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Network representation and analysis of energy coupling mechanisms in cellular metabolism by a graph-theoretical approach

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Abstract

Mechanisms coupling the chemical reactions of oxidation and ATP synthesis in cellular metabolism by the fundamental biological process of oxidative phosphorylation (OX PHOS) in mitochondria provide > 90% of the energy requirements in living organisms. Mathematical graph theory methods have been extensively used to characterize various metabolic, regulatory, and disease networks in biology. However, networks of energy coupling mechanisms in OX PHOS have not been represented and analyzed previously by these approaches. Here, the problem of biological energy coupling is translated into a graph-theoretical framework, and all possible coupling schemes between oxidation and ATP synthesis are represented as graphs connecting these processes by various intermediates or states. The problem is shown to be transformed into the hard problem of finding a Hamiltonian tour in the networks of possible constituent mechanisms, given the constraints of a cyclical nature of operation of enzymes and biological molecular machines. Accessible mathematical proofs of three theorems that guarantee sufficient conditions for the existence of a Hamiltonian cycle in simple graphs are provided. The results of the general theorems are applied to the set of possible coupling mechanisms in OX PHOS and shown to (1) unequivocally differentiate between the major theories and mechanisms of energy coupling, (2) greatly reduce the possibilities for detailed consideration, and (3) deduce the biologically selected mechanism using additional constraints from the cumulative experimental record. Finally, an algorithm is constructed to implement the graph-theoretical procedure. In summary, the enormous power and generality of mathematical theorems and approaches in graph theory are shown to help solve a fundamental problem in biology.

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References

  • Allegretti M, Klusch N, Mills DJ, Vonck J, Kühlbrandt W, Davies KM (2015) Horizontal membrane-intrinsic α-helices in the stator a-subunit of an F-type ATP synthase. Nature 521:237–240

    Article  CAS  PubMed  Google Scholar 

  • Applegate DL, Bixby RE, Chvátal V, Cook WJ (2006) The traveling salesman problem. Princeton University Press, Princeton

    Google Scholar 

  • Archbold GPR, Farrington CL, Lappin SA, McKay AM, Malpress FH (1979) Oxygen-pulse curves in rat liver mitochondrial suspensions. Biochem J 180:161–174

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Bal W, Kurowska E, Maret W (2012) The final frontier of pH and the undiscovered country beyond. PLoS ONE 7:e45832

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Barabasi AL, Gulbahce N, Loscalzo J (2011) Network medicine: a network-based approach to human disease. Nat Rev Genet 12:56–68

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Beard DA (2005) A biophysical model of the mitochondrial respiratory system and oxidative phosphorylation. PLoS Comp Biol 1:e36

    Article  CAS  Google Scholar 

  • Bhat R, Lerea KM, Peng H, Kaltner H, Gabius H-J, Newman SA (2011) A regulatory network of two galectins mediates the earliest steps of avian limb skeletal morphogenesis. BMC Dev Biol 11:6

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Bondy A, Chvátal V (1976) A method in graph theory. Discrete Math 15:111–135

    Article  Google Scholar 

  • Calabrese S, Chakrawal A, Manzoni S, Van Cappellen P (2021) Energetic scaling in microbial growth. Proc Natl Acad Sci USA 118:e2107668118

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Caputo J-G, Girardin V, Knipple A, Nguyen MH, Niquil N, Noguès Q (2021) Analysis of trophic networks: an optimisation approach. J Math Biol 83:53

    Article  PubMed  Google Scholar 

  • Casey R, de Jong H, Gouzé J-L (2006) Piecewise linear models of genetic regulatory networks: equilibria and their stability. J Math Biol 52:27–56

    Article  PubMed  Google Scholar 

  • Chance B, Mela L (1966) A hydrogen ion concentration gradient in a mitochondrial membrane. Nature 212:369–372

    Article  CAS  PubMed  Google Scholar 

  • Channakeshava C (2011) New paradigm for ATP synthesis and consumption. J Biosci 36:3–4

