Modeling More Complex Systems ABE 5646 Week 6, Spring 2010

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Modeling More Complex Systems ABE 5646 Week 6, Spring 2010 Week Description Reading Material 6 Modeling Temperature Effects on Biological Systems Effects on chemical reaction rates Effects on

1 Modeling More Complex Systems ABE 5646 Week 6, Spring 2010 Week Description Reading Material 6 Modeling Temperature Effects on Biological Systems Effects on chemical reaction rates Effects on biological activity, general Effects on developmental processes in plants and other biological organisms Degree-day models: basis for and use of Keen & Spain (1992), Ch. 12 Handout; degree-days

2 Weeks 6 Objectives 1) To review how temperature affects biological and agricultural systems 2) To gain insight into modeling temperature effects on biological and biophysical systems a) Continuous processes b) Discrete events

3 Topics on Temperature Effects Temperature in the environment Heat flow, heat balances Temperature control (physical, biological) Chemical reactions Enzyme activity Biological activity Developmental processes in biological systems Degree-days

4 Temperature in the Environment Affects all biological systems, Thus it is usually an important environment variable that changes over time (exceptions?) It interacts with other environmental variables Measurements of T are normally required May be possible to compute T vs. time for certain model uses or times May interact with other model components and thus may need to be modeled dynamically as a state variable

5 Estimating Temperature of the Environment Annual variation in daily mean temperature T ( t t ) t) = ( Tannual + Tamp sin[2π ] 365 c day (

6 Estimating Temperature of the Environment Temperature variations during a day Parton and Logan A model for diurnal variation in soil and air temperature. Agric. For. Meteorol. 23: Daytime ( t 12 + d / 2) T ( t) = Tmin + ( Tmax Tmin ) sin[ π ] d + 2 p d = daylength, h p = time from noon when maximum temperature occurs, h Night time Tmin Tsunset exp( n / TC ) + ( Tsunset Tmin T ( t) = 1 exp( n / T ) C ) exp( ( t t sunset ) / T C ) n = night length (24-d), h T C = nocturnal time coefficient, about 4 h

7 Temperature during a day

8 Temperature May vary with system dynamics May have to model temperature dynamically

9 Heat Loss and Changes in From Week 3 lectures: Temperature dt ( t) dt Where: T(t) TE k = k[ T ( t) T E ] is temperature of an object, o C is environment temperature, oc is a heat transfer or cooling constant, with units 1/t

10 Cooling Simulation, Ex. 1-6 Temperature of Body Temperature, C Environmental Temperature Time, hours Exact Solution T ( t) = T + ( T (0) T ) e E E kt Numerical solution T(t)= T(t-1)-k*[T(t-1)-T E ] For t > 0.0; initial conditions T(0) exact approximate

11 But, Prior Equations Do Not Show Heat Flow Starting with heat: Q and dq dt = = mc mc p p T dt dt Where ΔQ = change in heat of an object, calories m = mass of the object, g c p = specific heat, a property of the mass, calories g -1 o C -1 T = temperature, o C and heat is the extensive variable that flows from one object to another; temperature is an intensive variable related to the heat content of a body

12 Written differently m c or p dt ( t) dt dt ( t) dt k = m c = k[ T ( t) T p [ T ( t) T E E ] ] with, k = the conduction heat transfer coefficient and now, units of k are (calories o C -1 time -1 )

13 Set Point Temperature Control Heater (on/off) h=0 if T>T set + tol h max if T<T set - tol h max h=heater Heat T set, tol dt ( t) dt = h m c p k m c p [ T ( t) T E ] Heat Loss T T E

Proportional Temperature Control Proportional control (lags in sensing temperature & heater output) h max T c h = h max (T c - T s ) if T s <T c otherwise, h = 0 T s = temperature of sensor T

14 Proportional Temperature Control Proportional control (lags in sensing temperature & heater output) h max T c h = h max (T c - T s ) if T s <T c otherwise, h = 0 T s = temperature of sensor T c = control or desired temperature dt ( t) = dt dts ( t) = dt h m c ks m c s p k m c sp p [ T ( t) T [ T ( t) T s ( t)] E ks ] m c s sp [ T ( t) T s ( t)] T s h=heater Heat Sensor Heat Loss T

15 Implications Simple examples show lags in temperature control Both demonstrate very simple forms of feedback control Biological control systems are similar Feedback control of body temperature Usually, more like proportional control such that internal heat production, for example, varies in response to a biological setpoint or range Lags also exist between a state and output from the biological control system (sensing, metabolism, muscle response, etc.) What are other examples of biological control systems?

16 Temperature and Chemical Reactions Arrhenius equation Q 10 approach Also used directly in some enzymesubstrate reactions

17 Arrhenius Equation Consider the equation: fi Flow S S P where f i = flow rate of S into reaction K(T) = rate constant, dependent on temperature k ds dt = f k( T ) i S

18 Arrhenius Equation Assumes that substrate (S) must reach an activated state (E a ) for the reaction to occur The faster that S reaches the activation state, the faster the reaction rate See Thornley and Johnson (Chapter 5)

19 Arrhenius Equation A = a constant E a = Energy of activation R = gas constant T = Absolute Temperature, 0 K k = ds dt Ea Aexp( ) RT = f k( T ) S i Ea/R typically in range of 5,000 to 15,000 K

20 Q 10 Approximation Q 10 is the factor at which the rate constant, k, increases for a temperature increment of 10 0 K Empirical, based on observations Also 3 parameters to estimate k = k r Q [( T 10 T r )/10]

21 Conceptual Pools of Enzymes From Curry and Feldman, Texas A&M University Press

22 Enzyme Reactions Have optimal temperature range Enzymes assumed to be in inactive state (denatured), initial active state, and activated state (Ea), depending on temperature

