An Information Theoretical Analysis of Kinase Activated Phosphorylation Dephosphorylation Cycle
AAn Information Theoretical Analysis of KinaseActivated Phosphorylation Dephosphorylation Cycle
Hong Qian (cid:0) and Sumit Roy (cid:2)(cid:0)
Department of Applied MathematicsUniversity of Washington, Seattle, WA 98195-2420 [email protected] (cid:2)
Department of Electrical EngineeringUniversity of Washington, Seattle, WA 98195-2500 [email protected]
February 8, 2010
Abstract
Signal transduction, the information processing mechanism in biological cells, is carriedout by a network of biochemical reactions. The dynamics of driven biochemical reactions canbe studied in terms of nonequilibrium statistical physics. Such systems may also be studiedin terms of Shannon’s information theory. We combine these two perspectives in this study ofthe basic units (modules) of cellular signaling: the phosphorylation dephosphorylation cycle(PdPC) and the guanosine triphosphatase (GTPase). We show that the channel capacity is zeroif and only if the free energy expenditure of biochemical system is zero. In fact, a positivecorrelation between the channel capacity and free energy expenditure is observed. In termsof the information theory, a linear signaling cascade consisting of multiple steps of PdPC canfunction as a distributed “multistage code”. With increasing number of steps in the cascade,the system trades channel capacity with the code complexity. Our analysis shows that whilea static code can be molecular structural based; a biochemical communication channel has tohave energy expenditure.
Cellular biochemical signal transductions are communication processes on a molecular level throughchemistry. While the medium for the information processing is very different from electronic and1igure 1: A simple PdPC with non-saturated kinase and phosphatase. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5) and (cid:0)(cid:6)(cid:7)(cid:8)(cid:5) are the con-centrations of the kinase and phosphatase. Note that an enzyme has to catalyze both the forwardand backward reactions. Dephosphorylation, (cid:9)(cid:10)(cid:11)(cid:12)(cid:13) (cid:9)(cid:14)(cid:15) (cid:7)(cid:16)(cid:17) , is not the reverse reaction of thephosphorylation reaction, (cid:9)(cid:18)(cid:15)(cid:19)(cid:20)(cid:21)(cid:22) (cid:7) (cid:13) (cid:9) (cid:11) (cid:15)(cid:19)(cid:20)(cid:23)(cid:24) (cid:7) . Identical kinetics arises in GTPase signaling,where (cid:9) , (cid:9)(cid:25)(cid:11) , (cid:3) , and (cid:7) correspond to GDP (cid:26) GTPase, GTP (cid:26)
GTPase, GEF and GAP, respectively(see main text).optical, the fundamental principles in the theory for communications apply. Currently there is agrowing fascination with Shannon’s information theory in cellular biochemistry [4, 1, 14, 18].Biological information, either originating from DNA or from extra-cellular signals, are en-coded in the activities of signaling proteins and delivered as the biochemical activity propagatesthrough reaction pathways. Since all these processes are carried out by molecules, chemical ther-modynamic analysis has been developed to quantify the energetic aspect of signal transductionprocesses. We have recently shown that free energy expenditure is an important, but often over-looked, aspect of cellular signal transduction [20, 21, 22].To quantify the functional aspect of signal transduction as encoding of biochemical states andtransmission of biochemical information over channels, in this paper we introduce a formal Shan-non information theoretic analysis [13, 26, 5] into the most basic unit of cellular signal transduc-tion: the phosphorylation-dephosphorylation cycle (PdPC) and the GTPase. These two signalingmodules are kinetically isomorphic, with GDP and GTP bound GTPases correspond to the dephos-phorylated and phosphorylated states of a protein, and guanine-nucleotide exchange factor (GEF)and GTPase accelerating protein (GAP) correspond to the kinase and the phosphatase [22, 2]. Theanalysis developed below for the PdPC can be equally applied to the GTPase.2
PdPC Switch as an Informational Transfer Channel
