What the egg can tell about its hen

1. What the egg can tell about its hen Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Bas@bio.vu.nl http://www.bio.vu.nl/thb/ Rennes, 2007/12/18

2. What the egg can tell about its hen • Contents: • Egg development • Maintenance ratio < 1 • Reductions of initial reserve • Body size scaling implications • Hormesis effects of toxicants Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Bas@bio.vu.nl http://www.bio.vu.nl/thb/ Rennes, 2007/12/18

3. DEBtool/animal/get_pars Functions get_pars_* obtain compound DEB parameters from easy-to-observe quantities and the functions iget_pars_* do the reverse, which can be used for checking. The routines are organized as follows: get_pars iget_pars food level one several one several Constraint kJ = kM kJ != kM kJ = kM kJ = kM kJ != kM kJ = kM growth get_pars_gget_pars_hget_pars_iiget_pars_giget_pars_higet_pars_i growth & reprod get_pars_rget_pars_sget_pars_tiget_pars_riget_pars_siget_pars_t Functions for several food levels do not use age at birth data. If one food level is available, we have to make use of the assumption of stage transitions at fixed amounts of structure (k_M = k_J). If several food levels are available, we no longer need to make this assumption, but it does simplify matters considerably. Functions elas_pars_g and elas_pars_r give elasticity coefficients. Function get_pars_u converts compound parameters into unscaled primary parameters at abundant food. Theory in KooySous2008

4. DEBtool/animal/get_pars g r h get_pars_ s u g r iget_pars_  h s red quantities depend on food level, green do not Theory in KooySous2008

5. Maximum age at birth

6. Respiration ontogeny in birds 3.7.1 ml O2 d-1 ml CO2 d-1 altricial Troglodytes aëdon precocial Gallus domesticus age, d age, d • Observations: just prior to hatching • respiration shows a plateau in precocial, not in altricial birds • pore size and frequency in egg shell is such that O2 flux has constant resistance • Conclusion: ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990) • DEB theory: reserve dynamics controls ontogeny (same pattern in species without shells) • Minimization of water loss causes observed constant flux resistance

7. Embryonic development 3.7.1 Crocodylus johnstoni, Data from Whitehead 1987 embryo yolk O2 consumption, ml/h weight, g time, d time, d : scaled time l : scaled length e: scaled reserve density g: energy investment ratio ;

8. Foetal development 3.7.1 Foetes develop like eggs, but rate not restricted by reserve (because supply during development) Reserve of embryo “added” at birth Initiation of development can be delayed by implantation egg cell Nutritional condition of mother only affects foetus in extreme situations weight, g Mus musculus time, d Data: MacDowell et al 1927

9. Maintenance ratio

10. Initial reserve of an egg • Follows from: • maturity at birth equals a given value • reserve density at birth equals that of mother • State variables: • Parameters: • Problem: Given parameter values, find Theory in Kooy2008

11. DEBtool/animal/initial_scaled_reserve • The routine calculates the initial scaled reserve mass UE0 = ME0/ {JEAm}. • The constraint [UEb] = f [UEm] applies. • Inputs: • n-vector with scaled functional response • 5-vector with parameters • VHb, d.mm^2, scaled maturity at birth: M_H^b/ ((1 - kap) {J_EAm}) with kap is fraction allocated to soma • g, -, energy investment ratio • kJ, 1/d, maturity maintenance rate coefficient • kM, 1/d, somatic maintenance rate coefficient • v, mm/d, energy conductance • optional scalar or n-vector with initial estimates for Lb • Outputs: • n-vector with initial scaled reserve: M_E^0/ {J_EAm} • n-vector with length at birth Lb • n-vector with indicators for success (1) or failure (0) • Example of use (for Daphnia magna at 20 C): • p_Dm = [.8 .42 1.7 1.7 3.24 .012]; initial_scaled_reserve(1,p_Dm). Theory in Kooy2008

