BYOM, byom_debtox_daphnia.m, DEBtox with Daphnia

BYOM is a General framework for simulating model systems in terms of ordinary differential equations (ODEs). The model itself needs to be specified in derivatives.m, and call_deri.m may need to be modified to the particular problem as well. The files in the engine directory are needed for fitting and plotting. Results are shown on screen but also saved to a log file (results.out).

The model: DEBtox following Jager & Zimmer (2012), http://dx.doi.org/10.1016/j.ecolmodel.2011.11.012.

This script: The water flea Daphnia magna exposed to fluoranthene; data set from Jager et al (2010), http://dx.doi.org/10.1007/s10646-009-0417-z. Published as case study in Jager & Zimmer (2012). Simultaneous fit on growth, reproduction and survival. Parameter estimates differ slightly from the ones in Jager & Zimmer as we here take the two controls as seperate treatments (with the same parameters).

Contents

Initial things

Make sure that this script is in a directory somewhere below the BYOM folder.

clear, clear global % clear the workspace and globals
global DATA W X0mat % make the data set and initial states global variables
global glo          % allow for global parameters in structure glo
global pri zvd      % global structures for optional priors and zero-variate data
diary off           % turn of the diary function (if it is accidentaly on)
% set(0,'DefaultFigureWindowStyle','docked'); % collect all figure into one window with tab controls
set(0,'DefaultFigureWindowStyle','normal'); % separate figure windows

pathdefine % set path to the BYOM/engine directory
glo.basenm  = mfilename; % remember the filename for THIS file for the plots
glo.saveplt = 0; % save all plots as (1) Matlab figures, (2) JPEG file or (3) PDF (see all_options.txt)

The data set

Data are entered in matrix form, time in rows, scenarios (exposure concentrations) in columns. First column are the exposure times, first row are the concentrations or scenario numbers. The number in the top left of the matrix indicates how to calculate the likelihood:

% scaled internal concentrations
DATA{1} = [0]; % there are never data for this state

% scaled reserve density
DATA{2} = [0]; % there are never data for this state

% body length (mm) on each observation time (d), concentrations in uM
DATA{3} = [1 0 0 0.213	0.426	0.853
    0	0.88	0.88	0.88	0.88	0.88
    2	1.38	1.34	1.4     1.44	1.32
    4	1.96	1.72	1.9     1.88	1.74
    6	2.28	2.38	2.14	2.14	2.08
    8	2.34	2.52	2.56	2.36	2.16
    10	2.64	2.56	2.46	2.46	2.48
    12	2.66	2.56	2.6     2.58	2.7
    14	2.68	2.7     2.76	2.78	2.8
    16	2.88	2.78	2.82	NaN     NaN
    18	2.94	2.84	2.92	3.08	2.9
    20	3.06	3.02	3.02	3.22	2.76
    21	3.26	3.04	3.06	3       2.84];

W{3} = 5 * ones(size(DATA{3})-1); % weights: 5 animals in each treatment

% cumulative reproduction (nr. offspring) (mm) on each observation time (d), concentrations in uM
DATA{4} = [1 0 0 	0.213	0.426	0.853
    0	0.0	0.0	0.0	0.0	0.0
    2	0.0	0.0	0.0	0.0	0.0
    4	0.0	0.0	0.0	0.0	0.0
    6	0.0	0.0	0.0	0.0	0.0
    8	1.2     2.1     0.0     0.0 	0.0
    10	6.5     7.7     10.0	0.8     1.7
    12	17.9	25.4	16.3	2.0     1.7
    14	31.8	29.5	33.3	6.4     1.7
    16	50.6	48.8	56.2	9.4     1.7
    18	61.3	62.5	62.5	13.8	1.7
    20	81.0	76.4	64.1	16.4	1.7
    21	81.0	76.4	64.1	16.4	1.7];

W{4} = [10	10	10	10	10
	10	10	10	10	10
	10	10	10	10	10
	10	10	10	10	10
	10	10	10	10	10
	10	10	10	10	9
	10	10	10	10	7
	10	10	10	10	5
	10	10	10	10	5
	10	9	10	10	3
	10	9	10	9	3
	10	8	10	9	2]; % weights: nr. of mothers alive

% in the next part, I subtract 2.5 days from the time vector for the repro
% data; reason is that the eggs are produced some 2.5 days before the
% offspring are counted.
DATA{4}(2:end,1) = DATA{4}(2:end,1) - 2.5;
DATA{4}([2 3],:) = [];
W{4}([1 2],:) = [];

% survivors on each observation time (d), concentrations in uM
DATA{5} = [-1	0	0	0.213	0.426	0.853
    0	10	10	10	10	10
    2	10	10	10	10	10
    4	10	10	10	10	10
    6	10	10	10	10	10
    8	10	10	10	10	10
    10	10	10	10	10	9
    12	10	10	10	10	7
    14	10	10	10	10	5
    16	10	10	10	10	5
    18	10	9	10	10	3
    20	10	9	10	9	3
    21	10	8	10	9	2];

Initial values for the state variables

Initial states, scenarios in columns, states in rows. First row are the 'names' of all scenarios.