We consider a signaling molecule, a protein, which can be chemically modified via phosphoryla-tion and dephosphorylation, catalyzed by a protein kinase (cid:0) and protein phosphatase (cid:2) respec-tively. Let (cid:3) and (cid:3) (cid:4) be the dephosphorylated and phosphorylated states of the protein. Assumingthat neither the kinase nor the phosphatase are saturated. Hence we have a simple kinetic schemeshown in Fig. 1 [2, 22].The PdPC shown in Fig. 1 has been extensively studied, from kinetic and thermodynamicperspective, in [20, 22, 23, 8, 2]. When the kinase and the phosphatase are operating in the non-saturated linear regime, the (cid:0) ’s and (cid:2) ’s are the ratio of corresponding (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) to (cid:0)(cid:10)(cid:11) . Moreover, the (cid:0)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:0)(cid:17)(cid:18)(cid:13) (cid:5) (cid:6)(cid:7)(cid:8) (cid:2)(cid:9)(cid:10) , (cid:0)(cid:19)(cid:20)(cid:21)(cid:15)(cid:22)(cid:0)(cid:17)(cid:18)(cid:20) (cid:5) (cid:6) (cid:11) (cid:2)(cid:9)(cid:10) , and (cid:2) (cid:20)(cid:23)(cid:15) (cid:2) (cid:18)(cid:20) (cid:5)(cid:12)(cid:2)(cid:13)(cid:14) (cid:10) . Therefore, (cid:24) (cid:15) (cid:0)(cid:12)(cid:13) (cid:2) (cid:13)(cid:0)(cid:25)(cid:20) (cid:2) (cid:20) (cid:15) (cid:0)(cid:10)(cid:26)(cid:27)(cid:28)(cid:4)(cid:29) (cid:5) (cid:6) (cid:8) (cid:2)(cid:13)(cid:10)(cid:5) (cid:6)(cid:15)(cid:11) (cid:2)(cid:9)(cid:10) (cid:5)(cid:16)(cid:2)(cid:13)(cid:14) (cid:10) (cid:15)(cid:16)(cid:30)(cid:31) !" (1)where (cid:0)(cid:10)(cid:26)(cid:27)(cid:28)(cid:4)(cid:29) is the equilibrium constant of the ATP hydrolysis reaction; ()* is the ATP hydrolysisfree energy. Note that the hydrolysis free energy is in the sustained high concentration of ATP andlow concentrations of ADP and Pi, away from their chemical equilibrium. The useful hydrolysisfree energy is not in the phosphate bond of the ATP molecule.The fraction of the protein in the phosphorylated state (cid:3)(cid:17)(cid:4) is [21, 22, 2] + (cid:15) (cid:5) (cid:3)(cid:18)(cid:4) (cid:10)(cid:5) (cid:3) (cid:10) (cid:19) (cid:5) (cid:3) (cid:4) (cid:10)(cid:15) (cid:0)(cid:12)(cid:13) (cid:5)(cid:16)(cid:0)(cid:20)(cid:10) (cid:19) (cid:2) (cid:20) (cid:5)(cid:16)(cid:2)(cid:13)(cid:10)(cid:0)(cid:12)(cid:13) (cid:5)(cid:16)(cid:0)(cid:20)(cid:10) (cid:19),(cid:0)(cid:25)(cid:20) (cid:5)(cid:16)(cid:0)(cid:20)(cid:10) (cid:19) (cid:2) (cid:13) (cid:5)(cid:16)(cid:2)(cid:13)(cid:10) (cid:19) (cid:2) (cid:20) (cid:5)(cid:16)(cid:2)(cid:13)(cid:10)(cid:15) - (cid:19)./0 (cid:19) - (cid:19)./ (cid:19) -1234 (cid:24) /(cid:25)5 ’ (2)where - (cid:15) (cid:0)(cid:12)(cid:13) (cid:5)(cid:16)(cid:0)(cid:20)(cid:10)(cid:2) (cid:13) (cid:5)(cid:16)(cid:2)(cid:9)(cid:10) ’ /6(cid:15) (cid:2) (cid:20)(cid:2) (cid:13) ’ and (cid:24) (cid:15) (cid:0)7(cid:13) (cid:2) (cid:13)(cid:0)(cid:19)(cid:20) (cid:2) (cid:20) (3)The parameter - characterizes the level of “upstream signal”, the parameter / characterizes thelevel of background in the absence of the kinase, and the parameter (cid:24) characterizes the amountof energy available from ATP hydrolysis. The ()* in Eq. (1) is known as the phosphorylationpotential, which is different from the standard state free energy ()*9(cid:18):(cid:15);< (cid:8)6=>? (cid:0)(cid:10)(cid:26)(cid:27)(cid:28)@(cid:29) . A normalcell has ()* on the order of 13 kcal/mol, which corresponds to (cid:24) (cid:15) 0(cid:31)A (cid:13)BC .From a thermodynamic perspective of the biochemical reactions, there is always a certainamount of phosphorlation even without the kinase. And likewise, there is always less than 100%3f phosphorylation even with the presence of the kinase. This is the physical origin of the ‘channelnoise’ from information theoretical perspective. So far, we have presented the PdPC as a biochemical kinetic system. We now consider it as alogical system for information transfer, i.e., signaling transduction.The PdPC is widely considered as a ‘switch’ in cellular biology: An increase in the kinase con-centration (or activity) will lead to the phosphorylation of (cid:0)(cid:2)(cid:3) (cid:0) (cid:4) . We now define the amplitudeof the switch ( (cid:5)(cid:0)(cid:2) (cid:3) ) [21]: (cid:5)(cid:4)(cid:2) (cid:3) (cid:0) (cid:2)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) (cid:2)(cid:5)(cid:6)(cid:7)(cid:8) (cid:3)(cid:4)(cid:0) (cid:5) (cid:6)(cid:7) (cid:6) (cid:5) (cid:6) (cid:12) (cid:6)(cid:7) (cid:6)(cid:8)(cid:6) (cid:9) (4)In fact, characteristics of PdPC activation by the kinase can be represented by a matrix (cid:13) (cid:0) (cid:14)(cid:15) (cid:10)(cid:10)(cid:16)(cid:17)(cid:18)(cid:19) (cid:19)(cid:10)(cid:16)(cid:17)(cid:18)(cid:19)(cid:10)(cid:20) (cid:19)(cid:21)(cid:17) (cid:10) (cid:20) (cid:19)(cid:20) (cid:19)(cid:21)(cid:17) (cid:10) (cid:22)(cid:23) (5)where (cid:19)(cid:10)(cid:16)(cid:17)(cid:18)(cid:19) is the probability of the signaling protein being activated in the complete absence of itskinase; and (cid:20) (cid:19)(cid:10)(cid:16)(cid:17) (cid:20) (cid:19) is the probability in the presence of sufficiently amount of kinase.It has been shown that the amplitude of the switch is independent of whether the kinase andphosphatase are operating under the linear or saturated regimes [22, 2]. Therefore, the resultsobtained in our present work apply equally well to PdPC with saturated enzyme kinetics. The matrix (cid:13) in Eq. (5) is analogous to the