Thermodynamics of Computational Copying in Biochemical Systems

Ouldridge, Thomas E.; Govern, Christopher C.; Wolde, Pieter Rein ten

doi:10.1103/physrevx.7.021004

Cited by 86 publications

(140 citation statements)

References 63 publications

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“…For example, we saw that a naive version of Protocol 1 was intrinsically irreversible and therefore unable to reach the ultimate thermodynamic bounds based on starting and ending states. In recently published work [69], Ouldridge et al argue that realistic biochemical networks similarly cannot reach these fundamental bounds. In that case, the authors trace the extra dissipation to a failure to exploit all correlations generated between the measuring device and the physical system (receptors and readouts).…”

Section: Discussionmentioning

confidence: 99%

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here, for the first time, we measure experimentally this function. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase a fraction of a bit of information and show that it is bounded by the Shannon entropy for a twostate system. Further, by measuring directly the reversibility of slow protocols, we can distinguish unambiguously between protocols that can and cannot reach the expected thermodynamic bounds. INTRODUCTIONBeginning with the foundational work of Clausius, Maxwell, and Boltzmann in the 19th c., the concept of entropy has played a key role in thermodynamics. Yet, despite its importance, entropy is an elusive concept [1][2][3][4][5][6][7][8], with no unique definition; rather, the appropriate definition of entropy depends on the scale, relevant thermodynamic variables, and nature of the system, with ongoing debate existing over the proper definition even for equilibrium cases [9]. Moreover, entropy has not been directly measured but is rather inferred from other quantities, such as the integral of the specific heat divided by temperature. Here, by measuring the work required to erase a fraction of a bit of information, we isolate directly the change in entropy in an open nonequilibrium system, showing that it is compatible with the functional form proposed by Gibbs and Shannon, giving it a physical meaning in this context. Knowing the relevant form of entropy is crucial for efforts to extend thermodynamics to systems out of equilibrium.For a continuous classical system whose state in phase space x is distributed as the probability density function ρ(x), the Gibbs-Shannon entropy is [10,11] where k B is Boltzmann's constant. For quantum systems, von Neumann introduced, in 1927, the corresponding expression in terms of the density matrix [12]. Historically, the system in Eq. 1 has typically been assumed to be in thermal equilibrium. * email: johnb@sfu.caThe physical relevance of Eq. 1 for a nonequilibrium distribution ρ(x) has often been questioned (e.g., [13][14][15][16][17]). One concern is that S is constant on an isolated Hamiltonian system and can change only when evaluated on subsystems, such as those picked out by coarse graining. With many ways to choose subsystems or to coarse grain, is the associated notion of irreversibility intrinsic to the description of the system?In another approach to entropy, advanced in the context of communication and information theory, Shannon [11,18] proved that, up to a multiplicative constant, S is the only possible function satisfying three intuitive axioms. Alternatively, one can start from an...

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Section: Discussionmentioning

confidence: 99%

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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“…In addition, erasure in asymmetric states can be interpreted as situations where the initial system is out of global thermodynamic equilibrium. Since nonequilibrium settings are ubiquitous in biological systems, it is important to establish basic scenarios in simpler settings to understand more clearly, for example, why common biological systems may not be able to reach ultimate thermodynamic limits [26].…”

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confidence: 99%

Erasure without Work in an Asymmetric Double-Well Potential

Gavrilov

Bechhoefer

2016

Phys. Rev. Lett.

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According to Landauer's principle, erasing a memory requires an average work of at least kT ln 2 per bit. Recent experiments have confirmed this prediction for a one-bit memory represented by a symmetric double-well potential. Here, we present an experimental study of erasure for a memory encoded in an asymmetric double-well potential. Using a feedback trap, we find that the average work to erase can be less than kT ln 2. Surprisingly, erasure protocols that differ subtly give measurably different values for the asymptotic work, a result we explain by showing that one protocol is symmetric with the respect to time reversal, while the other is not. The differences between the protocols help clarify the distinctions between thermodynamic and logical reversibility.Introduction. Landauer's principle states that erasing a one-bit memory requires an average work of at least kT ln 2 [1, 2], with the lower bound achieved in the quasistatic limit. It plays a key role in sharpening our understanding of the second law of thermodynamics and of the interplay between information and thermodynamics, an issue first raised Maxwell [3], developed in important contributions by Szilard [4] and Bennett [5] but also subject to a long, sometimes confused discussion [6]. Recent experiments have confirmed Landauer's prediction in simple systems: in a one-bit memory represented by a symmetric double-well potential [7, 8], in memory encoded by nanomagnetic bits [9, 10], and even in quantum bits [11]. These successes have helped to create an extended version of stochastic thermodynamics [12, 13] that views information as another kind of thermodynamic resource, on the same footing as heat, chemical energy, and other sources of work [14]. This new way of looking at thermodynamics has led to experimental realizations of information engines ("Maxwell demons") [15][16][17][18].Despite its success in simple situations, Landauer's principle remains untested in more complex cases, such as systems where the symmetry between states is broken. This case, briefly mentioned but not pursued in Landauer's original paper [1], was followed up in later theoretical work [2,[19][20][21][22][23][24][25]. In addition, erasure in asymmetric states can be interpreted as situations where the initial system is out of global thermodynamic equilibrium. Since nonequilibrium settings are ubiquitous in biological systems, it is important to establish basic scenarios in simpler settings to understand more clearly, for example, why common biological systems may not be able to reach ultimate thermodynamic limits [26].Here, we explore the broken-symmetry case experimentally by studying erasure in a memory represented by an asymmetric, double-well potential. While we find broad agreement with the main predictions of theoretical work,