X0mat = [0 0.213 0.426 0.853 % the scenarios (here nominal concentrations)
         0 0   0   0  % initial values state 1
         1 1   1   1  % initial values state 2
         0 0   0   0  % initial values state 3 (overwritten by L0)
         0 0   0   0  % initial values state 4
         1 1   1   1];% initial values state 5

Initial values for the model parameters

Model parameters are part of a 'structure' for easy reference.

% global parameters as part of the structure glo
glo.moa = 4; % mode of action (see derivatives)

% syntax: par.name = [startvalue fit(0/1) minval maxval optional:log/normal scale (0/1)];
par.ke  = [0.039  1 0 10  0]; % elimination rate constant (d-1)
par.c0  = [0.063  1 0 1e6 1]; % no-effect concentration sub-lethal (uM)
par.cT  = [7e-3   1 0 1e6 1]; % tolerance concentration (uM)
par.c0s = [0.15   1 0 1e6 1]; % no-effect concentration survival (uM)
par.b   = [1.1    1 0 1e6 0]; % killing rate (1/(d uM))

par.L0 = [0.88   0 1e-3 100 1]; % initial body length (mm)
par.Lp = [1.6    1 1e-3 100 1]; % length at puberty (mm)
par.Lm = [3.1    1 1e-3 100 1]; % maximum length (mm)
par.rB = [0.14   1 0 100 1];    % von Bertalanffy growth rate (d-1)
par.Rm = [11     1 0 1e6 1];    % maximum reproduction rate (eggs/d)
par.g  = [0.422  0 0 100 1];    % energy-investment ratio (-)
par.f  = [1      0 0 2   1];    % scaled functional response (-)
par.h0 = [0.0024 1 0 1e6 1];    % background hazard rate (d-1)

Time vector and labels for plots

Specify what to plot. If time vector glo.t is not specified, a default is used, based on the data set

% specify the y-axis labels for each state variable
glo.ylab{1} = ['scaled internal concentration (',char(181),'M)'];
glo.ylab{2} = 'scaled reserve density (-)';
glo.ylab{3} = 'body length (mm)';
glo.ylab{4} = 'cumul. repro. (predicted egg formation)';
glo.ylab{5} = 'survival fraction (-)';

% specify the x-axis label (same for all states)
glo.xlab    = 'time (days)';
glo.leglab1 = 'conc. '; % legend label before the 'scenario' number
glo.leglab2 = [char(181),'M']; % legend label after the 'scenario' number

prelim_checks % script to perform some preliminary checks and set things up
% Note: prelim_checks also fills all the options (opt_...) with defauls, so
% modify options after this call, if needed.

Calculations and plotting

Here, the function is called that will do the calculation and the plotting. Options for the plotting can be set using opt_plot (see prelim_checks.m). Options for the optimsation routine can be set using opt_optim. Options for the ODE solver are part of the global glo.

For the demo, the iterations are turned off (opt_optim.it = 0).

glo.eventson     = 1; % events function on (1) or off (0)

opt_optim.fit    = 1; % fit the parameters (1), or don't (0)
opt_optim.it     = 0; % show iterations of the optimisation (1, default) or not (0)
opt_plot.bw      = 0; % if set to 1, plots in black and white with different plot symbols
opt_plot.annot   = 2; % annotations in multiplot for fits: 1) box with parameter estimates 2) single legend
% opt_plot.statsup = [1 2]; % vector with states to suppress in plotting fits

% optimise and plot (fitted parameters in par_out)
par_out = calc_optim(par,opt_optim); % start the optimisation
calc_and_plot(par_out,opt_plot); % calculate model lines and plot them
Provisional goodness-of-fit measures for each data set
       NaN         0    0.9767    0.9850    0.9977

Warning: Goodness of fit measure for survival data is meaningless when more data
sets are fitted simultaneously! 
 