transition probability matrix relating the input ( (cid:24) )and output ( (cid:25) ) random variables, of a discrete, memoryless channel. This forms the bedrock ofthe theory of information transmission over a noisy channel, developed by Shannon [13, 26, 5].We provide a very brief discussion of this theory for the case of binary inputs and outputs as isappropriate for this scenario. Without loss of generality, we label the input/output states as (cid:11) (cid:12) (cid:7) respectively; hence the elements of the matrix (cid:13) in Eq. (5) can be interpreted as the conditionalprobabilities (cid:26)(cid:27)(cid:28)(cid:29)(cid:30) (cid:31) (cid:0) (cid:7) ! (cid:25) (cid:0)" where .Shannon introduced the notion of entropy to capture the average uncertainty in a discrete ran-dom variable (cid:24) , defined by * (cid:13) (cid:24) (cid:14)(cid:15)(cid:0) (cid:12)+, (cid:31)-. (cid:31)(cid:16)(cid:17) (cid:18) . (cid:31) where the random variable (cid:24) (cid:0)/&0(cid:31) withprobability . (cid:31) . For a binary (cid:13) (cid:11) (cid:12) (cid:7) (cid:14) random variable (cid:24) , this is * (cid:13) (cid:24) (cid:14)(cid:19)(cid:0) (cid:12) . (cid:3) (cid:17)(cid:20)(cid:18) . (cid:3) (cid:12)(cid:21)(cid:13) (cid:7) (cid:12) . (cid:3) (cid:14)(cid:22)(cid:17) (cid:18) (cid:13) (cid:7) (cid:12) . (cid:3) (cid:14) (6)4here (cid:0) (cid:0) (cid:2) (cid:0)(cid:2)(cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:4) . In the theory of (noiseless) source coding [5], the entropy (cid:5) (cid:4) (cid:3) (cid:5) equals the number of bits/symbol required to represent and transmit a sequence of symbols chosenrandomly and independently from the source (cid:3) ; clearly this is at maximum, equal to (cid:6)(cid:4)(cid:5)(cid:6)(cid:7) (cid:7) (cid:6) (cid:3) (cid:6) where (cid:6) (cid:3) (cid:6) is the cardinality of (cid:3) . However, as we have seen, the PdPC is noisy and we apply thetheory of information transmission over a noisy channel [5].Shannon’s information theory provides a limit - called the channel capacity - on the informationcarrying capability of any such noisy channel, that represents the maximum rate (in bits/symbolor channel use) at which the source symbols may be communicated with arbitrarily low rates oferror over such a channel. This is best understood in terms of the concept of mutual information between the input (cid:3) and output (cid:7) pair for a channel, defined as (cid:8) (cid:4) (cid:3)(cid:9)(cid:10) (cid:7)(cid:8)(cid:5)(cid:9)(cid:2) (cid:5) (cid:4) (cid:3) (cid:5) (cid:8) (cid:5) (cid:4) (cid:3) (cid:6) (cid:7) (cid:5)(cid:10)(cid:2) (cid:5) (cid:4) (cid:7)(cid:8)(cid:5) (cid:8) (cid:5) (cid:4) (cid:7) (cid:6) (cid:3) (cid:5) (cid:11) (7)where (cid:5) (cid:4) (cid:3) (cid:6) (cid:7)(cid:12)(cid:5) denotes the conditional entropy of (cid:3) given (cid:7) , defined as (cid:5) (cid:4) (cid:3) (cid:6) (cid:7) (cid:5)(cid:9)(cid:2) (cid:8)(cid:11)(cid:12) (cid:9) (cid:0) (cid:9) (cid:12) (cid:10) (cid:11) (cid:10)(cid:12)(cid:13) (cid:9)(cid:13)(cid:6) (cid:14) (cid:11) (cid:10)(cid:12)(cid:13) (cid:9) (cid:15)
It can be shown that mutual information is symmetric and non-negative, (cid:8) (cid:4) (cid:3)(cid:9)(cid:10) (cid:7)(cid:8)(cid:5)(cid:9)(cid:2) (cid:8) (cid:4) (cid:7)(cid:13)(cid:10)(cid:14)(cid:3) (cid:5)(cid:14)(cid:15)(cid:3) . It represents the amount of information transmitted by the channel; this is reflected in the reduc-tion of the uncertainty of the original source symbols H(X) via observing a related variable (cid:7) (thechannel output), quantified by (cid:5) (cid:4) (cid:3) (cid:6) (cid:7)(cid:12)(cid:5) . The capacity of such a discrete, memoryless channel isgiven by maximizing (cid:8) (cid:4) (cid:3)(cid:16)(cid:10) (cid:7) (cid:5) over all possible input distributions (cid:0) (cid:9) . In the context of PdPC, thatamounts to the probability of (cid:0) being phosphorylated with, and without the kinase. An alternateinterpretation of the mutual information (and hence channel capacity) is obtained by noting that (cid:8) (cid:4) (cid:3)(cid:9)(cid:10) (cid:7) (cid:5) is also a divergence measure between the joint probability mass function (p.m.f.) (cid:0) (cid:4) (cid:15) (cid:11) (cid:16)(cid:16)(cid:5) of the input and output with the product of the input (cid:0) (cid:4) (cid:15) (cid:5) and output (cid:11) (cid:4) (cid:16)(cid:17)(cid:5) marginal p.m.f’s, i.e. (cid:8) (cid:4) (cid:3)(cid:9)(cid:10) (cid:7) (cid:5)(cid:10)(cid:2) (cid:12) (cid:9) (cid:13) (cid:10) (cid:0) (cid:4) (cid:15)(cid:17)(cid:9) (cid:11) (cid:16)(cid:18)(cid:10) (cid:5)(cid:17)(cid:6) (cid:14) (cid:0) (cid:4) (cid:15)(cid:17)(cid:9) (cid:11) (cid:16)(cid:18)(cid:10) (cid:5)(cid:0) (cid:4) (cid:15)(cid:17)(cid:9)(cid:18)(cid:5)(cid:19)(cid:11) (cid:4) (cid:16)(cid:20)(cid:10) (cid:5) (8)This says that if the output (cid:7) is independent of the input (cid:3) , i.e., (cid:0) (cid:4) (cid:15) (cid:9) (cid:11) (cid:16)(cid:20)(cid:10) (cid:5)(cid:10)(cid:2) (cid:0) (cid:4) (cid:15)(cid:17)(cid:9)(cid:18)(cid:5)(cid:12)(cid:11) (cid:4) (cid:16)(cid:18)(cid:10) (cid:5) , the output (cid:7) provides no information about the input (cid:3) , and hence (cid:5) (cid:4) (cid:3) (cid:6) (cid:7)(cid:12)(cid:5) (cid:2)(cid:21)(cid:5) (cid:4) (cid:3) (cid:5) . In all such cases, thechannel carries no information, i.e. (cid:8) (cid:4) (cid:3)(cid:16)(cid:10) (cid:7) (cid:5)(cid:10)(cid:2) (cid:3) , and hence has zero capacity [16].5 .4 The capacity of a binary, non-symmetric channel
