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Notes on The Energy Equivalence of Information Cherif F. Matta, and Lou Massa J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09528 • Publication Date (Web): 02 Nov 2017 Downloaded from http://pubs.acs.org on November 6, 2017

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The Journal of Physical Chemistry

Notes on The Energy Equivalence of Information *Chérif F. Matta1-4 Lou Massa,5 1

Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, NS, Canada B3M 2J6; 2Department of Chemistry, Dalhousie University, Halifax, NS, Canada, B3H 4J3; 3Department of Chemistry, Saint Mary's University, Halifax, Nova Scotia, Canada B3H 3C3; 4Département de chimie, Université Laval, Québec, Québec Canada G1V 0A6. 5Hunter College and the Graduate School, City University of New York, New York, NY 10065, USA. Corresponding author: Cherif Matta: [email protected] Abstract: Maxwell’s demon makes observations and thereby collects information. As Brillouin points out such information is the negative of entropy (negentropy) and is the equivalent of a cost in energy. The energy cost of information can be quantified in the relationship E = kTln2, where k is the Boltzmann constant, T is the absolute temperature, and the factor ln2 arises from the existence of two possibilities for a “yes/no” circumstance, as for example in the passage of a proton through a barrier controlled by a Maxwell’s demon. This paper considers further conclusions which follow from the quantification of the energy cost of information. First, consideration of the minimum uncertainty in the measurement of energy cost of information leads to an expression for the uncertainty in the corresponding time of the measurement, which depends inversely upon temperature at which the measurements occur. Second, because of the universal connection between energy and mass, an almost imperceptible mass accompanies the accumulation of information. And third, in order to account for the total free energy change which describes the action of ATP synthase, an additional term is suggested to be appended to the Mitchell chemiosmotic equation which describes this process. The additional term accounts for the energy cost of sorting away from background ions those protons allowed to enter the ATP synthase.

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1.

Introduction

In his “Theory of Heat”1 Maxwell poses a thought experiment that appears to violate the Second Law of thermodynamics later termed the “Maxwell’s demon” by William Thomson (Lord Kelvin).2-4 Maxwell imagined an intelligent being capable, by operating a shutter, of sorting molecules in a partitioned insulated container allowing the fast (hot) molecules to pass to one side and the cold (slow) one to the other. After some time, the demon would be able to separate the cold from the hot molecules and hence to create a temperature difference inside an insulated thermodynamic system starting from its initial thermal equilibrium state creating an improbable micro-configuration from a more probable one with an apparent zero energetic cost. This apparent violation of the Second Law is an example of “a perpetual-motion machine of the second kind”5 an example of which would be “an air-conditioner that needs no power supply”.6 The Second Law requires the entropy change in an isolated system to increase with the passage of time defining the direction of time’s arrow itself. By creating order out of disorder, the operation of Maxwell’s demon appears to effectively reverse the arrow of time by running the movie backwards. Thus, the demon is an effective time reversal machine or, equivalently, a resuscitator from thermal death. The resolution of this paradox has been incrementally worked out by Szilard,7 Gabor,8 Brillouin9,10 and several others2,3 and lies in the necessity of including the demon into the thermodynamic system to preserve the Second Law. The paradox is sidestepped by realizing that the demon must “observe” the incoming molecules and determine their momentum and path to choose the opportune instant of time to open the shutter separating the two compartments. Every act of observation by the demon of a molecule in a gas (by any conceivable mechanism) increases the entropy of the demon.9 While it is true that the total entropy of the gas molecules in the two compartments decreases with time, it is the combined entropy of the gas and the demon that increases with time with every selection.9 The demon’s entropy will rapidly increase quickly damaging its capacity to continue to perform its sorting function. To continue functioning, the demon must dissipate this entropy in the form of heat to the surroundings. Hence, the paradox is only apparent and there are no violations of the Second Law.

