Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam What the egg can tell about its hen Rennes, 2007/12/18

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 Rennes, 2007/12/18 What the egg can tell about its hen

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_g get_pars_h get_pars_i iget_pars_g iget_pars_h iget_pars_iget_pars_gget_pars_hget_pars_iiget_pars_giget_pars_higet_pars_i growth & reprod get_pars_r get_pars_s get_pars_t iget_pars_r iget_pars_s iget_pars_tget_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.elas_pars_gelas_pars_r Function get_pars_u converts compound parametersget_pars_u into unscaled primary parameters at abundant food. Theory in KooySous2008

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

Maximum age at birth

Respiration ontogeny in birds age, d ml CO 2 d -1 ml O 2 d -1 altricial Troglodytes aëdon precocial Gallus domesticus 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 O 2 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

Embryonic development time, d weight, g O 2 consumption, ml/h ;  : scaled time l : scaled length e: scaled reserve density g: energy investment ratio Crocodylus johnstoni, Data from Whitehead 1987 yolk embryo

Foetal development weight, g time, d Mus musculus 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 Data: MacDowell et al 1927

Maintenance ratio

Follows from: 1.maturity at birth equals a given value 2.reserve density at birth equals that of mother State variables: Parameters: Problem: Given parameter values, find Initial reserve of an egg Theory in Kooy2008

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 = [ ]; initial_scaled_reserve(1,p_Dm). Theory in Kooy2008

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.get_lb_foetus 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

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.get_lb Output scalar with scaled age at birth: taub = ab kM An example of use is given in mydata_ue0 Theory in Kooy2008

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_foetusget_tb_foetus optional scalar with scaled age at birth. Default calls get_tb_foetus but then the input parameter should have 3 elements.get_tb_foetus Output scalar with scaled length at birth: lb = Lb/Lm An example of use is given in mydata_ue0_foetus Theory in Kooy2008

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

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

Reduction of initial reserve scaled age scaled maturity scaled struct volume scaled reserve

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

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

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

DEB tele course 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 Cambridge Univ Press 2000 Audience : thank you for your attention Organizers : thank you for the invitation