    Article  CAS  PubMed  Google Scholar 

  • Choi KP, Kaur G, Wu T (2021) On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees. J Math Biol 83:40

    Article  PubMed  PubMed Central  Google Scholar 

  • Collienne L, Elmes K, Fischer M, Bryant D, Gavryushkin A (2021) Discrete coalescent trees. J Math Biol 83:60

    Article  PubMed  PubMed Central  Google Scholar 

  • de Jong H, Geiselmann J, Hernandez C, Page M (2003) Genetic network analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19:336–344

    Article  PubMed  CAS  Google Scholar 

  • Diestel R (2000) Graph theory. Springer, New York

    Google Scholar 

  • Dirac GA (1952) Some theorems on abstract graphs. Proc Lond Math Soc 2:69–81

    Article  Google Scholar 

  • Golas SM, Nguyen AN, Rietman EA, Tuszynski JA (2019) Gibbs free energy of protein-protein interactions correlates with ATP production in cancer cells. J Biol Phys 45:423–430

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Gould JM (1979) Respiration-linked proton transport, changes in external pH, and membrane energization in cells of Escherichia coli. J Bacteriol 138:176–184

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Gould JM, Cramer WA, van Thienen G (1976) The effect of colicin E1 on proton extrusion and the H+/O ratio in Escherichia coli. Biochem Biophys Res Commun 72:1519–1525

    Article  CAS  PubMed  Google Scholar 

  • Green DE (1981) A critique of the chemosmotic model of energy coupling. Proc Natl Acad Sci USA 78:2240–2243

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Grognard F, de Jong H, Gouzé J-L (2007) Piecewise-linear models of genetic regulatory networks: theory and example. Lecture notes in control and information sciences, vol 357. Springer, Berlin, Heidelberg, pp 137–159

    Google Scholar 

  • Guo H, Bueler SA, Rubinstein JL (2017) Atomic model for the dimeric FO region of mitochondrial ATP synthase. Science 358:936–940

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Hansen LD, Tolley HD, Woodfield BF (2021) Transformation of matter in living organisms during growth and evolution. Biophys Chem 271:106550

    Article  CAS  PubMed  Google Scholar 

  • Heiske M, Letellier T, Klipp E (2017) Comprehensive mathematical model of oxidative phosphorylation valid for physiological and pathological conditions. FEBS J 284:2802–2828

    Article  CAS  PubMed  Google Scholar 

  • Hill TL (1989) Free energy transduction and biochemical cycle kinetics. Springer, New York

    Book  Google Scholar 

  • Hochachka PW (2019) Muscle as molecular and metabolic machines. CRC Press, Boca Raton, FL, USA

    Book  Google Scholar 

  • Ivanischev VV (2018) Problems in the mathematical description of the chemiosmotic theory. News Tula State Univ Natl Sci 3:129–135

    Google Scholar 

  • Jain S, Murugavel R, Hansen LD (2004) ATP synthase and the torsional mechanism: resolving a 50-year-old mystery. Curr Sci 87:16–19

    Google Scholar 

  • Ji S (2018) The cell language theory: connecting mind and matter. World Scientific, Singapore (Chapter 3)

    Book  Google Scholar 

  • Jost J (2007) Dynamical networks. In: Feng J, Jost J, Qian M (eds) Networks: from biology to theory. Springer, London, pp 35–62

    Chapter  Google Scholar 

  • Jost J (2014) Mathematical methods in biology and neurobiology. Springer, Berlin (Chapter 2)

    Book  Google Scholar 

  • Jost J (2021) Biology, geometry and information. Theory Biosci. https://doi.org/10.1007/s12064-021-00351-9

    Article  PubMed  PubMed Central  Google Scholar 

  • Jou D, Llebot JE (1990) Introduction to the thermodynamics of biological processes. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Juretić D (2022) Bioenergetics: a bridge across life and universe. CRC Press, Boca Raton

    Google Scholar 

  • Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum Press, New York, pp 85–103