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Each sorting operation by the demon is equivalent to an acquisition of 1 bit of information (as defined in Shannon’s information theory).9,11-14 It has been shown that the acquisition or erasure of 1 bit of information is always accompanied by a minimum increase in entropy:9 ∆S min = k ln 2 ,

(1)

which in units of energy is

ε min = T ∆Smin = kT ln 2 .

(2)

Eq. (2) gives the minimal energy acquired by an information gathering or erasing device for a bit of information. The amount specified by Eq. (2) is a lower bound, i.e., a minimum below which there can be no acquisition or erasure of information, but in actual practice this amount is not bounded from above (i.e., could be higher). This energy needs to be dissipated at the same rate as accumulated to ensure the steady-state of operation of the devise.7,9 The same minimum energy has been shown by Landauer to necessarily be dissipated with the erasure of one bit from a physical computer memory.15-18 The minimal energy specified by Eq. (2) accompanies a bit of information irrespective of the precise physical mechanism by which the information is acquired or erased. The Landauer principle limits computer speeds by the rate of heat dissipation needed to keep the temperature of its electronics within a working range. This heat dissipation rate limitation is, however, far from representing a practical limitation on the operation of today’s computers.

2.

A Temporal Landauer-Type of Uncertainty

In addition to the heating kinetic limitation on computing, it is argued here that Heisenberg’s energy-time indeterminacy relation suggests that also a temporal limit on computing speed may be at play since ∆E ∆t ≥

h , 2

(3)

implies a spread in the interaction lifetime between the observed object and the observing Maxwell’s demon. Eq. (2) gives a free energy term, i.e., an energy, and not an uncertainty in energy. But this, as with every measurement, has an uncertainty that goes with it as required by

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Eq. (3). Let us consider the energy E gained by the demon as a result of a Yes/No question. Suppose we allow the demon to make erroneous decisions, for example, forbidding a cold molecule to pass to the cold compartment or vice versa. That would be tantamount to getting a wrong answer, which can be the result of a measurement error. Getting the answer wrong (measurement error) would seem to indicate an error of magnitude kTln2. If that error can be called an uncertainty, in the Heisenberg sense, then we have the following and inserting Eq. (2) into (3), the range of time uncertainty scale for reading or erasing 1 bit is given by: ∆t ≥

h . 2kT ln 2

(4)

This is the temporal uncertainty. It asserts that a measurement of a bit, equivalent to an energy ε min , occurs at time t ± ∆t , and that the range of temporal uncertainty is inversely related to temperature. This is listed numerically in Table 1 and represented graphically (at temperatures near) 0K in Fig. 1 showing the tendency to infinity at T = 0 K. While the absolute zero is a limit that cannot be reached it can be approached arbitrarily closely by cutting-edge technologies such as laser cooling and trapping19,20 reaching record low temperatures of Einstein-Bose condensates. The condensates can reach temperatures below a microkelvin21 and, more recently, reached the unprecedented temperature of ~ 50 picokelvins in an experiment in Bremen (Germany) and are expected to drop even lower when the experiment is conducted on the space station.22 At temperatures of the order of a microkelvin the temporal uncertainty per bit is of the order of a µs, while at a picokelvin this uncertainty is of the order of an entire 5.4 s, a million and 1012 times larger than the ps uncertainty around 3 K, respectively. From Table 1, at temperatures of living organisms (25-37oC) the temporal uncertainty of the bit is of the order of ~ 10 fs. This is a time scale characteristic of fast chemical reactions. At exactly T = 0 K, the absolute temperature limit, the time uncertainty for acquiring or erasing a unit of Shannon’s information explodes to infinity (Eq. (4)). Stated differently, the change in information content or exchange of information, could occur at any time at all, since the error band around t is infinite.

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3.