23 Enzyme Reaction Rate k( T ) = 1+ Aexp( E / RT a exp( S / R H ) / RT ) ΔH = enthalpy of reaction ΔS = entropy of reaction Others same as in Arrhenius equation

24 Enzyme Reaction Rates vs. Temperature

25 Enzyme Reaction Rates vs. Parameters must be estimated (4) from data [A, Ea/R, ΔH/R, ΔS/R] Temperature

26 Estimation of Enzyme Parameters From Curry and Feldman, Texas A&M University Press

27 Other Equations for Temperature Effects on Biological Systems O Neill et al. (1972) k T = with k T U = T T V = T max max max max T exp( xv ) 2 2 W ( / W ) x = 400 W = ( Q 1)( T T ) 10 U x T T opt T opt opt max opt Keen and Spain, pp193-4

28 Other Equations for Temperature Effects on Biological Systems Logan (1988) Keen and Spain, pp193-4 opt T T T T T T T with k k k = = + = max min max 2 min ) ( ) exp( τ τ τ τ τ

29 Temperature Effects on Enzyme- Mediated Reactions

30 Discussion Arrhenius Q 10 Enzyme, biological reactions Practical implications Parameter estimation Normalization Implementation

31 Temperature and Development Temperature has major influence on developmental processes of plants, insects, other organisms Duration of time between egg and pupae in insects may vary between 8 and 40 days, depending on temperature Similarly, duration of time from planting to emergence, time from emergence to 1 st flower, time needed to complete its lifecycle to maturity, etc. depend strongly on temperature

32 Experiments to Measure Development Typically, temperature treatments, each having a constant temperature Time measured until a particular development stage (duration)

33 Development Rates Computed as Reciprocal of Duration Development rate is the inverse of days required to complete a phase k d (T) = [1/(Dur(T)] Units are (1/time)

34 Modeling Development If temperature varies with time, one can integrate development rate (which is dependent on temperature) until cumulative development is equal to 1.0 The general relationship for computing a development index, h ij is: h ij = t t i j k d dt h ij = index of progress along the development pathway from i to j. The length of the phase is t i t j, also referred to as Developmental time h ij starts at 0.0 at t i and progresses until 1.0 at t j when stage j occurs

35 Modeling Development Another way to write this equation is: dh ij = dt k d Where k d is a function of Temperature (or k d (T), and T may vary with time (then k d =k d (T(t)). This is in the same form as our rate equations, and can be solved using Euler. This will be shown later on, with a Δt=1 How would k d vary with Temperature? How would Temperature vary with time?

36 Modeling Germination From figure 5.12 in Thornley and Johnson, k d = slope*(t-t B ) k d = *(T-276) (approx.) where T units are 0 K and slope units are 1/( 0 C-t) or k d = *(T-3.0) If T is 0 C

37 Modeling Germination h or h ij ij = t t k dt in our example : t i j d j = 0 [ ( T T B )] dt

38 Modeling Germination h which can be approximated via h ij ij = t 0 j [ ( T t T B )] dt j = o [ ( T 3)] t Euler : When the summation (h ij ) = 1.0, germination occurs T B = base temperature, 0 K or 0 C

39 Modeling Germination The can be moved outside the summation sign (it is constant), And if Δt=1, then we can re-write the equation as: hij t j = o ( T ) 1 When h ij = 1, germination occurs, and at this time, the left side of the equation above is (1/.0588)=17.0 Note that units of the left side of the equation are in 0 C-time; if time is in days, units are 0 C-days

40 Modeling Development Using Degree-Days The previous equation can be generalized to show the widely-used degree-day model DD = t o j ( T TB ) Where DD = degree days and the summation is carried out until a Threshold is reached When the threshold is reached (on day t j ), development is complete and the germination (or other event) occurs on that day In the example above, the threshold would be 17.0

41 Degree Days Check units on DD, what are they? What are the assumptions of this method? Can it be further generalized? How can you compute the parameters needed to use DD to model time to first flower, time to germination, time to maturity, time between egg and larvae,???

42 Modeling Development h ij = t t i j k d dt What if k d is not linear? We have seen how enzyme kinetics depend on temperature, and the internal biological reactions that lead to development are enzyme reactions

43 Modeling Development h ij = t t i j k d dt What if k d is not linear? In many models, k d is approximated by a piecewise linear function:

44 Development in Varying Temperature Environments Assumption is that development continues at the temperature it experiences, with no lags, memory effect, etc. Choice of using daily average vs. hourly temperatures will affect results

45 Discussion Temperature in the environment Heat flow, heat balances Temperature control (physical, biological) Chemical reactions Enzyme activity Biological activity Developmental processes in biological systems Degree-days

46 Homework Set 4 ABE 5646 Due on February 19, Work camel body temperature simulation problem 12-3 in Keen & Spain, pp Work problem 12-4 in Keen & Spain on enzyme-catalyzed reactions, p Use the O Neill temperature function with the following parameters: Q 10 = 2.1 T max = 40 T opt = 28 k max = (units of d -1 ) a. Plot rate (k T ) vs. Temperature (T) over the range of 0 to 40 o C. b. Assume that we want to use a linear approximation to this function for temperatures below 30 o C, what are the slope and intercept values? c. Explain how you could use this approximation with a degree-day model. What would the threshold be for development?

47 Questions?

FST 123 Problem Set 3 Spring, V o at [S] 0 = 10 mm (mm/min)

FST 123 Problem Set 3 Spring, V o at [S] 0 = 10 mm (mm/min) FST 23 Problem Set 3 Spring, 2009 Name. A student needed to know the activity versus ph profile for an enzyme that he was going to use in a later application. He collected the appropriate data and determined