According to the information theory [13, 26, 5], a binary non-symmetric noisy channel is repre-sented by the conditional probability matrix (cid:0) (cid:0)(cid:0)(cid:2) (cid:2) (cid:2)(cid:3)(cid:4) (cid:4)(cid:5) (cid:2) (cid:2) (cid:5)(cid:6)(cid:7) (cid:3) (9)where (cid:4) (cid:0) (cid:0) (cid:3) (cid:4) (cid:0) (cid:2) (cid:5) (cid:6) (cid:0) (cid:4) (cid:7) and (cid:5) (cid:0) (cid:0)(cid:2)(cid:3) (cid:3) (cid:4) (cid:0) (cid:4) (cid:5) (cid:6) (cid:0) (cid:2) (cid:7) , (cid:4)(cid:8)(cid:9)(cid:0) (cid:5) , represent the two errorsintroduced by the channel, and hence (cid:4) (cid:5) (cid:5)(cid:10)(cid:11) (cid:2) is the regime of interest. Let us further assumethe input signal has a probability distribution (cid:6) (cid:8) (cid:5) (cid:2) (cid:2) (cid:8) (cid:7) . Then the joint probabilities of input andoutput are the elements of (cid:2) (cid:8) (cid:6) (cid:2) (cid:2)(cid:3)(cid:4) (cid:7) (cid:8) (cid:4)(cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:5) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7) (cid:7) (cid:3) (10)Using the divergence formulation in Eq. (8) above, the mutual information (cid:4) (cid:6) (cid:6)(cid:5)(cid:6) (cid:4)(cid:8)(cid:7) can be writtenin terms of the input p.m.f variable (cid:8) as follows: (cid:4) (cid:6) (cid:8) (cid:7) (cid:0) (cid:8) (cid:6) (cid:2) (cid:2)(cid:12)(cid:4) (cid:7)(cid:9)(cid:10) (cid:7) (cid:8) (cid:11) (cid:2) (cid:2)(cid:12)(cid:4)(cid:8) (cid:6) (cid:2) (cid:2)(cid:12)(cid:4) (cid:7) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:5) (cid:2) (cid:8) (cid:4) (cid:10) (cid:7)(cid:9)(cid:8) (cid:11) (cid:4)(cid:8) (cid:4) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7) (11) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:5) (cid:10) (cid:7) (cid:8) (cid:11) (cid:5)(cid:8) (cid:6) (cid:2) (cid:2)(cid:12)(cid:4) (cid:7) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:5) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7)(cid:9)(cid:10) (cid:7) (cid:8) (cid:11) (cid:2) (cid:2) (cid:5)(cid:8) (cid:4) (cid:2) (cid:6) (cid:2) (cid:2) (cid:8) (cid:7) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7) (cid:3) Hence, the channel capacity among all possible (cid:8) is obtained by (cid:13) (cid:0)(cid:14)(cid:15)(cid:6)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23) (cid:12) (cid:9) (cid:13)(cid:24)(cid:25) (cid:4) (cid:6) (cid:8) (cid:7) (cid:3) (12)Some straightforward algebra leads to the optimal (cid:8)(cid:8) (cid:3) (cid:0) (cid:2) (cid:2) (cid:5) (cid:6) (cid:2) (cid:2)(cid:26)(cid:27)(cid:14)(cid:7)(cid:6) (cid:2) (cid:2)(cid:12)(cid:4) (cid:2) (cid:5) (cid:7) (cid:6) (cid:2) (cid:2)(cid:26)(cid:27)(cid:14)(cid:7) (cid:5) (13)in which (cid:27) (cid:0)(cid:0)(cid:2) (cid:5)(cid:28)(cid:29) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7) (cid:13)(cid:30)(cid:31) (cid:29)(cid:4) (cid:15) (cid:6) (cid:2) (cid:2)(cid:3)(cid:4) (cid:7) (cid:13)(cid:30)(cid:31) (cid:15) (cid:7) (cid:24)!(cid:20)" (14)Therefore, the channel capacity is (cid:13) (cid:0) (cid:4) (cid:6) (cid:8) (cid:3) (cid:7)(cid:16)(cid:0) (cid:10)(cid:10)(cid:7)(cid:9)(cid:8) (cid:11) (cid:6) (cid:2) (cid:2)(cid:3)(cid:27)(cid:14)(cid:7) (cid:2) (cid:2)(cid:2) (cid:2)(cid:3)(cid:4) (cid:2) (cid:5) (cid:10) (cid:7) (cid:8) (cid:11) (cid:2) (cid:4) (cid:15)(cid:28)% (cid:13)(cid:30)(cid:31) (cid:29)&’ (cid:6) (cid:2) (cid:2)(cid:3)(cid:4) (cid:7) %(cid:17)(cid:13)(cid:30)(cid:31) (cid:15) ’ %(cid:17)(cid:13)(cid:30)(cid:31) (cid:29)&’(cid:5) (cid:29) (cid:15) (cid:6) (cid:2) (cid:2) (cid:5) (cid:7) %(cid:17)(cid:13)(cid:30)(cid:31) (cid:29)&’ (cid:15) (cid:7) (cid:3) (15)6igure 2: Channel capacity (cid:0) given in Eq. (15), and amplitude of switch ( (cid:0)(cid:0)(cid:2) (cid:3) ) given in Eq. (4), ofPdPC, as functions of (cid:0)(cid:2)(cid:3) (cid:4) (cid:5) (cid:2) (cid:6) (cid:7) (cid:8) and (cid:9) . (A) Channel capacity (solid curves) and (cid:0)(cid:0)(cid:2) (cid:3) (dashedcurves) from left to right with (cid:9) (cid:4) (cid:10) (cid:11) (cid:10)(cid:12)(cid:13) (cid:14) (cid:13)(cid:15)(cid:10) (cid:2)(cid:0)(cid:2) (cid:14) (cid:13) (cid:10) (cid:2)(cid:3)(cid:4) . (B) Channel