The Mass Equivalent of Information

The cost of the usual unit of information (the bit) is less than 0.02 eV at the range of temperatures relevant to life (Table 1). The energy equivalent of the bit drops with temperature ( ε min ∝ T ) until it becomes zero at the absolute zero. Since information is associated with energy, and because of the relativistic energy-mass equivalence, a mass equivalent of the bit (m) can be estimated from Einstein celebrated formula E = mc2. This estimate of m (at some temperatures of interest as listed in Table 1) comes from the mass expressed as m=

kT ln 2 , c2

(5)

obtained by inserting Eq. (2) into the Einstein energy-mass formula. At the temperature range between 0 and 100 oC the mass equivalent of the bit is in the range 3 − 4 × 10−38 kg (Table 1), seven orders of magnitude smaller than the rest mass of an electron (~ 9.1×10 −31 kg). To have a more convenient unit, the mass of a terabyte (in multiples of the proton rest mass) is listed in Table 1 at selected temperatures, where one terabyte of information is defined as 13 gigabytes = 1012 bytes = 8 × 1012 bits. The mass equivalent of a terabyte at room temperature is 2.5 ×10 −25 kg = 250 yoktograms (1 yg = 10 −24 (a septillionth of a gram)). Only a carbon nanotube mechanical resonator balance is sufficiently sensitive at the yoktogram level of precision, a balance that has been used to weigh molecules and cells.23 At the current state of technological advance, there is no way of compressing a terabyte into the confines of small volumes that can be weighed by nanomechanical balances and, hence, it seems unlikely that an experimental setup can be designed to verify this prediction in the foreseeable future, but the future is boundless and inherently unpredictable. One may conclude that information is effectively massless for all practical purposes at the present stage of scientific development.

4.

The Energy Cost of Proton Sorting in the Function of ATP Synthase

Molecular motors have been recognized as realizations of the Maxwell demon when they perform sorting work.24-27 The work presented above evolved from information theoretic

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considerations of the thermodynamic efficiency of ATP synthase, the enzyme embedded in the inner mitochondrial membrane and which is responsible of harvesting chemiosmotic energy into adenosine 5’-triphoshpate (ATP) energy rich bonds.26,27 According to this work, the efficiency of ATP synthase enzyme has been revised to a figure ~ 90%26-28 rather than its typical textbook value of ~ 60%.29 This revision follows the lead of Horton A. Johnson and Knud D. Knudsen30-32 who factored the energy equivalent of information into the thermodynamic efficiency of the kidney. Johnson and Knudsen solved the long-held mystery of the apparent thermodynamic inefficiency of the kidney (compared to other body organs) by showing that the information-theoretic cost of ions sorting by the kidney is two orders of magnitudes greater than its osmotic work.30-32 It is imperative to account for the work spent in sorting ions as the kidney’s primary function is as a regulatory organ akin to a valve or a transistor allowing the flow of a given ion in only one direction.30 Similarly, mitochondrial ATP synthase can be viewed as primarily a regulatory molecular motor.33-38 The enzyme must first determine whether an ion is or is not a proton to allow protons only in its FO channels to be passed from the mitochondrial intermembrane gap (the compartment of low pH) to the inner matrix of the mitochondrion (the compartment of higher pH). For every ca. 3 or 4 protons entering ATP synthase, one adenosine 5’-diphosphate (ADP) molecule is phosphorylated into an ATP. The phosphorylation of ADP is driven to completion by the Gibbs energy released as the protons pass from a higher to a lower chemiosmotic potential. By acting as a regulatory valve, deciding an ion is or is not a proton, ATP synthase behaves as a Maxwell’s demon. In this case the demon operates in the reverse direction of the normal Maxwell’s demon, i.e., instead of accumulating protons on one side and depleting them from another, here the enzymes is selecting protons from the proton-rich compartment and allowing them in a direction that depletes the concentration and electrical gradients. In this sense, ATP synthase is a reverse Maxwell’s demon, while when it operates as ATPase, i.e., splitting ATP and using the ensuing energy to pump proton creating the gradient it can be regarded as a “forward” or “normal” demon. Whether the demon is forward or reverse is immaterial since the energy cost is associated

