior can be captured mathematically by introducing a time-dependent random noise variable (t) into the chemical kinetics equation (Eq. 1),

(11)

written here for the total number of molecules in a fixed volume. When this number is large and on a time scale longer than the microscopic time scale (time scales of order 1/(r₊ + r), on which single molecules are produced), (t) is a Gaussian random function of time with a zero average (t) = 0 and a "white noise" autocorrelation:

(12)

(where, (z) is a Dirac delta function whose value is zero everywhere except in the infinitesimal vicinity of z = 0 and whose integral over z is 1. The coefficient of the delta function, which is the strength of the noise, is determined by equating ^t dt(t) ^t dt'(t') to (_aX + _dX*)t, which is the total variance of increment of X* during the interval t. Equation 11 could be written more generally for the spatially dependent concentrations with the inclusion of molecular diffusion, but here we will neglect this effect. Small stochastic fluctuations about the uniform steady state (Eq. 3) X* are governed by the linearization of Eq. 11:

(13)

This equation is very similar in structure to Eq. 5, except that E_a stands for the noise in the input. The contribution to the variance of the fluctuations, X*(t), due to (t) is

(14)

with   representing the average over the noise . This expression can be obtained using Eq. 12, and the fact that  = X_tot*d/(_a + _d) and * = X_tot*a/(_a + _d). Note that here we have not included the fluctuations in the number of the activating/deactivating enzymes. If substrate saturation can be neglected in the enzymatic rates, expression (14) for the variance reduces to (1/X + 1/X*)¹, which also holds in equilibrium. Since X* serves as a "readout," the left-hand side of Eq. 14 is identified as the output noise N_out. Note that the r.m.s. fluctuation normalized to the mean /* decreases with increasing total messenger number, X_tot. Of course, just like an increase in the gain-bandwidth product, noise reduction comes at a price of increasing energy dissipation, because the number of activation/deactivation events per unit time increases with X_tot.

In addition to the above output noise, the total variance of X* includes the contribution of the fluctuation in the enzyme number, E_a, which is amplified by the gain factor and should be thought of as the input noise of the amplifier. The total variance is given by

(15)

Here, N_out is given by Eq. 14, |Ê_a()|² is the power spectrum of the fluctuations in number of active input enzymes, and g(), given by Eq. 8, is the frequency-dependent gain. For example, if the activating enzyme is itself governed by the push-pull process with a time constant _Ea, one would have |Ê_a()|² = 2_Ea E_a²/(1 + ² _Ea²). The frequency integral in Eq. 15 reflects the low-pass filtering property of the X* response: the magnitude of the gain |g()|² = g₀²/(1 + ²²) decreases with . If the bandwidth of the amplifying stage, ¹, is small compared with the bandwidth of input fluctuations, _Ea¹, input noise variance will be suppressed by a factor of _Ea/. This is just the effect of time averaging, because, in that case, the amplifier response sums over /_Ea independent samples of the input. Noise reduction can be achieved at the price of sluggish response, i.e., by increasing .

	ENZYMATIC CASCADE

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Why does cellular signal transduction often involve multiple steps? The primary engineering benefit of having a cascade of amplifiers is the ability to achieve higher gain without compromising the time constant of the response. Consider, for example, a cascade constructed from a sequence of enzymatic loops (Chock and Stadman, 1977) (Eq. 1), with the identification of the activated messenger output X⁽ⁿ⁾* of the nth stage with the "input" enzyme of the n + 1st stage, E_a⁽ⁿ⁺¹⁾. Each stage is endowed with its own set of kinetic parameters k_a,d⁽ⁿ⁾ (and K_a,d⁽ⁿ⁾) and tunable X_tot⁽ⁿ⁾ and E_d⁽ⁿ⁾. With the latter two parameters per stage one can control both the time constant _n (via E_d⁽ⁿ⁾) and the static gain, g₀⁽ⁿ⁾ = _n(k_a⁽ⁿ⁾[⁽ⁿ⁾]/(1 + (K_a⁽ⁿ⁾)¹[⁽ⁿ⁾])) in each stage. If the cascade performance specifications require a certain overall zero frequency gain, ₀, how should the parameters of the individual stages be set to achieve maximum bandwidth for the cascade?