capacity (solid curve) and (cid:0)(cid:0)(cid:2) (cid:3) (dashed curve) as function of (cid:9) with (cid:8) (cid:4)(cid:5)(cid:6) . One sees a general agreement between the channelcapacity (cid:0) and (cid:0)(cid:0)(cid:2) (cid:3) . (cid:7) and (cid:8)(cid:9)(cid:10)(cid:11)(cid:12) of PdPC Applying the calculation for channel capacity, given in Eq. (15), to the information transfer matrixgiven in Eq. (5), we have (cid:3) (cid:4) (cid:9)(cid:13) (cid:3) (cid:9) and (cid:4) (cid:4) (cid:13)(cid:13) (cid:3) (cid:8) (cid:9) (cid:11) (16)Fig 2 shows the channel capacity (cid:0) as a function of (cid:8) and (cid:9) . In comparison, we also havedrawn the amplitude of the switch ( (cid:0)(cid:4)(cid:2) (cid:3) ) defined in Eq. (4) [21]. A general agreement between (cid:0) and (cid:0)(cid:0)(cid:2) (cid:3) is observed. Note that in terms of the (cid:3) and (cid:4) in Eq. (9), (cid:0)(cid:0)(cid:2) (cid:3) (cid:4) (cid:13) (cid:5) (cid:3)(cid:6)(cid:5) (cid:4) . In fact, (cid:0)(cid:4)(cid:2) (cid:3) (cid:4) (cid:10) or (cid:3) (cid:3) (cid:4) (cid:4) (cid:13) if and only if (cid:0) (cid:4) (cid:10) as can be readily verified from Eq. 11, when (cid:0)(cid:2)(cid:3) (cid:4) (cid:10) (equivalently (cid:8) = 1).Fig. 2A also shows that the mid-points of the dashed curves are all near their corresponding (cid:16)(cid:7) .In fact, solving the mid-point of the (cid:0)(cid:4)(cid:2) (cid:3) from Eq. (4), we have the mid-point located at (cid:8) (cid:17)(cid:13)(cid:14) (cid:15)(cid:18)(cid:4)(cid:16)(cid:8)(cid:9)(cid:16)(cid:17)(cid:10)(cid:7)(cid:7) (cid:5)(cid:19)(cid:16)(cid:6)(cid:2) (cid:7) (cid:7) . So when the (cid:9) (cid:8) (cid:13) , (cid:8) (cid:17)(cid:13)(cid:14) (cid:15) (cid:4) (cid:16)(cid:7) . Therefore, the critical condition for the PdPC to functionwell as a communication channel is when (cid:8) (cid:9)(cid:18)(cid:19) (cid:13) . An insight from the above analysis is that, fromthe information transmission perspective, a PdPC is more energetically efficient, i.e., requires lessamount of free energy expenditure, when (cid:9) is larger. The parameter (cid:13) (cid:20) (cid:9) is the equilibrium constantfor the dephosphorylation reaction. For epidermal growth factor receptor (EGFR) the (cid:9) is indeedvery large; it has been reported in the range of 0.5-1.6 [9].Since, PdPC is a binary channel, the maximum information rate or capacity per channel use is7igure 3: The optimal probability for kinase being active in order to fully utilize the channelcapacity according to the information theory. (cid:0) (cid:0) (cid:2) bit. This is in fact the implications in almost all the writing of cellular biologists: “Whenthe kinase is activated, (cid:0) is phosphorylated; when the kinase is not activated, the (cid:0) is in its non-phosphorylated state.” We now realize that this assumption implies that (cid:3) (cid:0) (cid:4) and (cid:3) (cid:5) (cid:0) (cid:0) .As we have stated, the channel capacity (cid:0) is defined for the optimal scenario of a channelusage. For the PdPC, the optimal usage is when the upstream kinase with probability (cid:0) (cid:0) beinginactive, and (cid:6) (cid:2) (cid:2) (cid:0) (cid:0) (cid:7) being active, where (cid:0) (cid:0) is given in Eq. (13). It is useful to determine thevalue of (cid:0) (cid:0) in the limit (cid:5) (cid:2) (cid:0) for a fixed (cid:3) as shown in Fig 3. Since (cid:2) (cid:0) (cid:2) (cid:8)(cid:9)(cid:6) (cid:2) (cid:3) (cid:5) (cid:3)(cid:10)(cid:7) , it followsthat (cid:2) (cid:2) (cid:4) when (cid:5) (cid:2) (cid:0) with fixed (cid:3) . Furthermore from Eq. (14), we have (cid:11) (cid:0) (cid:3)(cid:3)(cid:2)(cid:3) (cid:4) (cid:4)(cid:12)(cid:0) (cid:5) (cid:6) (cid:7)(cid:8)(cid:9)(cid:10) (cid:7)(cid:2) (cid:2)(cid:11)(cid:5) (cid:0) (cid:6) (cid:2) (cid:3)(cid:13)(cid:3)(cid:14)(cid:7) (cid:15)(cid:5)(cid:6)(cid:7)(cid:8)(cid:3) (cid:8) (cid:16) (17)Hence, in Eq. (13) (cid:11) (cid:0) (cid:3)(cid:3)(cid:2)(cid:3) (cid:4) (cid:0) (cid:0) (cid:0) (cid:6) (cid:2) (cid:3)(cid:13)(cid:3)(cid:10)(cid:7)(cid:17)(cid:3) (cid:8)(cid:3) (cid:8) (cid:3) (cid:6) (cid:2) (cid:3) (cid:3)(cid:10)(cid:7) (cid:15)(cid:5)(cid:6)(cid:7)(cid:8) (cid:16) (18)It follows that (cid:11) (cid:0) (cid:3)(cid:8) (cid:3) (cid:18) (cid:11) (cid:0) (cid:3)(cid:3)(cid:2)(cid:3) (cid:4) (cid:0) (cid:0) (cid:0) (cid:2)(cid:4) (cid:16) Fig. 3 shows that the optimal scenario for PdPC is when the kinase is little more than 50% of thetime being active, and a little less than 50% of the time being inactive. In fact with (cid:5) (cid:0) (cid:0) , Eq.(18) shows that (cid:2) (cid:2) (cid:0) (cid:0) (cid:0) (cid:3)(cid:12)(cid:0) (cid:4) (cid:16) (cid:4)(cid:19)(cid:2) (cid:20) (cid:2) (cid:4) (cid:6)(cid:2)(cid:3) , and (cid:2) (cid:4) (cid:6)(cid:4)(cid:5) respectively.