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with its regulatory valve-like function. ATP synthase by acting as a reverse Maxwell’s demon erases information and hence, by Landauer’s principle, must accumulate entropy during every act of picking of a proton. Therefore, an ATP synthase molecule must rid itself of a build-up of entropy in the form of heat emission to the surroundings. In other words, not all free energy stored into the proton electrochemical gradient across the inner mitochondrial membrane is available for useful conversion into ATP phosphoanhydride bonds. Some of that stored energy must be dissipated into heat to the surroundings for the very reason of keeping ATP synthase operational. The chemiosmotic theory’s central equation, as it stands, accounts for the chemical and electrical gradients as a means to store Gibbs energy. Recently, it has been proposed to add an information-theoretic term to that equation to account for the dissipation of heat by ATP synthase.26 With this additional (third) information theoretic term, the equation becomes26

∆G = 2.3 RT ∆pH + F ∆ψ + nkT ln32 , 424 14243 123 1 ∆Gchem.

∆Gelec.

(6)

∆Ginfo.

where ∆pH is the difference between the pH on the two sides of the inner mitochondrial membrane, F

is the Faraday constant (96,485 J.V−1.mol−1) if G is in molar units, ∆ψ

is the electric voltage across the inner mitochondrial membrane due to the proton gradient. In Eq. (6), the first term on the R.H.S. accounts for the Gibbs energy arising from the purely chemical gradient ( ∆Gchem. ), the second term is due to the electrical gradient and is usually the dominant term ( ∆Gelec. ), and the last term ( ∆Ginfo. ) – that is added to the textbook expression – accounts for the energy that must be dissipated to sort protons from their chemical background. The positive sign of the information theoretic term indicates the conversion of Gibbs energy into a low-grade heat unavailable for driving ATP synthesis. Without such heat dissipation, the enzyme would soon lose its ability to function. But what is n in Eq. (6)? It can be argued that it is the number of protons recognized by and traversing ATP synthase from the inter-membrane space to the mitochondrial matrix. In this view, an ion, say Na+ or K+, or Mg2+, which is denied entry 7



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is not counted and would constitute part of the background noise. In this case it would be no different than say a water molecule that also bounces off and is not admitted by ATP synthase. ATP synthase acquires the energetic equivalent of a minimum of one bit in every act of sorting to select a proton. Eq. (4) implies a time spread for measuring one bit, which at physiological temperature is ~ 10 fs. If the phosphorylation of an ADP molecule into ATP requires, say, 3 or 4 protons to pass, then the minimal temporal uncertainty for the formation of an ATP molecule is of the order of 40 fs if limited by proton recognition. This time interval is some 12 orders of magnitude shorter than the actual time per ATP formed given a turnover number of ca. 270 ± 40 s-1.39 Hence, proton recognition is not the rate limiting step that controls ATP synthase catalysis. The entry of a proton through the rotor of ATP synthase is initiated by the protonrecognition step mediated by an aspartic acid residue acting as a proton “antenna” in the inter-membrane gap.33 To obtain a sense of the time-scale of this protonation step, we find, for example, that proton transfer between H2O and •OH (adsorbed on a Pt(111) surface) occurs at a time scales 5.2±0.9 ns and 48±12 ns at 140 K (depending on the specific pathway),40 6-orders of magnitude greater than the proposed 10 fs lower limit. The concerted double proton hopping in the dimer of formic acid happens at a time-scale of ~ 100 ps,41 while the first step of the biphasic double-proton transfer in the dimer of 7azaindole occurs on a 660 fs time-scale followed by the second step on a picosecond time-scale.42 Even the first step of the latter reaction is still at least one order of magnitude slower than the information theoretic limit proposed here through Eq. (4) (i.e., ~ 20 fs at room and physiologic temperatures).

5.

