THE STOCHASTIC DYNAMICS OF BIOCHEMICAL SYSTEMS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2013 By Joseph Daniel Challenger School of Physics and Astronomy Contents Abstract 8 Declaration 9 Copyright 10 Acknowledgements 11 Publications 12 1 Introduction 13 1.1 Stochastic effects in biochemical systems . 15 1.2 The role of noise in oscillatory behaviour . 17 1.3 Theoretical and numerical techniques . 18 1.4 Our approach . 19 1.5 Introducing COPASI . 20 1.6 Outlook . 22 2 Background 23 2.1 The master equation . 23 2.1.1 An example . 26 2.2 Deterministic dynamics . 27 2.3 Simulating the master equation . 28 2.3.1 The Gillespie algorithm . 29 2.3.2 Other simulation algorithms . 33 2.4 The van Kampen expansion . 34 2.5 The Fokker-Planck equation . 37 2.5.1 Solving the Lyapunov equation . 39 2.6 The Langevin equation . 41 2 2.7 Stochastic oscillations & power spectra . 43 2.8 Summary . 46 3 Fluctuation Analysis and COPASI 50 3.1 Using COPASI . 50 3.2 Formalism . 53 3.3 Details of the implementation . 59 3.4 Conservation relations . 60 3.5 The linear noise approximation in COPASI . 63 3.6 A MAPK signalling system . 66 3.7 Summary . 69 4 Multi-compartment LNA 73 4.1 A two-compartment system . 74 4.2 The general case . 77 4.3 Relation between A^ and A~ ...................... 81 4.4 A model of metabolism in cardiac muscle . 83 4.5 A yeast glycolysis model . 86 4.6 Discussion . 91 5 Synchronisation of stochastic oscillators 92 5.1 Collective behaviour in Dictyostelium ................ 99 5.2 Synchronisation of glycolytic oscillations . 106 5.3 Discussion . 110 6 Conclusions 112 Bibliography 118 A Ill-Conditioned Systems 128 B Preliminary Code 133 C Form of the complex coherence function 140 Word Count: 28,981 3 4 List of Tables 3.1 The covariance matrix for the Michaelis-Menten reaction system . 65 3.2 Reaction scheme for the Kholodenko MAPK signalling model . 67 3.3 Covariance matrix for the MAPK signalling model . 70 3.4 A comparison of the optimisation algorithms in COPASI . 71 4.1 The three-compartment reaction system . 80 4.2 Covariances of the fluctuations for the three-compartment system 81 4.3 The creatine kinase model at the steady-state . 85 4.4 Covariances for the creatine kinase model . 86 4.5 Reaction scheme for the yeast glycolysis model . 88 4.6 Covariances for the yeast glycolysis model . 90 5 List of Figures 1.1 The user interface for the software package COPASI. 21 2.1 Flow chart illustrating the Gillespie algorithm. 31 2.2 Output from the Gillespie algorithm . 32 2.3 A comparison of the stochastic (dark blue) and deterministic (light blue) dynamics . 32 2.4 Time series of stochastic oscillations . 47 2.5 Power spectra for species Y1, Y2 and Y3................ 48 3.1 List of chemical reactions, as displayed in COPASI . 52 3.2 The COPASI display for the LNA task. 64 3.3 Reaction Scheme for the Kholodenko model . 67 3.4 Oscillations observed in MAPKK . 68 3.5 The variance of MKKK fluctuations around the steady state . 70 3.6 Two-dimensional parameter scan of the covariance of the fluctua- tions of MKKK and MKK-P . 71 4.1 The proposed geometry of the two-compartment model . 75 4.2 Reaction Scheme of the creatine kinase model . 84 4.3 Reaction Scheme of the yeast glycolysis model . 87 4.4 Stochastic oscillations in the yeast glycolysis model . 89 5.1 Absolute value of the CCF for species Y and Z . 97 5.2 Parametric plot of the CCF for species Y and Z . 98 5.3 Plot of the phase spectrum for species Y and Z . 98 5.4 Theoretical power spectra for the internal (larger peak) and exter- nal cAMP ............................... 100 5.5 Magnitude for the (theoretical) CCF for the internal and external cAMP fluctuations. 101 6 5.6 Parametric plot of the (theoretical) CCF for the internal and ex- ternal cAMP fluctuations . 101 5.7 Diagram from Kim et. al. illustrating the three-cell model . 102 5.8 Time series for cAMP i for a three-cell model . 103 5.9 Theoretical power spectra for cAMP i in each cell of the two-cell model . 103 5.10 The magnitude of the CCF for the cAMP i in each cell of the two cell model . 104 5.11 Parametric plot of the CCF for the cAMP i in each cell of the two-cell model . 104 5.12 Phase lag for oscillations of cAMP i in each cell for the two-cell model . 105 5.13 Results for a two-cell model, where the cell volumes are not identical107 5.14 Results for a two-cell model, where the cell volumes are not identical108 5.15 Phase spectrum for the cAMP fluctuations in a two-cell model for non-equal cell volumes . 109 5.16 Results for oscillations in species A3 ................. 110 5.17 Results for oscillations in species A3 for increased coupling strength 111 7 Abstract The stochastic dynamics of biochemical systems Joseph Daniel Challenger A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy, 2013 The topic of this thesis is the stochastic dynamics of biochemical reaction systems. The importance of the intrinsic fluctuations in these systems has become more widely appreciated in recent years, and should be accounted for when modelling such systems mathematically. These models are described as continuous time Markov processes and their dynamics defined by a master equation. Analytical progress is made possible by the use of the van Kampen system-size expansion, which splits the dynamics into a macroscopic component plus stochastic correc- tions, statistics for which can then be obtained. In the first part of this thesis, the terms obtained from the expansion are written down for an arbitrary model, enabling the expansion procedure to be automated and implemented in the soft- ware package COPASI. This means that the fluctuation analysis may be used in tandem with other tools in COPASI, in particular parameter scanning and optimisation. This scheme is then extended so that models involving multiple compartments (e.g. cells) may be studied. This increases the range of models that can be evaluated in this fashion. The second part of this thesis also concerns these multi-compartment models, and examines how oscillations can synchronise across a population of cells. This has been observed in many biochemical processes, such as yeast glycolysis. How- ever, the vast majority of modelling of such systems has used the deterministic framework, which ignores the effect of fluctuations. It is now widely known that the type of models studied here can exhibit sustained temporal oscillations when formulated stochastically, despite no such oscillations being found in the deter- ministic version of the model. Using the van Kampen expansion as a starting point, multi-cell models are studied, to see how stochastic oscillations in one cell may influence, and be influenced by, oscillations in neighbouring cells. Analytical expressions are found, indicating whether or not the oscillations will synchronise across multiple cells and, if synchronisation does occur, whether the oscillations synchronise in phase, or with a phase lag. 8 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 9 Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, De- signs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproduc- tions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/ DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University's policy on presentation of Theses 10 Acknowledgements First of all, I would like to thank my supervisor Alan McKane, for giving me the opportunity to work with him for the last three years. Alan's knowledge of the field and his intuitive ability to identify and refine research ideas have been extremely useful, and working with him has been a very rewarding experience. I would also like to thank Tobias Galla for his support, and for encouraging me to present my work wherever possible.