    Chapter  Google Scholar 

  • Korzeniewski B, Zoladz JA (2001) A model of oxidative phosphorylation in mammalian skeletal muscle. Biophys Chem 92:17–34

    Article  CAS  PubMed  Google Scholar 

  • Koutrouli M, Karatzas E, Paez-Espino D, Pavlopoulos GA (2020) A guide to conquer the biological network era using graph theory. Front Bioeng Biotechnol 8:34. https://doi.org/10.3389/fbioe.2020.00034

    Article  PubMed  PubMed Central  Google Scholar 

  • Kühlbrandt W, Davies KM (2016) Rotary ATPases: a new twist to an ancient machine. Trends Biochem Sci 41:106–115

    Article  PubMed  CAS  Google Scholar 

  • Lebon G, Jou D, Casas-Vázquez J (2008) Understanding non-equilibrium thermodynamics: foundations, applications. Frontiers, Springer, Berlin

    Book  Google Scholar 

  • Levy WB, Calvert VG (2021) Communication consumes 35 times more energy than computation in the human cortex, but both costs are needed to predict synapse number. Proc Natl Acad Sci USA 118:e2008173118

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Marciniak-Czochra A, Karch G, Suzuki K (2017) Instability of turing patterns in reaction-diffusion-ODE systems. J Math Biol 74:583–618

    Article  PubMed  Google Scholar 

  • Martyushev LM (2021) Maximum entropy production principle: History and current status. Phys Usp 64:558–583

    Article  Google Scholar 

  • Mehta R, Singh J, Nath S (2020) Time-resolved oxygen exchange measurements offer novel mechanistic insights into enzyme-catalyzed ATP synthesis during photophosphorylation. J Phys Chem B 124:5139–5148

    Article  CAS  PubMed  Google Scholar 

  • Mercker M, Brinkmann F, Marciniak-Czochra A, Richter T (2016) Beyond turing: mechanochemical pattern formation in biological tissues. Biol Direct 11:22

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Mitchell P (1966) Chemiosmotic coupling in oxidative and photosynthetic phosphorylation. Biol Rev 41:445–502

    Article  CAS  PubMed  Google Scholar 

  • Mitchell P (1981) Bioenergetic aspects of unity in biochemistry: Evolution of the concept of ligand conduction in chemical, osmotic and chemiosmotic reaction mechanisms. In: Semenza G (ed) Of oxygen, fuels and living matter, part 1. Wiley, New York, pp 30–56

    Google Scholar 

  • Murray JD (1977) Nonlinear differential equation models in biology. Clarendon Press, Oxford

    Google Scholar 

  • Murray JD (2002) Mathematical biology I: an introduction, 3rd edn. Springer, New York

    Book  Google Scholar 

  • Murray JD (2003) Mathematical biology II: spatial models and biomedical applications, 3rd edn. Springer, New York

    Book  Google Scholar 

  • Nath S (2002) The molecular mechanism of ATP synthesis by F1F0-ATP synthase: a scrutiny of the major possibilities. Adv Biochem Eng Biotechnol 74:65–98

    CAS  PubMed  Google Scholar 

  • Nath S (2003) Molecular mechanisms of energy transduction in cells: engineering applications and biological implications. Adv Biochem Eng Biotechnol 85:125–180

    CAS  PubMed  Google Scholar 

  • Nath S (2004) The torsional mechanism of energy transduction and ATP synthesis as a breakthrough in our understanding of the mechanistic, kinetic and thermodynamic details. Thermochim Acta 422:5–17

    Article  CAS  Google Scholar 

  • Nath S (2008) The new unified theory of ATP synthesis/hydrolysis and muscle contraction, its manifold fundamental consequences and mechanistic implications and its applications in health and disease. Int J Mol Sci 9:1784–1840

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Nath S (2010a) Beyond the chemiosmotic theory: Analysis of key fundamental aspects of energy coupling in oxidative phosphorylation in the light of a torsional mechanism of energy transduction and ATP synthesis—invited review part 2. J Bioenerg Biomembr 42:301–309