Conclusions

The main point of this article is that the energy cost of information leads to follow-on consequences. One notes that quantum mechanical uncertainty is unified with thermodynamics through Eq. (4) for any act of receiving or erasing of (Shannon’s) information. The existence of an energetic equivalent of the unit of information implies an associated uncertainty of the time necessary to measure it specified by Heisenberg’s principle. If it can then be asserted that the measured time is t ±∆t , then ∆t is the time in

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which the measurement resulting into the communication of information occurs. Said differently, the time of the measurement t can occur anywhere within the range of the time period ∆t . Now we have the result that the period of time, ∆t , is inversely proportional to the temperature and tends to infinity at 0 K. It is that inverse relation that is shown in Fig. 1 and in Table 1. That inverse relation is one of the points of this paper. Another point is that information acquisition is accompanied by a gain of mass equivalent. Because c2 is big the mass equivalent of information m is extremely small, making the mass almost imperceptible. Not even the most accurate state-of-the-art nanobalances can come close to the precision needed to detect the mass of a terabyte which is in the hundreds of yoktograms order of magnitude. But where is the mass? Presumably, that is like asking: Where has the information accumulated? The question suggests it is the Maxwell’s demon that picks up mass along with the information. Presumably the Demon’s mass difference could in principle be measured before and after accumulation of information. It seems to be a novel observation that since information is equivalent to energy, and that energy is equivalent to mass, there exists an equivalency between information and mass. In symbolic terms, one may write, information → energy → mass . It seems to have gone unnoticed that in the total summation of free energy terms which describe the function of ATP synthase there is an energy cost required to sort protons out of the background in which they reside. Johnson and Knudsen were the first to point out this type of sorting cost in their analysis of the kidney function. When their reasoning is invoked in consideration of ATP synthase function it follows that a sorting cost of information must be appended to the Mitchell equation, as in Eq. (6). In summary, to recognize the energy cost of information leads to the conclusions that have been captured in Eqs. (4), (5), and (6).

Acknowledgements L.M. was funded by the US Naval Research Lab (project 47203-00 01) and by the Professional Staff Congress, CUNY (63842-00 41). C.F.M acknowledges the Natural Sciences and Engineering Research Council of Canada (NSERC), Canada Foundation for Innovation (CFI), The Lady Davis Trust and the Hebrew University of Jerusalem, and 9



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Mount Saint Vincent University for funding.

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References (1) Maxwell, J. C. Theory of Heat; Longmans, Green, and Co.: London, 1872. (2) Leff, H. S.; Rex, A. F.; (Eds.) Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing; Institute of Physics Publishing: Bristol, 2003. (3) Leff, H. S.; Rex, A. F.; (Eds.) Maxwell's Demon: Entropy, Information, Computing; Princeton University Press: Princeton, 1990. (4) Bennett, C. H. Demons, Engines and the Second Law. Sci. Am. 1987, 257, 108-116. (5) Wiener, N. Cybernetics; The M.I.T. Press and John Wiley & Sons, Inc.: New York, 1961. (6) Department of Chemistry, U. o. C. I. Maxwell's Demon. https://www.chem.uci.edu/undergraduate/applets/bounce/demon.htm 2017. (7) Szilard, L. Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen Z. Phys. 1929, 53, 840-856. (English translation: On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings, Behavioral Sci.1961, 9, 301-310. (8) Gabor, D. IV Light and Information. Progress in Optics 1961, 1, 109-153. (9) Brillouin, L. Science and Information Theory (Second Ed.); Dover Publications, Inc.: Mineola, New York, 2004. (10) Brillouin, L. Life, Thermodynamics, and Cybernetics. Am. Sci. 1949, 37, 554-568. (11) Shannon, C.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, 1963. (12) Cover, T. M.; Thomas, J. A. Elements of Information Theory; Wiley-Interscience , John Wiley & Sons, Inc.: New Jersey, 2006. (13) Volkenstein, M. V. Entropy and Information; Birkhäser Verlag AG: Basel, 2009. (14) Gatlin, L. L. Information Theory and the Living System; Columbia University Press: New York, 1972. (15) Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 1961, 5, 183-191. (16) Smith, E. Thermodynamics of natural selection III: Landauer's principle in computation and chemistry. J. Theor. Biol. 2008, 252, 213-220. (17) Bennett, C. H. Notes on Landauer's principle, reversible computation, and Maxwell's demon. Stud. Hist. Phil. Mod. Phys. 2003, 34, 501-510. (18) Bennett, C. H. Notes on Landauer's principle, reversible computation and Maxwell's demon. Cornell University Library: arXiv.org, physics, arXiv:physics/0210005 2002. (19) Cohen-Tannoudji, C. Atomic motion in laser light; Elsevier Science Publishers B. V.: 1992. (20) Metcalf, H. J.; van der Straten, P. Laser Cooling and Trapping; Springer: New York, 1999. (21) Stellmer, S. P. B.; Grimm, R.; Schreck, F. Laser cooling to quantum degeneracy. Phys. Rev. Lett. 2013, 110, 263003. (22) Cho, A. Trapped in orbit: Atomic physicists will soon make ultracold states of 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(23)