As long as we consider only linear response to small inputs, the overall gain of the cascade is just the product over the stages:

(16)

where n_c is the number of cascade stages. Our requirement for the overall gain implies _n=1^n_c g₀⁽ⁿ⁾ = ₀. Because each of the stages obeys the bound (Eq. 10), we obtain a constraint on the time constants,

(17)

We can generally define the overall time constant as the maximum of the time constants of the individual stages, i.e.,

(18)

The total bandwidth, ¹, is maximized, under the constraint of Eq. 17, by making all time constants equal:
Thus, the maximum bandwidth, which is achieved by setting all of the time constants to be equal, is
When the catalytic velocities, k_a⁽ⁿ⁾K_a⁽ⁿ⁾, are all comparable, increasing the number of stages, n_c, increases the bandwidth or equivalently decreases the response time. The "speed" comes at a price of higher energy dissipation in the case of the cascaded amplifier because every stage requires an energy supply.

Another hidden "cost" of the cascade is the noise. As we have seen in the previous section, the gain in each stage has to be sufficiently large for the signal-to-noise not to deteriorate because of the shot noise introduced in every stage. That precludes the temptation to build a cascade with a large number of steps and a small gain per stage.

	ENZYMATIC AMPLIFIER WITH FEEDBACK

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Response characteristics of the amplifier may be controlled and modified via feedback. Imagine, for example, that the output [X*] of the push-pull circuit affects the "production" or influx of molecular species C:

(19)

which in turn regulates the activity of, say, deactivating enzyme, so that E_d = E_d^totH([C]) (i.e., only a fraction, H([C]), of the total number E_d^tot are active). Function F in (19) denotes the influx (or production) of C, and _C¹ denotes the rate of its outflux (or destruction). In phototransduction, as well as in many other cases, the feedback signal is Ca²⁺ (see Appendix A), which regulates enzymatic activity via an intermediary Ca-binding proteins. Including the C dependence in Eq. 1 and linearizing it together with (19) yields

(20)

(21)

with g_xc = k_d[E_d^tot]*dH/d[C] and g_C = _CdF/d[X*]. The above equations can be solved using Fourier transforms. The response of X as a function of E_a in Fourier space is given by
with the effective gain, g_f, given by

(22)

At very low frequencies the gain is

(23)

Therefore the static gain is divided by a gain reduction factor,

(24)

Negative feedback corresponds to either g_xc or g_c negative, so that g_xcg_c < 0 and  > 1, in which case the effective static gain is reduced.

Because of the additional dynamical variable, C, the temporal response of X* becomes more complex and involves two time constants. Consider the response to a small step in E_a. Suppose for simplicity _C . In that case the feedback effect is slow and the response peaks at X^*_peak  g₀E_a (the static response value without feedback) at the time of order . Relaxation to the lower, asymptotic value, X^*_s = g_f(0)E_a, occurs as the feedback switches on, on the time scale of _fbk = _C/. In the opposite limit of fast feedback, _C , there is no peak in the step response, which goes directly toward X^*_s with a time constant /. The two limits are compared in Fig. 4. (For   _c the system has damped oscillatory response.)

View larger version (11K): [in this window] [in a new window]   FIGURE 4   Frequency response with feedback: amplitude of the frequency-dependent gain, |gf()|, as a function of frequency, . Slow feedback response, i.e., large c, is shown by the solid line; the fast feedback response is shown by the dashed line; and the case with no feedback is shown by the dot-dashed line.

The static input-output map X*(E_a) and the dependence of the differential gain on the signal level involve the details of feedback coupling, F([X*]) and E_d^totH([C]).

We saw here that the feedback loop is characterized by two parameters: the feedback factor  and the time constant _C. In the case of Ca feedback (discussed in Appendix A) the latter is controlled by the number of Na/K/Ca exchangers which pump Ca out of the cell. The gain, on the other hand, is controlled by the number of Ca-binding proteins which mediate its effect on the push-pull loop enzyme (guanylyl cyclase in the case of phototransduction). Most significantly, the introduction of feedback allows one to decouple the fast and slow responses by introducing a slow time scale. In the case of phototransduction, the slow time scale is associated not with Ca recovery _c (as in the above example) but with the intermediate Ca-binding proteins acting as Ca buffers (see Appendix B).

	MINIMAL REQUIRED GAIN AND MINIMAL MESSENGER CONCENTRATION

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How much gain should a signal transduction cascade have? The input signal must generate a significant change at the output, which means a change that is unlikely to be produced by a spontaneous fluctuation of the output substance. Hence, amplification must be sufficiently strong for the signal to be larger than the root mean square (r.m.s.) noise of the output, . On the other hand, the minimal significant input signal is set by the r.m.s. input noise . (Here, N_in is not quite the fluctuations in the input but includes the frequency dependence of amplification and is defined as (d/2)|Ê()|²/(1 + ²²).) Detectability of this signal requires

(25)

which puts a lower bound on required gain. Of course, the noise may always be reduced by increasing the time constant  of the amplifier, but this comes at a price of a sluggish response to interesting stimuli. Therefore in our discussion we assume  to be fixed at its upper bound determined by the temporal response requirements. Under this condition, both signal and noise in X* fluctuate with the same time scale, namely . Thus, further filtering of this output does not improve signal detection.

