STUDIES ON THE CONTROL OF TIME-DEPENDENT METABOLIC PROCESSES Luis Acerenza Ph.D University of Edinburgh May, 1991 1 Abstract Sensitivity analysis studies how changes in the parameters affect the systems variables. Its application to metabolic systems Metabolic Control Analysts, MCA ) was traditionally developed under certain assumptions: i ) the steady state is stable ( the effect on the steady state values only is studied ). ii ) each reaction is catalyzed by one enzyme, the rates being proportional to the corresponding enzyme concentration. iii ) the parameters are changed by a small ( strictly speaking Infinitesimal ) amount. In the present work MCA is extended to deal with the instantaneous values of time-dependent metabolite concentrations and fluxes. Their summation and connectivity relationships are derived. In some cases it is more convenient to characterize the time courses by time-invariant variables ( such as period and amplitude in oscillating systems ). Summation relationships for time-invariant variables are also derived. Stability analysis shows that a linear chain of four enzyme- catalized reactions, where the third metabolite is a negative effector of the first enzyme constitutes a minimal' oscillator. The model is used to gain insight in the control of oscillations. The control exerted by enzyme concentrations and other parameters that are not proportional to the rate is appropriately described by parameter-unspecified coefficients ( C, ). A proof of the theorems of steady-state MCA in terms of C. is given. By a similar procedure an attempt is made to derive the theorems in terms of C v for time-dependent systems, which is only successful for the particular case of constant ,r-matrix. The effect that a simultaneous change in all the enzyme concentrations by the same factor oe ( Coordinate-Control Operation. CCO ) has on the variables of time-dependent metabolic systems is investigated. This factor a can have any arbitrary large value. The metabolic variables are classified according to the relationships they fulfil when the CCO is applied. A method is given to test these relationships In experimental systems and quantify deviations from the predicted behaviour. 11 Declaration This thesis was composed by myself and describes my own work except where otherwise stated in the Acknowledgements or in the text. Luis Acerenza 111 Acknowledgements I would like to take this opportunity to thank two scientists to whom I am indebted: Henrik Kacser and Eduardo Mizraji. I wish to thank Henrik Kacser for many enlightening discussions during my stay in Edinburgh. Whether the topic was metabolic control, genetics, how to communicate science, 'proper' english or many other aspects of culture. I learned these in a very enjoyable way. I also wish to thank Eduardo Mizraji for his tutorial guide during my previous years of research in Montevideo. It was in this period when my simultaneous interests in quantitative theories, biological phenomena and scientific inquiry met. I can only hope that the following pages reflect some of the many things I have learnt from them. I would also like to thank William G. Hill for his sound advice. Many thanks are due to Herbert Sauro for fruitful discussions and, in particular, for his help to use and implement new features in his computer package SCAMP. Thanks are also due to Henrik Kacser. Herbert Sauro, Rankin Small, Sheila Carmichael and Peter Keightley for reading and comments on previous versions of this thesis. Last but not least I am grateful to my wife Alicia and my son Nicols for their patience when hours due to family were given to research. I received financial support from the Commission of the European Communities, the University of Edinburgh and the Universidad de la Republica ( Montevideo ) for which I am grateful. iv Table of Contents Abstract j Declaration...........................................................................................ii Acknowledgements..............................................................................iii Tableof contents ...............................................................................iv Abbreviations.......................................................................................vii Chapter 1: General Introduction 1.1 Some thoughts concerning the status of Biology today 1 1.2 Historical background ............................................................5 1.3 Methodological Background ...................................................9 1.3.1 Thermodynamics and kinetics ......................................9 1.3.2 Differential equations and stability ............................13 1.3.3 Minimal and mimical models .....................................16 1.3.4 Sensitivity analysis ........................................................21 1.4 Technical Background to MCA of the steady state ...........28 1.4.1 The isolated enzyme-catalized reaction ......................29 1.4.2 Bienzyme system ...........................................................31 1.4.3 Multienzyme system .....................................................39 1.4.4 Transition time ..............................................................42 Chapter 2 : Control analysis of time-dependent metabolic variables. 2.1 Introduction ............................................................................44 2.2 The system .............................................................................45 V 2.3 The change in time scale ...................................................... 46 2.4 Control, elasticity and time coefficients ............................. 49 2.5 Summation relationships ........................................................ 53 2.6 The invariance of rates .......................................................... 57 2.7 Connectivity relationships ...................................................... 59 2.8 Summation and connectivity relationships ( matrix equations ) .............................................................................. 62 2.8.1 Preliminary equations .................................................... 62 2.8.2 Summation relationships .............................................. 64 2.8.3 Connectivity relationships ............................................ 67 2 .9 Example ................................................................................... 68 2 .10 Discussion ................................................................................ 76 Chapter 3 : Enzyme-enzyme Interactions and parameter-unspecified control coefficients 3.1 Introduction .............................................................................83 3.2 Parameter-unspecified control coefficients: the steady state.........................................................................................84 3.3 Parameter-unspecified control coefficients: time- dependentsystems .................................................................89 Chapter 4 : On the consequences of large changes: the Co-ordinate Control Operation 4 .1 Introduction ............................................................................94 4.2 Parameters and variables .......................................................94 4.3 The Co-ordinate Control Operation .....................................96 4.4 Assumptions ............................................................................97 4.5 Co-ordinate Control of time-dependent variables ..............99 vi 4.6 Co-ordinate Control of time-invariant variables ................ 101 4.6.1 with dimension of time ................................................ 101 4.6.2 of a transition to a stable steady state ................... • 101 4.6.3 of sustained oscillations .............................................. 102 4.7 Classification of the variables .............................................. 105 4.8 Test of the general relationships ......................................... 107 4.8.1 Time-invariant variables ................................................ 107 4.8.2 Time-dependent variables ............................................. 108 4.9 Breakdown of the assumptions ............................................. 111 4.10 Quantification of the deviations .......................................... 113 4.10.1 RPS-plot ........................................................................ 114 4 .10.2 PPS-plot ........................................................................ 114 4.11 Example .................................................................................. 116 4 .12 Discussion ............................................................................... 120 4.13 Appendix: Relationship of the CCO to control analysis: summation relationships ........................................................ 128 Chapter 5 : A minimal oscillator with negative feedback 5.1 Introduction .............................................................................132 5.2 The oscillator ..........................................................................133 5.3 The control of period ............................................................135 5.4 Discussion ................................................................................137 5.5 Appendix: The steady-state metabolite concentrations 139 Chapter 6 : Final remarks . ................................................................ 142 References............................................................................................145 Published papers vii Abbreviations CCO :