(24) (25)

(26)

(27)

(28) (29) (30) (31) (32) (33)

(34) (35)

(36)

(37) (38)

(39)

matter on the space station. Science 2017, 357, 986-989. Chaste, J.; Eichler, A.; Moser, J.; Ceballos, G.; Rurali, R.; Bachtold, A. A nanomechanical mass sensor with yoctogram resolution. Nature Nanotech. 2012, 7, 301-304. Kurzynski, M.; Chelminiak, P. Do biological molecular machines act as Maxwell's demons? Cornell University Library: arXiv.org, physics, arXiv:1412.3981 2014. Shu, L.; Li, Y.; Liu, X.; Qian, X.; Huang, S.; Jin, S.; Yang, B. Aquaporin-1 can work as a Maxwell's demon in the body. Cornell University Library: arXiv.org, physics, arXiv:1511.07177 2015. Matta C. F.; Massa, L. “Information theory and the thermodynamic efficiency of biological sorting systems: Case studies of the kidney and of mitochondrial ATPsynthase”, Chapter 1 in: Sustained Energy for Enhanced Human Functions and Activity; Elsevier: The Netherlands, 2017. (pp. 3-29). Matta, C. F.; Massa, L. Energy equivalence of information in the mitochondrion and the thermodynamic efficiency of ATP synthase. Biochemistry 2015, 54, 53765378. Nath, S. The thermodynamic efficiency of ATP synthesis in oxidative phosphorylation. Biophys. Chem. 2016, 219, 69-74. Garrett, R. H.; Grisham, C. M. Biochemistry (5th Edition); Brooks/Cole, Cengage Learning: Belmont, 2013. Johnson, H. A.; Knudsen, K. D. Renal efficiency and information theory. Nature 1965, 206, 930-931. Johnson, H. A. Information theory in biology after 18 years. Science 1970, 168, 1545-1550. Johnson, H. A. Thermal noise and biological information. Quarter. Rev. Biol. 1987, 62, 141-152. Romanovsky, Y. M.; Tikhonov, A. N. Molecular energy transducers of the living cell. Proton ATP synthase: a rotating molecular motor. Phys.-Uspekhi 2010, 53, 893-914. Boyer, P. D. Molecular motors: What makes ATP synthase spin? Nature 1999, 402, 247-248. Suzuki, T.; Ueno, H.; Mitome, N.; Suzuki, J.; Yoshida, M. Fo of ATP synthase is a rotary proton channel. Obligatory coupling of proton translocation with rotation of c-subunit ring. J. Biol. Chem. 2002, 277, 13281-13285. Nakanishi-Matsui, M.; Kashiwagi, S.; Hosokawa, H.; Cipriano, D. J.; Dunn, S. D.; Wada, Y.; Futai, M. Stochastic high-speed rotation of escherichia coli ATP synthase F1 sector: The  subunit-sensitive rotation. J. Biol. Chem. 2006, 281, 4126–4131. Lau, W. C. Y.; Rubinstein, J. L. Subnanometre-resolution structure of the intact Thermus thermophilus H1-driven ATP synthase. Nature 2012, 481, 214-218. Symersky, J.; Pagadala, V.; Osowski, D.; Krah, A.; Meier, T.; Faraldo-Gómez, J. D.; Mueller, D. M. Structure of the c10 ring of the yeast mitochondrial ATP synthase in the open conformation. Nature Struct. Mol. Biol. 2012, 19, 485-491. Etzold, C.; Deckers-Hebestreit, G.; Altendorf, K. Turnover number of Escherichia