12. DEBtool/animal/get_lb • Obtains scaled length at birth, given the scaled reserve density at birth. • A Newton Raphson scheme is used with Euler integration, starting from an optional initial value. • The default initial value is the exact one for maintenance ratio 1. • Consider the application of get_lb_foetus for an alternative initial value. • Comparable functions: • get_lb1 uses a Newton Raphson scheme with advanced integration (but is rather slow), • get_lb2 uses a shooting method (in one variable; and is faster than get_lb1). • Inputs • 3-vector with parameters • g: energy investment ratio • k: maintenance ratio kJ/ kM • vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma • optional scalar with scaled reserve density at birth (default 1) • optional scalar with initial value for scaled length at birth • Outputs • scalar with scaled length at birth: lb = Lb/ Lm • indicator for success (1) or failure (0) • An example of use is given in mydata_ue0 Theory in Kooy2008

13. DEBtool/animal/get_tb • Obtains scaled age at birth, given the scaled reserve density at birth. • Multiply the result with the somatic maintenance rate coefficient to arrive at age at birth. • Inputs • 1- (if third input is specified) or 3 -vector with parameters • g: energy investment ratio • k: maintenance ratio kJ/ kM • vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma • optional scalar with scaled reserve density at birth (default 1) • optional scalar with scaled length at birth. • Default calls get_lb but then the first input should have 3 rather than 1 elements. • Output • scalar with scaled age at birth: taub = ab kM • An example of use is given in mydata_ue0 Theory in Kooy2008

14. DEBtool/animal/get_lb_foetus Obtains the scaled length at birth of a foetus, which is not restricted by reserve availability. Inputs 1 or 3-vector with energy investment ratio g, see get_tb_foetus optional scalar with scaled age at birth. Default calls get_tb_foetus but then the input parameter should have 3 elements. Output scalar with scaled length at birth: lb = Lb/Lm An example of use is given in mydata_ue0_foetus Theory in Kooy2008

15. DEBtool/animal/get_tb_foetus • Obtains scaled age at birth, given the scaled reserve density at birth. • Multiply the result with the somatic maintenance rate coefficient to arrive at age at birth. • Inputs • 3-vector with parameters • g: energy investment ratio • k: maintenance ratio kJ/ kM • vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma • optional scalar with initial value for scaled age at birth. • Default exact value for maintenance ratio 1. • Output • scalar with scaled age at birth: taub = ab kM. • indicator for succes (1) of failure (0). • An example of use is given in mydata_ue0_foetus Theory in Kooy2008

16. Effects of nutrition scaled length at birth scaled age at birth scaled res density at birth scaled res density at birth scaled initial reserve scaled res density at birth

17. Reduction of initial reserve scaled maturity 1 scaled reserve 0.8 0.5 scaled age scaled age scaled struct volume scaled age

18. Scaling relationships log scaled initial reserve log scaled age at birth log zoom factor, z log zoom factor, z approximate slope at large zoom factor log scaled length at birth log zoom factor, z

19. indirect effects Effects on reproduction mg L-1 0, 320 assimilation 560 1000 cum # offspring/♀ Phenol on Daphnia magna at 20°C 1800 3200 direct effects maintenance cost/offspring cum # offspring/♀ growth hazard cum # offspring/♀ time, d time, d

20. Metamorphosis The larval malphigian tubes are clearly visible in this emerging cicada They resemble a fractally-branching space-filling tubing system, according to Jim Brown, but judge yourself …. Java, Nov 2007

21. DEB tele course 2009 Cambridge Univ Press 2000 http://www.bio.vu.nl/thb/deb/ Free of financial costs; some 250 h effort investment Program for 2009: Feb/Mar general theory April symposium in Brest (2-3 d) Sept/Oct case studies & applications Target audience: PhD students We encourage participation in groups who organize local meetings weekly Software package DEBtool for Octave/ Matlab freely downloadable Slides of this presentation are downloadable from http://www.bio.vu.nl/thb/users/bas/lectures/ Audience: thank you for your attention Organizers: thank you for the invitation