J. theor. Biol. (1975) 53, 51-65
Kinetics and Thermodynamics of Isothermal Seed Germination G. T. HAGESETH
AND
R. D.
JOYNER
Physics Department, The University of North Carolina, Greensboro, North Carolina 27412, U.S.A. (Received 9 July 1973, and in revisedform
14 February
1974)
We have been successful in building a mathematical model that fits both the germination rate and the total number of seeds that germinate as a function of time. This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes play an important role. The model gives values for the initial concentration of two enzymes. From these initial enzyme concentrations an equilibrium constant is calculated and the thermodynamic model gives the change in enthalpy, entropy, free energy and the activation energy. A plot of the natural logarithm of the equilibrium constant as a function of the reciprocal of the absolute temperature gives two straight lines. The change of enthalpy for the process below 33°C differs considerably to the change above 33°C. The free energy as a function of the absolute temperature gives a straight line from which the change in entropy is calculated. The activation energy is determined from the slope of the natural logarithm of the rate constant as a function of the reciprocal of the absolute temperature.
1. Introduction Literature on seed germination reveals only a few attempts at fitting a mathematical model to the process of seed germination. A method of recording germination curves by fitting the linear and two non-linear parameters of a normal function to germination data for Veronica arvensis L. has been developed (Janssen, 1973). Other attempts (Nichols & Heydecker, 1968; Thompson, 1970) have been made to relate the germination count as a function of time. We have observed the number of turnip seeds that germinate per hour and also the total number of seeds that germinate as a function of time. All these experiments were performed at constant temperature. We have built a mathematical model that fits both the differential germination rate and the total number of seeds germinated as a function of time (Hageseth, 1974). This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes 51
52
G.
T.
HAGESETH
AND
R.
D.
JOYNER
play an important role. The model is successful in describing the germination of turnip seeds throughout the temperature range 18-4o”C. It gives values for the initial concentration of two enzymes. From these proposed initial enzyme concentrations we calculate an equilibrium constant. Since the equilibrium constant is known as a function of temperature we can calculate the changes in enthalpy, entropy and free energy. The laws of thermodynamics predict that the natural logarithm of the equilibrium constant plotted as a function of the reciprocal of the absolute temperature and the change of free energy as a function of temperature should be straight lines. This condition is satisfied as indicated by the data and thus supports the thermodynamic model. These experiments were highly reproducible for a given set of conditions. As a result we repeated all of the experiments but this time added a 100 dB, 4000 Hz sound field to see if it had any effect. The 4000 Hz sound affected the reaction rates and also the enzyme concentration, but the thermodynamic parameters of the process below 31°C are the same even if the kinetic parameters are not. The 4000 Hz sound affected a change in the thermodynamic parameters for the process that occurs above 31°C. 2. Theoretical (A)
THE
KINETICS
OF
Model
SEED
GERMINATION
The autocatalytic reaction describes many biochemical reactions in which enzymes play an important role. The reaction is summarized below and is available in most chemical kinetic textbooks (Stevens, 1970). The main concept of autocatalysis is that the products of the reaction catalyse the reaction. If the first reaction is: A+B+AB (1) and the second reaction is: AB e F+B. (2) A and B are substrates that go together to form the complex AB. The complex AB reacts further to form the final enzyme product F. The autocatalytic reaction is given by: AB+F-+2F+B. (3) The net result of the series of enzyme reactions is to transform enzyme A into enzyme F. If the reactions described in equations (1) and (2) are consecutive then the initial rate is:
dCF1 ~ = dt
k,[A]
(4)
SEED
KINETICS
AND
THERMODYNAMICS
53
where [F] is the concentration of F and where [A] is the concentration of A and k, is the rate constant. If the reactions described in equations (1) and (3) are consecutive then the rate becomes
adt = MCAlo - CFl>
Kinetics and Thermodynamics of Isothermal Seed Germination G. T. HAGESETH
AND
R. D.
JOYNER
Physics Department, The University of North Carolina, Greensboro, North Carolina 27412, U.S.A. (Received 9 July 1973, and in revisedform
14 February
1974)
We have been successful in building a mathematical model that fits both the germination rate and the total number of seeds that germinate as a function of time. This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes play an important role. The model gives values for the initial concentration of two enzymes. From these initial enzyme concentrations an equilibrium constant is calculated and the thermodynamic model gives the change in enthalpy, entropy, free energy and the activation energy. A plot of the natural logarithm of the equilibrium constant as a function of the reciprocal of the absolute temperature gives two straight lines. The change of enthalpy for the process below 33°C differs considerably to the change above 33°C. The free energy as a function of the absolute temperature gives a straight line from which the change in entropy is calculated. The activation energy is determined from the slope of the natural logarithm of the rate constant as a function of the reciprocal of the absolute temperature.
1. Introduction Literature on seed germination reveals only a few attempts at fitting a mathematical model to the process of seed germination. A method of recording germination curves by fitting the linear and two non-linear parameters of a normal function to germination data for Veronica arvensis L. has been developed (Janssen, 1973). Other attempts (Nichols & Heydecker, 1968; Thompson, 1970) have been made to relate the germination count as a function of time. We have observed the number of turnip seeds that germinate per hour and also the total number of seeds that germinate as a function of time. All these experiments were performed at constant temperature. We have built a mathematical model that fits both the differential germination rate and the total number of seeds germinated as a function of time (Hageseth, 1974). This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes 51
52
G.
T.
HAGESETH
AND
R.
D.
JOYNER
play an important role. The model is successful in describing the germination of turnip seeds throughout the temperature range 18-4o”C. It gives values for the initial concentration of two enzymes. From these proposed initial enzyme concentrations we calculate an equilibrium constant. Since the equilibrium constant is known as a function of temperature we can calculate the changes in enthalpy, entropy and free energy. The laws of thermodynamics predict that the natural logarithm of the equilibrium constant plotted as a function of the reciprocal of the absolute temperature and the change of free energy as a function of temperature should be straight lines. This condition is satisfied as indicated by the data and thus supports the thermodynamic model. These experiments were highly reproducible for a given set of conditions. As a result we repeated all of the experiments but this time added a 100 dB, 4000 Hz sound field to see if it had any effect. The 4000 Hz sound affected the reaction rates and also the enzyme concentration, but the thermodynamic parameters of the process below 31°C are the same even if the kinetic parameters are not. The 4000 Hz sound affected a change in the thermodynamic parameters for the process that occurs above 31°C. 2. Theoretical (A)
THE
KINETICS
OF
Model
SEED
GERMINATION
The autocatalytic reaction describes many biochemical reactions in which enzymes play an important role. The reaction is summarized below and is available in most chemical kinetic textbooks (Stevens, 1970). The main concept of autocatalysis is that the products of the reaction catalyse the reaction. If the first reaction is: A+B+AB (1) and the second reaction is: AB e F+B. (2) A and B are substrates that go together to form the complex AB. The complex AB reacts further to form the final enzyme product F. The autocatalytic reaction is given by: AB+F-+2F+B. (3) The net result of the series of enzyme reactions is to transform enzyme A into enzyme F. If the reactions described in equations (1) and (2) are consecutive then the initial rate is:
dCF1 ~ = dt
k,[A]
(4)
SEED
KINETICS
AND
THERMODYNAMICS
53
where [F] is the concentration of F and where [A] is the concentration of A and k, is the rate constant. If the reactions described in equations (1) and (3) are consecutive then the rate becomes
adt = MCAlo - CFl>