=================================================================================
Results of the parameter estimation with BYOM version 4.0
=================================================================================
   Filename      : byom_debtox_daphnia 
   Analysis date : 28-Apr-2017 (13:59) 
   Data entered  :
     data state 1: 0x0, no data.
     data state 2: 0x0, no data.
     data state 3: 12x5, continuous data, no transf.
     data state 4: 10x5, continuous data, no transf.
     data state 5: 12x5, survival data.
   Search method: Nelder-Mead simplex direct search, 2 rounds. 
     The optimisation routine has converged to a solution
     Total 206 simplex iterations used to optimise. 
     Minus log-likelihood has reached the value 439.698 (AIC=899.397). 
=================================================================================
ke        0.04003 (fit: 1, initial: 0.039) 
c0        0.06318 (fit: 1, initial: 0.063) 
cT        0.00732 (fit: 1, initial: 0.007) 
c0s        0.1492 (fit: 1, initial: 0.15) 
b           1.104 (fit: 1, initial: 1.1) 
L0           0.88 (fit: 0, initial: 0.88) 
Lp          1.629 (fit: 1, initial: 1.6) 
Lm          3.107 (fit: 1, initial: 3.1) 
rB         0.1387 (fit: 1, initial: 0.14) 
Rm          10.81 (fit: 1, initial: 11) 
g           0.422 (fit: 0, initial: 0.422) 
f               1 (fit: 0, initial:  1) 
h0       0.002448 (fit: 1, initial: 0.0024) 
=================================================================================
Parameters ke and b are fitted on log-scale.
=================================================================================
Time required: 16.4 secs
Plots result from the optimised parameter values. 

Profiling the likelihood

By profiling you make robust confidence intervals for one or more of your parameters. Use the name of the parameter as it occurs in your parameter structure par above. You do not need to run the entire script before you can make a profile.

Options for the profiling can be set using opt_prof (see prelim_checks). For this demo, the level of detail of the profiling is set to 'course' and no sub-optimisations are used. However, consider these options when working on your own data (e.g., set opt_prof.subopt=10).

In this case, the profile for c0 is ragged, and has all the signs of the profiling failing to find the best optimum at some points. However, sub-optimisations do not seem to solve this, so this case requires further scrutiny.

opt_prof.detail   = 2; % detailed (1) or a course (2) calculation
opt_prof.subopt   = 10; % number of sub-optimisations to perform to increase robustness

% % UNCOMMENT FOLLOWING LINE(S) TO CALCULATE
% % run a profile for selected parameters ...
% calc_proflik(par_out,'ke',opt_prof);  % calculate a profile
calc_proflik(par_out,'c0',opt_prof);  % calculate a profile

% % UNCOMMENT FOLLOWING LINE(S) TO CALCULATE
% % or run profiles for all fitted parameters.
% all_profiles(par_out,opt_prof);

% % UNCOMMENT FOLLOWING LINE(S) TO CALCULATE
% % Automatically calculate profiles for all parameters, and redo
% % optimisation when a better value is found.
% opt_prof.subopt   = -1; % number of sub-optimisations to perform to increase robustness
% % Note: the -1 makes a comparison between 0 and 10 sub-optimisations.
% % In this example, sub-optimisations are helpful!
% par_out = auto_profiles(par_out,opt_prof,opt_optim); % Experimental!
  
95% confidence interval from the profile
=================================================================================
The confidence interval is a broken set (check likelihood profile to check these figures)
c0         
   interval:    0.02734 - 0.02977 
   interval:    0.03567 - 0.04554 
   interval:    0.04865 - 0.1312 
 
Time required: 1 hour, 23 mins, 27.6 secs

Population growth rate

The function calc_pop allows for a calculation of the intrinsic rate of population increase (Euler-Lotka equation). It calculates the growth rate at three food levels (set in the engine file calc_pop.m).

% Tpop = linspace(0,42,100); % time vector for the population calculations
% Cpop = linspace(0,1,50); % concentration range for population calculations
% % Cpop = -1; % use only concentrations as given in X0mat
% Th = 2.5; % time from fresh egg to t=0 in time vector (e.g., hatching time)
% % Here, use 2.5 days, as this value was earlier subtracted from the time
% % vector in the data set. So, reproduction now represents egg formation,
% % and we need to account for hatching time in the population calculation.
% trep = [0]; % vector with reproduction events (first one must be a zero)
% % here, zero to base calculation on continuous reproduction
% calc_pop(par_out,X0mat,[5 4],Tpop,Cpop,Th,trep)

Other files: derivatives

To archive analyses, publishing them with Matlab is convenient. To keep track of what was done, the file derivatives.m can be included in the published result.