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2010b) Beyond the chemiosmotic theory: analysis of key fundamental aspects of energy coupling in oxidative phosphorylation in the light of a torsional mechanism of energy transduction and ATP synthesis—invited review part 1. J Bioenerg Biomembr 42:293–300

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2017) Two-ion theory of energy coupling in ATP synthesis rectifies a fundamental flaw in the governing equations of the chemiosmotic theory. Biophys Chem 230:45–52

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2018a) Molecular mechanistic insights into uncoupling of ion transport from ATP synthesis. Biophys Chem 242:15–21

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2018b) Molecular mechanistic insights into coupling of ion transport to ATP synthesis. Biophys Chem 241:20–26

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2018c) Optimality principle for the coupled chemical reactions of ATP synthesis and its molecular interpretation. Chem Phys Lett 699:212–217

    Article  CAS  Google Scholar 

  • Nath S (2019a) Entropy production and its application to the coupled nonequilibrium processes of ATP synthesis. Entropy 21(746):1–22

    Google Scholar 

  • Nath S (2019b) Integration of demand and supply sides in the ATP energy economics of cells. Biophys Chem 252:106208

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2020) Consolidation of Nath’s torsional mechanism of ATP synthesis and two-ion theory of energy coupling in oxidative phosphorylation and photophosphorylation. Biophys Chem 257:106279

    Article  CAS  PubMed  Google Scholar 

  • Nath S (2021) Energy landscapes and dynamics of ion translocation through membrane transporters: a meeting ground for physics, chemistry, and biology. J Biol Phys 47:401–433

    Article  CAS  PubMed  Google Scholar 

  • Nath S, Jain S (2000) Breakthroughs and views: kinetic modeling of ATP synthesis by ATP synthase and its mechanistic implications. Biochem Biophys Res Commun 272:629–633

    Article  CAS  PubMed  Google Scholar 

  • Nath S, Villadsen J (2015) Oxidative phosphorylation revisited. Biotechnol Bioeng 112:429–437

    Article  CAS  PubMed  Google Scholar 

  • Nath S, Rohatgi H, Saha A (1999) The torsional mechanism of energy transfer in ATP synthase. Curr Sci 77:167–169

    CAS  Google Scholar 

  • Newman M (2018) Networks, 2nd edn. Oxford University Press, Oxford

    Book  Google Scholar 

  • Ore O (1960) Note on Hamilton circuits. Amer Math Mon 67:55

    Article  Google Scholar 

  • Painter DT, Daniels BC, Laubichler MD (2021) Innovations are disproportionately likely in the periphery of a scientific network. Theory Biosci 140:391–399

    Article  PubMed  Google Scholar 

  • Pasquini M, Angeli D (2021) On convergence for hybrid models of gene regulatory networks under polytopic uncertainties: a Lyapunov approach. J Math Biol 83:64

    Article  PubMed  PubMed Central  Google Scholar 

  • Pellissier-Tanon A, Morgado G, Jullien L, Lemarchand A (2021) Quasi–steady–state and partial–equilibrium approximations in chemical kinetics: one stage beyond the elimination of a fast variable. ChemRxiv. https://doi.org/10.26434/chemrxiv.14198768.v2

    Article  Google Scholar 

  • Pinke G, Zhou L, Sazanov LA (2020) Cryo-EM structure of the entire mammalian F-type ATP synthase. Nat Struct Mol Biol 27:1077–1085

    Article  CAS  PubMed  Google Scholar 

  • Roach TNF, Salamon P, Nulton J, Andresen B, Felts B, Haas A, Calhoun S, Robinett N, Rohwer F (2018) Application of finite-time and control thermodynamics to biological processes at multiple scales. J Non-Equilib Thermodyn 43:93–210

    Article  CAS  Google Scholar 

  • Rohatgi H, Saha A, Nath S (1998) Mechanism of ATP synthesis by protonmotive force. Curr Sci 75:716–718

