Maxwell's demon in biochemical signal transduction with feedback loop

Signal transduction in living cells is vital to maintain life itself, where information transfer in noisy environment plays a significant role. In a rather different context, the recent intensive research on ‘Maxwell's demon'—a feedback controller that utilizes information of individual molecules—have led to a unified theory of information and thermodynamics. Here we combine these two streams of research, and show that the second law of thermodynamics with information reveals the fundamental limit of the robustness of signal transduction against environmental fluctuations. Especially, we find that the degree of robustness is quantitatively characterized by an informational quantity called transfer entropy. Our information-thermodynamic approach is applicable to biological communication inside cells, in which there is no explicit channel coding in contrast to artificial communication. Our result could open up a novel biophysical approach to understand information processing in living systems on the basis of the fundamental information–thermodynamics link.

where a node represents a random variable and an edge represents a causal relationship.
Due to a general framework of information thermodynamics 4 , information of initial correlation ini is characterized by the mutual information between and , the information of final correlation fin is characterized by the mutual information between + and { , + }, and the transfer entropy tr from the subsystem to the other system is characterized by the conditional mutual information between and + under the condition of . These information quantities ini , fin , and tr give a lower bound of the entropy production in the subsystem .

Supplementary note 1 | Explicit expression of the information-thermodynamic dissipation.
We consider the coupled Langevin equations (2) in the main text, where ( = , ) is a white Gaussian noise with the variance : 〈 〉 = 0, and 〈 ′ ′ 〉 = 2 ′ ′ ′ ( − ′). In the model of E. coli bacterial chemotaxis given by Eqs. (1) and (2) with ̅ ( , ) = − , we can analytically calculate the information-thermodynamic dissipation in the stationary state: When this quantity becomes zero, the equality in inequality (5) in the main text is achieved. With the linear approximation ̅ ( , ) = − , we can explicitly calculate the stationary value of 〈 〉, 〈 〉, 〈 2 〉, 〈 〉 and 〈 2 〉 as The information-thermodynamic dissipation (3) then reduces to where the correlation coefficient ( ) 2 is given by In the limit of → 0 and / → 0, the information-thermodynamic dissipation (3) can be zero, and the equality in Eq. (5) in the main text is achieved such that This corresponds to the situation where the feedback loop does not work ( → 0) and the information flow vanishes, and relaxes infinitely fast ( / → 0).

Supplementary note 2 | Detailed derivation of the second law of information thermodynamics.
Here, we show the detailed derivation of the second law of information thermodynamics for Eqs. (1) and (2) [Eq. (4) in the main text]: The heat absorption 1 is defined as the ensemble average of the Stratonovich product of the force −̇ and the velocity ̇ such that The heat absorption can be rewritten by Eq. (3) in the main text: where we used the relation of the Stratonovich integral 1 〈 ( , , ) ∘ 〉 = 〈 ( , , )〉 for any function .
From the detailed fluctuation theorem 2 , / can be rewritten as a ratio of the probability distribution. Let the backward path-probability where is given by the path-integral expression: is the normalization constant, so that ∫ + ( + ; ; ) = 1 is satisfied.
Here, we show that the general result in Ref. 4 is tighter than the information-thermodynamic inequality (12).
We first consider the path probability of a single time step from ( , ), to We next consider a Bayesian network, which represents the stochastic process of Eq.
(22) (see Supplementary Fig. 7). This Bayesian network is given by the parents Let stochastic mutual information be  Supplementary Fig. 7:   (32)]. However, in the main text, we only focus on the role of the transfer entropy tr for the sake of simplicity, by applying the weaker inequality (32).
In the model of the E. coli bacterial chemotaxis, we have = 2 and