%% BYOM function derivatives.m (the model in ODEs)
%
%  Syntax: dX = derivatives(t,X,par,c)
%
% This function calculates the derivatives for the model system. It is
% linked to the script files <byom_debtox_daphnia.html
% byom_debtox_daphnia.m>. As input, it gets:
%
% * _t_   is the time point, provided by the ODE solver
% * _X_   is a vector with the previous value of the states
% * _par_ is the parameter structure
% * _c_   is the external concentration (or scenario number)
%
% Time _t_ and scenario name _c_ are handed over as single numbers by
% <call_deri.html call_deri.m> (you do not have to use them in this
% function). Output _dX_ (as vector) provides the differentials for each
% state at _t_.
%
% * Author: Tjalling Jager
% * Date: April 2017
% * Web support: <http://www.debtox.info/byom.html>
% * Back to index <walkthrough_debtox.html>

%% Start

function dX = derivatives(t,X,par,c)

global glo   % allow for global parameters in structure glo (handy for switches)

%% Unpack states
% The state variables enter this function in the vector _X_. Here, we give
% them a more handy name.

cV = X(1); % state 1 is the scaled internal concentration
e  = X(2); % state 2 is the scaled reserve density
L  = X(3); % state 3 is body length
Rc = X(4); % state 4 is cumulative reproduction
S  = X(5); % state 5 is the survival probability at previous time point

%% Unpack parameters
% The parameters enter this function in the structure _par_. The names in
% the structure are the same as those defined in the byom script file.
% The 1 between parentheses is needed as each parameter has 5 associated
% values.

ke  = par.ke(1);  % elimination rate constant, d-1
c0  = par.c0(1);  % no-effect concentration sub-lethal
cT  = par.cT(1);  % tolerance concentration
c0s = par.c0s(1); % no-effect concentration survival
b   = par.b(1);   % killing rate

L0 = par.L0(1);  % initial body length (mm)
Lp = par.Lp(1);  % length at puberty (mm)
Lm = par.Lm(1);  % maximum length (mm)
rB = par.rB(1);  % von Bertalanffy growth rate
Rm = par.Rm(1);  % maximum reproduction rate
g  = par.g(1);   % energy-investment ratio
f  = par.f(1);   % scaled functional response
h0 = par.h0(1);  % background hazard rate

%% Calculate the derivatives
% This is the actual model, specified as a system of ODEs:

kM = rB * 3 * (1+g)/g;    % maintenance rate coefficient
v  = Lm * kM * g;         % energy conductance
s  = (1/cT)*max(0,cV-c0); % stress factor

switch glo.moa % stress effect according to selected mechanism
    case 1 % effect on assimilation/feeding
        f = f*max(0,1-s);
    case 2 % effect on maintenance costs
        kM = kM * (1+s);
        Rm = Rm * (1+s);
    case 3 % effect on growth costs
        g  = g * (1+s);
        kM = kM /(1+s);
    case 4 % effect on repro costs
        Rm = Rm / (1+s);
    case 5 % hazard to early embryo
        Rm = Rm * exp(-s);
    case 6 % costs for growth and repro
        g  = g * (1+s);
        kM = kM /(1+s);
        Rm = Rm / (1+s); 
end

de = (f-e)*v/L; % change in scaled energy density
dL = (kM*g/(3*(e+g))) * (e*v/(kM*g) - L); % change in length
if L<Lp % before length at puberty is reached ...
    R = 0; % no reproduction
else
    R = (Rm/(Lm^3-Lp^3))*((v*(L^2)/kM + L^3)*e/(e+g) - Lp^3); % repro rate
    R = max(0,R); % make sure it is always positive
end
dRc = R; % change in cumulative repro is the repro rate
dcV = ke*(Lm/L)*(c-cV) - cV*(3/L)*dL; % change in scaled internal conc.

h  = b * max(0,cV-c0s); % calculate the hazard rate
dS = -(h + h0)* S;      % change in survival probability (incl. background mort.)

dX = [dcV;de;dL;dRc;dS]; % collect all derivatives in one vector dX

Other files: call_deri

To archive analyses, publishing them with Matlab is convenient. To keep track of what was done, the file call_deri.m can be included in the published result.