Fig. 2A shows a general trend of increasing channel capacity with increasing free energy expendi-ture in the PdPC, from ATP hydrolysis. We also see that If (cid:21)(cid:22)(cid:23) (cid:0) (cid:4) , that is (cid:5) (cid:0) (cid:2) and (cid:2) (cid:0) (cid:2) (cid:2)(cid:12)(cid:5) ,8hen a changing of upstream kinase activity has no effect on the down steam protein phosphoryla-tion. Hence (cid:0) (cid:0) (cid:2) . This result is well known in biochemistry: When a reaction is at its chemicalequilibrium, changing the amount of enzyme, the kinase, can only change the rate process but notequilibrium concentrations [22, 2].When the PdPC is kept away from chemical equilibrium, i.e. either with active ATP hydrolysis( (cid:3)(cid:0)(cid:2) (cid:4) ) or ATP synthesis ( (cid:3)(cid:0)(cid:3) (cid:4) ), the mutual information between the upstream kinase and thedownstream protein activities increases: The information then can be passed through the “chan-nel”. However, the greater mutual information transmission is achieved when (cid:3)(cid:4)(cid:2) (cid:4) : This isindeed the regime the cell biology operates. One can further ask the amount of mutual informationgain per unit of energy (cid:5) (cid:6) (cid:0)(cid:7)(cid:8) (cid:0)(cid:9)(cid:10)(cid:11)(cid:12) (cid:3) : (cid:5) (cid:0)(cid:5) (cid:5) (cid:6) (cid:0) (cid:3)(cid:8) (cid:0) (cid:5) (cid:0)(cid:5) (cid:3) (cid:0) (cid:0) (cid:13) (cid:3) (cid:14) (cid:4) (cid:0) (cid:2) (cid:15)(cid:8) (cid:0) (cid:14) (cid:4) (cid:2)(cid:16)(cid:13) (cid:3) (cid:15) (cid:17) (cid:10)(cid:2)(cid:3)(cid:4)(cid:5) (cid:17) (cid:2) (cid:3) (cid:14) (cid:2) (cid:4) (cid:2) (cid:14) (cid:4) (cid:0) (cid:2) (cid:15) (cid:3) (cid:15)(cid:14) (cid:4) (cid:0) (cid:3) (cid:15) (cid:14) (cid:2) (cid:14) (cid:4) (cid:0)(cid:5)(cid:4) (cid:15) (cid:2) (cid:14) (cid:4) (cid:0) (cid:2) (cid:15) (cid:3) (cid:15) (cid:6) (cid:18) (19)Fig. 4 shows the mutual information (cid:0) (cid:14) (cid:2) (cid:15) (Eq. 11) increase with both increasing and decreasingfrom (cid:5)(cid:19)(cid:6) (cid:0) (cid:2) . It also shows that after (cid:5)(cid:19)(cid:6) (cid:2)(cid:6)(cid:7) (cid:20)(cid:8)(cid:9) (cid:4)(cid:10)(cid:11) (cid:21)(cid:12)(cid:13) (cid:3) (cid:11) , the increase in the free energy leads torelatively little gain in (cid:0) . M u t u a l I n f o r m a ti on Phosphorylation Energy Δ G = RT ln γ (kcal/mol) Figure 4: The mutual information (cid:0) between kinase activity and the substrate protein phosphory-lation, shown by the solid curve. The dashed curve is the (cid:5) (cid:0) (cid:21) (cid:5) (cid:5) (cid:6) , showing how much mutualinformation gain per unit of energy. The parameters for the computation: (cid:13) = 0.001 and (cid:2)(cid:9)(cid:0) (cid:2) (cid:18) (cid:14) ,i.e., the probability for upstream kinase being active is 0.2.
We have discussed a single step PdPC in the previous sections. We now turn our attention to asequence of PdPC. In current cell biology literature, this is widely known as “signaling cascade”9igure 5: In cellular biochemistry, different PdPCs are often found to form a chain, called a signal-ing cascade pathway. This can be represented as (a). The relation between two successive kinasesare shown in (b). The matrix in (c) quantifies the information transfer from a upstream kinase, (cid:0) (cid:0) ,to a downstream kinase (cid:0) (cid:0) (cid:2) (cid:0) .[24]. Fig. 5 illustrates a chain of the PdPC, each intermediate is a kinase itself which can beactivated by its upstream kinase and activates its downstream kinase. For example, in the mitogen-activated protein kinase (MAPK) pathway, there are MAPKK(say Erk) and MAPKKK (say Raf)[28].However, most discussions on signaling cascade emphasize the temporal aspect of the multiplePdPC: One mainly is interested in the relation between the first kinase activation event and the lastprotein phosphorylation in the chain. From information theoretical perspective, this means thatone is interested in the information transfer matrix (cid:3)(cid:0)(cid:2) (cid:2) (cid:0) (cid:3) (cid:4)(cid:2)(cid:3) (cid:3)(cid:4) (cid:3) (cid:4) (cid:4) (cid:5) (cid:2) (cid:2) (cid:5)(cid:6) (cid:3) (cid:4) (cid:4)(cid:6)(cid:7) (cid:0)(cid:8)(cid:9)(cid:3)(cid:4)(cid:5)(cid:6) (cid:10)(cid:0)(cid:8)(cid:9)(cid:3)(cid:4) (cid:4) (cid:7)(cid:5)(cid:0)(cid:8)(cid:9)(cid:3)(cid:4)(cid:7)(cid:6) (cid:10)(cid:0)(cid:8)(cid:9)(cid:3)(cid:4)(cid:11) (cid:7) (cid:0)(cid:8)(cid:9)(cid:3)(cid:4) (cid:6) (cid:10)(cid:0)(cid:8)(cid:9)(cid:3)(cid:4) (cid:3) (cid:4) (cid:11) (cid:7) (cid:0)(cid:8)(cid:9)(cid:3)(cid:4) (cid:6) (cid:10)(cid:0)(cid:8)(cid:9)(cid:3)(cid:4) (cid:7)(cid:8) (cid:6) (20)where (cid:8) (cid:2) (cid:3) (cid:4) (cid:3) (cid:4) (cid:4) is in fact the (cid:2)(cid:9)(cid:10) (cid:11) for single step PdPC. This is still a binary, asymmetricnoisy channel. Note that since (cid:8) (cid:0) (cid:3) , the n-step (cid:2)(cid:9)(cid:10) (cid:11)(cid:2)(cid:9)(cid:10) (cid:11) (cid:7)(cid:9)(cid:10) (cid:8)(cid:9)(cid:2) (cid:3) (cid:4) (cid:3) (cid:7) (cid:3) (cid:4) (cid:8) (cid:2) (cid:8)(cid:3) (cid:4) (cid:8) (cid:4) (cid:4) (cid:7) (cid:3) (cid:4) (cid:8) (cid:2) (cid:8)(cid:3) (cid:4) (cid:8) (cid:2)(cid:11)(cid:8) (cid:2) (21)decreases geometrically with (cid:10) . This is reflected in Fig. 6, where the channel capacity decreaseswith the number of steps in the