For the push-pull enzymatic circuit, the input noise would be set by spontaneous fluctuations of the input enzyme concentration [E_a]² and the output noise by [X_*]². Because gain is proportional to the concentration of messenger molecules, Eq. 25 implies a lower bound on the required messenger concentration:

(26)

(with the saturation effect included, one finds that Eq. 26 can be satisfied only if does not exceed the maximal gain k_aK_a (Eq. 10). Note that although the variance of both input and output noise scales linearly with the total number of participating molecules (as appropriate for a Poisson process), their ratio depends only on concentrations and is independent of the cell volume. Let us estimate N_out according to Eq. 14 and assume for simplicity that the time constant of E_a fluctuations, _Ea, is equal to , so that N_in  E_a². In the regime below saturation, []  k_d[_d][X_tot] (according to Eq. 3), and one finds explicitly

(27)

Note that the right-hand side of Eq. 27 depends on the "operating point," i.e., the steady-state concentrations [_a]. In the limit of [_a]  0, ¹  k_d[_d] from Eq. 6 and with the Poisson statistics assumption (E_a² = _a), the bound reduces to [X_tot] > k_d[_d]/k_a. We shall return to this inequality and the role it plays in constraining the relative abundance of enzymes in a signal transduction cascade in the section Enzymatic Amplifier Cascade in Phototransduction.

	OPTIMIZATION OF INPUT/OUTPUT RELATION AND ADAPTATION

TOP ABSTRACT INTRODUCTION BASIC ENZYMATIC AMPLIFIER FLUCTUATIONS AND NOISE ENZYMATIC CASCADE ENZYMATIC AMPLIFIER WITH... MINIMAL REQUIRED GAIN AND... OPTIMIZATION OF INPUT/OUTPUT... ENZYMATIC AMPLIFIER CASCADE IN... CONCLUSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

In the previous section we established the lower bound on the gain necessary to resolve the smallest significant input. More generally, one must consider the performance of the transduction system over the full range of stimuli. It is typically desirable to transduce as broad a dynamic range of the input signal as possible. Setting the amplification gain too high is bad, as it will reduce the dynamic range by causing the output to saturate. While detectability of weak stimuli puts a lower bound on the differential gain at low background stimulus, the dynamic range consideration constrains the gain over the whole input range. Under conditions of a wide input dynamic range, a compromise between the two is required. The optimal input/output relation for a transduction system is determined by information theoretic considerations (Cover and Thomas, 1989), which formalize and extend the argument given in the previous section. Some of the details are relegated to Appendix D.

Generalizing the discussion in the previous section, we consider signal transduction as a mapping of an input variable, say y, measurable with an accuracy set by the r.m.s. noise to an output variable z = f(y) measurable with accuracy . In phototransduction, the input is the light intensity with the measurement uncertainty set by the photon shot noise, and the output is the neurotransmitter with uncertainty set by shot noise in the vesicle release. Information theoretically, the "quality" of signal transduction can be quantified via mutual information, which measures the degree of certainty about the input value y gained from observing output z. The optimal input/output mapping is the one which maximizes this mutual information. It depends not only on the noise properties but on the statistical distribution of inputs, i.e., probability P(y) of input value being between y and y + y. The r.m.s. noise levels, N_in^1/2(y) and N_out^1/2(z), define just noticeable differences in y and z, respectively, and provide the natural units for these quantities; e.g., dy/N_in^1/2(y) counts the number of distinguishable input states in a small interval dy. In the limit where the number of distinguishable output states is much smaller than the number of distinguishable input states, it has been demonstrated (Laughlin, 1981) that the optimal input/output mapping is the one which makes all distinguishable states of the output occur with equal probability. The latter is achieved if z(y) is chosen to satisfy dz/dy = cN_out^1/2(z)P(y) (with the constant c fixed by imposing the output dynamic range constraint:  dy dz/dy = z_Max).

To illustrate the relation of the input signal statistics with the optimal input/output relation, let us consider the case of phototransduction under the high light (photopic) conditions handled by the cones. It has been argued forcefully (e.g., see Shapley,