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coli FoF1 ATP synthase for ATP synthesis in membrane vesicles. Eur. J. Biochem. 1997, 243, 336-343. (40) Nagasaka, M.; Kondoh, H.; Amemiya, K.; Ohta, T. ; Iwasawa, Y. Proton transfer in a two-dimensional hydrogen-bonding network: Water and hydroxyl on a Pt(111) surface. Phys. Rev. Lett. 2008, 100, 106101. (41) Arabi, A. A.; Matta, C. F. Effects of external electric fields on double proton transfer kinetics in the formic acid dimer. Phys. Chem. Chem. Phys. (PCCP) 2011, 13, 13738-13748. (42) Folmer, D. E.; Poth, L.; Wisniewski, E. S.; Castleman Jr., A. W. Arresting intermediate states in a chemical reaction on a femtosecond time scale: Proton transfer in model base pairs. Chem. Phys. Lett. 1998, 287, 1-7.

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Table 1 Minimal energy ( ε min ) and the time ( ∆t ) per bit as a function of temperature, and the corresponding mass of a terabyte of information in multiples of the rest mass of the proton. Temperature (oC)

(K) 1.0 ×10 −6 -273 0.15 -272 1 -270 3 0 273 25 298 37 310 100 373 3,000 ~3,300 30,000 ~30,300

(J) 9.6 × 10−30 1.4 × 10−24 1.1× 10−23 3.0 × 10−23 2.6 × 10−21 2.9 × 10−21 3.0 × 10−21 3.6 × 10−21 3.2 × 10−20 2.9 × 10−19 * 1 Terabyte = 8 × 1012 bits, the

m

∆t

ε min (T ) (eV) 6.0 × 10−11 9.0 × 10−6 7.0 × 10−5 1.9 ×10 −4 0.016 0.018 0.019 0.022

(s) 5.5 ×10 −6 3.7 × 10−11 4.8 ×10 −12 1.8 × 10−12 2.0 × 10−14 1.9 × 10−14 1.8 × 10−14 1.5 × 10−14 0.20 1.7 × 10−15 1.80 1.8 × 10−16 proton mass is taken to

(kg/bit) 1.1× 10−46 1.6 ×10 −41 1.2 × 10−40 3.4 × 10−40 2.9 × 10−38 1.4 × 102 3.2 × 10−38 1.5 × 102 3.3 × 10−38 1.6 × 102 4.0 × 10−38 1.9 × 102 3.5 ×10−37 1.7 × 103 3.2 × 10−36 1.5 × 104 be 1.6726 × 10−27 kg (or 938.28

MeV/c2).

14

Mass of a terabyte* (proton mass/TB) 5.1×10 −7 7.6 × 10−2 5.9 × 10−1 1.6


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Fig. 1 Temporal uncertainty of the bit (in seconds) as a function of absolute temperature (in K) below 10 K on a semi-logarithmic plot.

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Notes on The Energy Equivalence of Information - The Journal of

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