cascade and asymptotically approaches zero (cid:12) (cid:7) (cid:3) (cid:3) (cid:4) (cid:8) (cid:10) (cid:8) (cid:10) (cid:2) (cid:11)(cid:12)(cid:7) (cid:2) (cid:3) (cid:4) (cid:8) .This is consistent with the well-known data-processing inequality in information theory [5] thatstates that any intermediate processing of data cannot increase mutual information.With this information theoretic perspective, one naturally asks - why there are so many path-ways with cascade? One widely given answer is that, with the multiple steps, a signaling process10igure 6: The channel capacity decreases with the number of steps in a sequential PdPC cascade,as shown in Fig. 5. Parameters used (cid:0) (cid:2) (cid:3) (cid:4) (cid:3)(cid:5)(cid:6) and (cid:7) (cid:2) (cid:0)(cid:0)(cid:2) (cid:6) (cid:3)(cid:3)(cid:4) . The result illustrates thedata-processing inequality in the information theory.can have a multiple regulatory “entry point”, and information is heavily integrated in the systemthrough feedback. We certainly believe this is a very reasonable idea which we shall call “regu-latability hypothesis”. However, we would like to suggest an alternative perspective, the conceptof distributed code [17]: If one moves away from focusing on the each and every kinase in thecascade chain one at a time, rather considers them together as a collection of different kinases thatcharacterizes the states of the biochemical system, then it is not enough to know only the temporalsequential events of one activation after another. Rather, question such as “When (cid:0)(cid:5)(cid:6) is activated,is (cid:0) (cid:0) still activated, or no longer so?” becomes meaningful. Indeed, one should consider the (cid:0) -dimensional binary variable (cid:8) (cid:0) (cid:9) (cid:10) (cid:0) (cid:11) (cid:10) (cid:2) (cid:2) (cid:2) (cid:10) (cid:0) (cid:2) (cid:12) as the dynamic variable [6]. Such information israrely provided in the experimental studies of cellular signal transduction.But this simultaneous information is crucial in understanding the function of a signaling cas-cade. For example, if one measures the fluctuating activity of a particular kinase as a function oftime, such dynamics could contain deterministic oscillation or only stochastic fluctuations. Onecan not even addressed this question without at least simultaneously measuring a pair of relevantsignaling molecules [27]. The biochemical state of a pathway is defined by the (cid:0) -dimensionalactivation “pattern”. We call this distributed code [17].The distributed code and regulatability clearly are not exclusive. They are different functionalaspects of a signaling cascade [19]. The former emphasizes the logical intra-relationship betweenthe “players”, while the latter emphasizes the system as a whole that define the biochemical statesof a module. It is interesting to recognize that in information engineering, the source coding (or11fficient information representation) and channel coding (or communication over a noisy chan-nel), as reflected in Shannon’s two major theorems, have been largely two separated communities.Signalling in biological cells suggests the need for a unification of both theories.
One can quantify the possible amount of biochemical information encoded in such a distributed,multistage code (see Fig. 5a). Note that considering the distributed code is a very different questionas the one treating the PdPC chain as multiple independent steps of a memoryless chain as in Eq.20. Rather, the entire PdPC signaling cascade as a whole represents a code that follows a Markovchain whose one-step transition probability matrix is given by the stochastic matrix (cid:0) in Eq. 5.Because the correlation between the activations of a kinase and its substrate protein (i.e., thematrix in Eq. 5), not all the proteins can code completely independent information. Therefore,the information theory computes the entropy per symbol of an infinite long (Markov) sequence (cid:0) (cid:0) (cid:2) (cid:3) (cid:0) (cid:4) (cid:2) (cid:2) (cid:0) (cid:3) is defined [5] as (cid:5) (cid:4) (cid:0) (cid:5)(cid:6)(cid:0) (cid:7) (cid:0) (cid:0)(cid:0) (cid:2) (cid:6) (cid:3)(cid:2) (cid:5) (cid:4) (cid:3) (cid:8) (cid:2) (cid:3) (cid:9) (cid:2) (cid:10)(cid:11)(cid:10)(cid:11)(cid:10) (cid:3) (cid:0) (cid:5) (22)In biology, the infinitely long cascade does not make sense. Still, it is interesting, from biochemicalinformation perspective, to obtain this parameter. It is know that for stationary, ergodic Markovprocesses, the entropy per symbol is given by (cid:5) (cid:4) (cid:0) (cid:0)(cid:2)(cid:3) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:2) (cid:3) (cid:2)(cid:4)(cid:5) (cid:0) (cid:0) (cid:0) (cid:3) (cid:2)(cid:4)(cid:5) (cid:0) (23)where (cid:3)(cid:6)(cid:2)(cid:4)(cid:5) (cid:0)(cid:4)(cid:3) (cid:0)(cid:7)(cid:8) (cid:9) (cid:3) (cid:10)(cid:11)(cid:2)(cid:12)(cid:13) (cid:14) (cid:3) (cid:15)(cid:16)(cid:0) (cid:17) (cid:3) (cid:18) (cid:0) (cid:5) (cid:2) are the 1-step transition probabilities of the chain givenby the elements of the matrix (cid:18) , and (cid:0) (cid:0) are elements of the stationary (row) vector (cid:0) of the chainthat satisfies: (cid:0) (cid:3) (cid:0) (cid:18) .For the Markov transition matrix in Eq. (9), the stationary distribution (cid:0) is readily computed as (cid:2) (cid:0) (cid:5) (cid:6) (cid:0) (cid:7) (cid:3) (cid:3) (cid:2) (cid:0)(cid:8) (cid:19) (cid:0) (cid:6) (cid:8)(cid:8) (cid:19) (cid:0)(cid:3)(cid:4) . Hence the (asymptotic) Shannon entropy per symbol for the Markov chain,as a distributed multistage code, is: (cid:5)(cid:20) (cid:3) (cid:21) (cid:0)(cid:0) (cid:22) (cid:5) (cid:5) (cid:7) (cid:0) (cid:0) (cid:0)(cid:2) (cid:22) (cid:5) (cid:5) (cid:7) (cid:3)(cid:6)(cid:2)(cid:4)(cid:5) (cid:0)(cid:9)(cid:10) (cid:2) (cid:3) (cid:11) (cid:3)(cid:6)(cid:2)(cid:4)(cid:5) (cid:0)(cid:3) (cid:21) (cid:12) (cid:13) (cid:21)(cid:2)(cid:3) (cid:0) (cid:4) (cid:10) (cid:2)(cid:4)(cid:3) (cid:11) (cid:12) (cid:13) (cid:21)(cid:5)(cid:3) (cid:0) (cid:4) (cid:3) (cid:4) (cid:10)(cid:5)(cid:2)(cid:4)(cid:3) (cid:11) (cid:3) (cid:4) (cid:3) (cid:4) (cid:10) (cid:2)(cid:4)(cid:3) (cid:11) (cid:4) (cid:4) (cid:3) (cid:12) (cid:13) (cid:21) (cid:4) (cid:0)(cid:14)(cid:10) (cid:2)(cid:4)(cid:3) (cid:11) (cid:12) (cid:13) (cid:21) (cid:4) (cid:0)(cid:3) (cid:4) (cid:4)(cid:3) (cid:12) (cid:13) (cid:4)(cid:15)(cid:16)(cid:17)(cid:0)(cid:14)(cid:10)(cid:5)(cid:2)(cid:4)(cid:3) (cid:11) (cid:12) (cid:13) (cid:4) (cid:16)(cid:17)(cid:0) (cid:4)(cid:15)(cid:16) (cid:10)(cid:5)(cid:2)(cid:4)(cid:3) (cid:11) (cid:18) (cid:4) (cid:16) (cid:12) (cid:13) (cid:4) (cid:18) (cid:16)(cid:19)(cid:0)(cid:14)(cid:10) (cid:2)(cid:4)(cid:3) (cid:11)(cid:6)(cid:7) (cid:13) (cid:4) (cid:7)(cid:23) (cid:24)(cid:8)(cid:9)(cid:13) (cid:4) (cid:0) (cid:16) (cid:4) (cid:18) (cid:16) (cid:11) (cid:20) (24)12igure 7: Entropy per symbol in a linear PdPC cascade as a multistage distributed code, (cid:0)(cid:0) , chang-ing with (cid:0) and (cid:2) according to Eq. (24).Fig. 7 shows that as a multistage distributed code, a PdPC cascade has the entropy per symbol, (cid:0)(cid:0) ,first increases with (cid:2) and then eventually decreases. Note that to have a high entropy per symbol,one needs to have high entropy in the stationary distribution (cid:2) , as well as weak correlation betweenthe successive stages. Fig. 7 shows that (cid:2) increases the entropy of the (cid:2) while also increases thecorrelation (i.e., mutual information). There is a competition between these two effects, leading toan optimal (cid:2) (cid:0) (cid:3) (cid:0) (cid:4) .For (cid:0) (cid:0) (cid:4) and (cid:0) (cid:2) (cid:0) (cid:4) , we have (cid:0)(cid:0)(cid:0)(cid:2) (cid:0) (cid:5) (cid:0) (cid:0)(cid:6)(cid:7)(cid:0)(cid:2)(cid:3)(cid:4) (cid:3) (cid:2) (cid:8) (cid:9) (cid:5) (cid:4) (cid:0) (cid:2) (cid:0) (cid:3) (cid:8) , which indeed first increasesand then decreases with increasing (cid:2) . In classical thermodynamics, energy is conserved. However, a spontaneous conversion of energyfrom one form to another, in real world, creats entropy. Entropy and free energy, in fact, wereinvented to account for the “usefulness” of energy. Ever since Shannon’s work on entropy, therehas been a continuous interest in the relation between information theory and thermodynamics,particularly in solid-state physics computers, in quantum theory, and in statistical mechanics [15,3, 10].Chemistry based cellular information processing provides another scenario where the funda-mental relation, if any, between information theory and thermodynamics can and should be further13nvestigated. In the present work, we are able to clearly show the following:
A communicationchannel consisting of a sequence of biochemical activities has zero capacity if and only if there isno free energy dissipation.
Furthermore, we have shown a clear positive correlation between thechannel capacity and the amount of free energy dissipation (Fig. 2A). Whether these quantitativerelations can be turned into something more universal remains to be further investigated.
In molecular biology, since the discovery of DNA as the information storage for heredity, there hasbeen a continuous fascination with Shannon’s information theory [12, 25]. There are two theses inShannon’s information theory, which correspond precisely to his two theorems: The informationstorage, i.e., encoding, and the information transmission, i.e., channels. Our present analysis showsthat in cellular signaling systems in terms of biochemical activities such as PdPC and GTPase, bothissues are present: signaling cascade certainly delivers information as a relay channel, but it couldalso serves as a distributed multistage code, as recently suggested [19, 17]. As a thermodynamicsystem that processes information, we have shown that free energy expenditure is an indispensablepart of biochemical communication inside a living cell.
Pursuing a possible unification of information theory and nonequilibrium thermodynamics mustbe based on the mathematical theory of probability with stochastic modeling. It is interesting topoint out that three terms have been used in connection to the nature of uncertainties. The terminformation is often used by engineers. Philosophers, on the other hand, prefer to use the term“propensity” [7]. The most understood term however is probability, which are used by mathemati-cians. In some sense, the term information is from an observer’s perspective [11], and the termpropensity emphasizes the “intrinsic nature” of the observed. And the term probability has a moreneutral connotation.
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