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“…Horowitz and Esposito showed that entropy production within a system X can be negative if X is coupled to a second system Y , and transitions in X decrease I(X; Y ) [18]. A third key result, essential to exorcising Maxwell's Demon [6,21], is that if X and Y are uncoupled from each other, yet coupled to heat baths at temperature T , then the total free energy is [22,23] HereF (X) = F eq (X) − kT x∈X p(x) ln(p eq (x)/p(x)) is the non-equilibrium free energy [20,23], with the tilde indicating the generalisation from the standard equilibrium free energy F eq (X). Systems X and Y could be two non-interacting spins, or two physically separated molecules.…”

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confidence: 99%

Biochemical Machines for the Interconversion of Mutual Information and Work

et al. 2017

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We propose a physically-realizable information-driven device consisting of an enzyme in a chemical bath, interacting with pairs of molecules prepared in correlated states. These correlations persist without direct interaction and thus store free energy equal to the mutual information. The enzyme can harness this free energy, and that stored in the individual molecular states, to do chemical work. Alternatively, the enzyme can use the chemical driving to create mutual information. A modified system can function without external intervention, approaching biological systems more closely.Organisms exploit correlations in their environment to survive and grow. This fact holds across scales, from bacterial chemotaxis, which leverages the spatial clustering of food molecules [1,2], to the loss of leaves by deciduous trees, which is worthwhile because sunlight exposure is highly correlated from day to day. Evolution itself relies on correlations across time and space, otherwise a mutation which is beneficial would immediately lose its utility and selection would be impossible.Biological systems also generate correlations. In particular, information transmission is an exercise in correlating input and output [3], and recent years have thus seen information theory applied to biological systems involved in sensing [4][5][6], signalling [7,8], chemotaxis [1, 2], adaption [9, 10] and beyond [11]. In the language of information theory, correlated variables X and Y have a positive mutual informationp(x)p(y) (measured in nats), with p(x, y) the joint probability of a given state and p(x), p(y) the marginals. The "information entropy" H(Y ) = − y∈Y p(y) ln p(y) quantifies Y 's uncertainty, and I(X; Y ) is the reduction in this entropy given knowledge of X: I(X; Y ) = H(Y ) − H(Y |X). The mutual information is symmetric, non-negative, and zero if and only if X and Y are statistically independent.Information theory is also deeply connected to thermodynamics [12][13][14][15][16][17][18][19][20]. Sagawa and Ueda [14], building on [12,13], showed that measurement cycles have a minimal work cost equal to the mutual information generated between data and memory. Horowitz and Esposito showed that entropy production within a system X can be negative if X is coupled to a second system Y , and transitions in X decrease I(X; Y ) [18]. A third key result, essential to exorcising Maxwell's Demon [6,21], is that if X and Y are uncoupled from each other, yet coupled to heat baths at temperature T , then the total free energy is [22,23] HereF (X) = F eq (X) − kT x∈X p(x) ln(p eq (x)/p(x)) is the non-equilibrium free energy [20,23], with the tilde indicating the generalisation from the standard equilibrium free energy F eq (X). Systems X and Y could be two non-interacting spins, or two physically separated molecules. For uncoupled systems, the partition function is separable and X and Y are independent in equilibrium, I eq (X; Y ) = 0. However, correlations induced by coupling between X and Y at earlier times could persist even after the coupling...

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Thermodynamics of Computational Copying in Biochemical Systems

Cited by 86 publications

References 63 publications

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Erasure without Work in an Asymmetric Double-Well Potential

Biochemical Machines for the Interconversion of Mutual Information and Work

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