ABC transporters are billion-year-old Maxwell Demons

ATP-Binding Cassette (ABC) transporters are a broad family of biological machines, found in most prokaryotic and eukaryotic cells, performing the crucial import or export of substrates through both plasma and organellar membranes, and maintaining a steady concentration gradient driven by ATP hydrolysis. Building upon the present biophysical and biochemical characterization of ABC transporters, we propose here a model whose solution reveals that these machines are an exact molecular realization of the autonomous Maxwell Demon, a century-old abstract device that uses an energy source to drive systems away from thermodynamic equilibrium. In particular, the Maxwell Demon does not perform any direct mechanical work on the system, but simply selects which spontaneous processes to allow and which ones to forbid based on information that it collects and processes. In its autonomous version, the measurement device is embedded in the system itself. In the molecular model introduced here, the different operations that characterize Maxwell Demons (measurement, feedback, resetting) are features that emerge from the biochemical and structural properties of ABC transporters, revealing the crucial role of allostery to process information. Our framework allows us to develop an explicit bridge between the molecular-level description and the higher-level language of information theory for ABC transporters.

Exchange processes correspond to the unbinding of one nucleotide(ATP or ADP), with the subsequent binding of the other (ADP or ATP, respectively).Hence, the rates are [1]: It follows that the ratio of two reverse exchange rates is given by Eq. 1b (and the same for the different superscripts associated to the states TS, T* and T*S).
All the rates in Figures 2A and 2B must satisfy thermodynamic constrains at equilibrium.In particular, all cycles have to fulfill the Kolmogorov condition, which corresponds to the requirement of detailed balance.Thus, not all rates are independent: as an example the constraint between hydrolysis-synthesis and exchange is given by hence leading to Eq. 1a.
In Supplementary Table 1 we list all the rates as a function of independent system parameters.
For the sake of clarity and until the end of this section, we introduce the parameters α := [ATP]/[ADP] and

Rate Expression
Supplementary Table 1: Expression of the rates as a function of independent parameters, after introducing detailed balance constrains as well as η, K e and K S e .All the terms in the right column "Expression" can be defined independently.
The evolution of the system is described by the following rate equation: Note that we keep the short notation for composite reactions defined in the S.I.Section .To the set of equations above, we must add the normalization condition, that is the conservation of the total concentration of transporters: so that the probability for an ABC transporter to be in the state X is P (X) = [X]/C tot .Note that we can write down the evolution for probabilities since we are only looking at the states of a single transporter, and treating substrate concentrations ([in] and [out]) as variables that converge to stationarity.At steady state, all time derivatives are equal to zero.Naming the stationary probability to be in X as P st (X), and introducing the following notation as described in the Methods, the solution can be written in a compact way (with a proportionality factor accounting for the normalization of the probability distribution) [2] Finally, we employ the condition that at stationarity there must be a steady rate of binding/unbinding, that is the absence of net flux of substrates both between T and TS, and D and DS.This corresponds to: The substitution of the expressions for P st (TS) and P st (T) (S11) into (S12) leads to Λ TS k T off,S Λ T + ( ( ( ( ( ( ( ( ( and finally: We will sketch the analytical development to obtain Eq. (3) from Eq. (S14).Introducing the symmetry constraints as well as the assymmetry induced by K e and K S e , we can rewrite Eq. (S14) as: Thus, the displacement from equilibrium is given by Splitting the total rates between T and D states (and similarly for their analogs) as a sum of hydrolyis/synthesis and exchange leads to: After a few rearrangements, the final result can be written in a more convenient way, with the with the explicit form for F 1 ({k}) (Eq.( 3)):

Supplementary Note 4: Ratio between import and export cycles
In the discussion of Eq. (3), we stress the role played by the ratio . This corresponds to the ratio between import and export paths going through one hydrolysis and one exchange.There are two such possible paths that move the system out of equilibrium: However, if on both sides the transport cycle goes through the states T * S and T * the symmetry of the system brings the system back to equilibrium.The same result holds for a cycle that avoids the states T * S and T * , only going through the four central states.
4/5 Supplementary Note 5: Heat released by feedback operations Here, we show that all cycles involving hydrolysis after measurement, and exchange processes after resetting exhibit the same dissipation.Indeed, there are four different possible sets of feedback operations, each one associated with a certain heat release into the environment, ∆S To compute the heat release, we employ the thermodynamic relations presented in Table 1.Hence, we have: [3]: a) TS → T * S hydrolysis − −−−−−− → DS, after measurement, i.e. with bound substrate S; D exchange − −−−−− → T, after resetting, i.e. without bound substrate.