    CAS  Google Scholar 

  • Rubinow SI (1975) Introduction to mathematical biology. Wiley, New York

    Google Scholar 

  • Saa A, Siqueira KM (2013) Modeling the ATP production in mitochondria. Bull Math Biol 75:1636–1651

    Article  CAS  PubMed  Google Scholar 

  • Schadt EE (2009) Molecular networks as sensors and drivers of common human diseases. Nature 461:218–223

    Article  CAS  PubMed  Google Scholar 

  • Segel LA (ed) (1980) Mathematical models in molecular and cellular biology. Cambridge University Press, Cambridge

    Google Scholar 

  • Slater EC (1953) Mechanism of phosphorylation in the respiratory chain. Nature 172:975–978

    Article  CAS  PubMed  Google Scholar 

  • Slater EC (1987) The mechanism of the conservation of energy of biological oxidations. Eur J Biochem 166:489–504

    Article  CAS  PubMed  Google Scholar 

  • Sporns O (2018) Graph theory methods: applications in brain networks. Dialog Clin Neurosci 20:111–120

    Article  Google Scholar 

  • Temkin ON, Zeigarnik AV, Bonchev D (1996) Chemical reaction networks: a graph-theoretical approach. CRC Press, Boca Raton

    Google Scholar 

  • Vigoreaux JO (2006) Nature’s versatile engines: insect flight muscle inside and out. Springer Science, New York

    Google Scholar 

  • Villadsen J, Nielsen J, Lidén G (2011) Bioreaction engineering principles, 3rd edn. Springer, New York (Chapter 4)

    Book  Google Scholar 

  • West DB (2001) Introduction to graph theory, 2nd edn. Pearson Education, Singapore

    Google Scholar 

  • Williams RJP (1979) Some unrealistic assumptions in the theory of chemi-osmosis and their consequences. FEBS Lett 102:126–132

    Article  CAS  PubMed  Google Scholar 

  • Williams RJP (2011) Chemical advances in evolution by and changes in use of space during time. J Theor Biol 268:146–159

    Article  CAS  PubMed  Google Scholar 

  • Witkov C (2018) Bond graphs: graphical cross-disciplinary systems modeling based on power and energy relations. GRC Physics Research and Education, Novel Research in Energy Topics, Smithfield

  • Wray V (2015) Commentary on “oxidative phosphorylation revisited.” Biotechnol Bioeng 112:1984–1985

    Article  CAS  PubMed  Google Scholar 

  • Xu Y, Asadi-Zeydabadi M, Tagg R, Shindell O (2021) Universality in kinetic models of circadian rhythms in Arabidopsis thaliana. J Math Biol 83:51

    Article  PubMed  Google Scholar 

  • Yamagishi JF, Hatakeyama TS (2021) Microeconomics of metabolism: the Warburg effect as Giffen behaviour. Bull Math Biol 83:120

    Article  PubMed  PubMed Central  Google Scholar 

  • Zanin M et al (2020) Mitochondria interaction networks show altered topological patterns in Parkinson’s disease. Syst Biol Appl 38:1–12

    Google Scholar 

  • Zhu J, Zhang Y-T, Alber MS, Newman SA (2010) Bare bones pattern formation: a core regulatory network in varying geometries reproduces major features of vertebrate limb development and evolution. PLoS ONE 5:e10892

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Żurawik TM et al (2016) Revisiting mitochondrial pH with an improved algorithm for calibration of the ratiometric 5(6)-carboxy-SNARF-1 probe reveals anticooperative reaction with H+ ions and warrants further studies of organellar pH. PLoS ONE 11:e0161353

    Article  PubMed  PubMed Central  CAS  Google Scholar 

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  1. Department of Biochemical Engineering and Biotechnology, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Sunil Nath

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Nath, S. Network representation and analysis of energy coupling mechanisms in cellular metabolism by a graph-theoretical approach. Theory Biosci. 141, 249–260 (2022). https://doi.org/10.1007/s12064-022-00370-0

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