%% BYOM function call_deri.m (calculates the model output)
%
%  Syntax: [Xout TE] = call_deri(t,par,X0v)
%
% This function calls the ODE solver to solve the system of differential
% equations specified in <derivatives.html derivatives.m>, or the explicit
% function(s) in <simplefun.html simplefun.m>. As input, it gets:
%
% * _t_   the time vector
% * _par_ the parameter structure
% * _X0v_   a vector with initial states and one concentration (scenario number)
%
% The output _Xout_ provides a matrix with time in rows, and states in
% columns. This function calls <derivatives.html derivatives.m>. The
% optional output _TE_ is the time at which an event takes place (specified
% using the events function). The events function is set up to catch
% discontinuities. It should be specified according to the problem you are
% simulating. If you want to use parameters that are (or influence) initial
% states, they have to be included in this function.
%
% * Author: Tjalling Jager
% * Date: April 2017
% * Web support: <http://www.debtox.info/byom.html>
% * Back to index <walkthrough_debtox.html>

%% Start

function [Xout TE] = call_deri(t,par,X0v)

global glo   % allow for global parameters in structure glo
global zvd   % global structure for zero-variate data

%% Initial settings
% This part extracts optional settings for the ODE solver that can be set
% in the main script (defaults are set in prelim_checks). The useode option
% decides whether to calculate the model results using the ODEs in
% <derivatives.html derivatives.m>, or the analytical solution in
% <simplefun.html simplefun.m>. Using eventson=1 turns on the events
% handling. Also modify the sub-function at the bottom of this function!
% Further in this section, initial values can be determined by a parameter
% (overwrite parts of X0), and zero-variate data can be calculated. See the
% example BYOM files for more information.

useode   = glo.useode; % calculate model using ODE solver (1) or analytical solution (0)
eventson = glo.eventson; % events function on (1) or off (0)
stiff    = glo.stiff; % set to 1 to use a stiff solver instead of the standard one

% Unpack the vector X0v, which is X0mat for one scenario
X0 = X0v(2:end); % these are the intitial states for a scenario
% % if needed, extract parameters from par that influence initial states in X0
L0 = par.L0(1);  % initial body length (mm)
X0(3) = L0; % put this parameter in the correct location of the initial vector

% % if needed, calculate model values for zero-variate data from parameter set
% if ~isempty(zvd)
%     zvd.ku(3) = par.Piw(1) * par.ke(1); % add model prediction as third value
% end

%% Calculations
% This part calls the ODE solver (or the explicit model in <simplefun.html
% simplefun.m>) to calculate the output (the value of the state variables
% over time). There is generally no need to modify this part. The solver
% ode45 generally works well. For stiff problems, the solver might become
% very slow; you can try ode15s instead.

c  = X0v(1);     % the concentration (or scenario number)
t  = t(:);       % force t to be a row vector (needed when useode=0)

% specify options for the ODE solver
options = odeset; % start with default options
if eventson == 1
    options = odeset(options, 'Events',@eventsfun); % add an events function
end
options = odeset(options, 'RelTol',1e-4,'AbsTol',1e-7); % specify tightened tolerances
% options = odeset(options,'InitialStep',max(t)/1000,'MaxStep',max(t)/100); % specify smaller stepsize
% smaller stepsize is a good idea for pulsed exposures; otherwise, stepsize
% may become so large that a pulse is missed completely

TE = 0; % dummy for time of events

if useode == 1 % use the ODE solver to calculate the solution
    % call the ODE solver (use ode15s for stiff problems, escpecially for pulses)
    if isempty(options.Events) % if no events function is specified ...
        switch stiff
            case 0
                [~,Xout] = ode45(@derivatives,t,X0,options,par,c);
            case 1
                [~,Xout] = ode113(@derivatives,t,X0,options,par,c);
            case 2
                [~,Xout] = ode15s(@derivatives,t,X0,options,par,c);
        end
    else % with an events functions ... additional output arguments for events:
        % TE catches the time of an event, YE the states at the event, and IE the number of the event
        switch stiff
            case 0
                [~,Xout,TE,YE,IE] = ode45(@derivatives,t,X0,options,par,c);
            case 1
                [~,Xout,TE,YE,IE] = ode113(@derivatives,t,X0,options,par,c);
            case 2
                [~,Xout,TE,YE,IE] = ode15s(@derivatives,t,X0,options,par,c);
        end
    end
else % alternatively, use an explicit function provided in simplefun!
    Xout = simplefun(t,X0,par,c);
end

if isempty(TE) || all(TE == 0) % if there is no event caught
    TE = +inf; % return infinity
end

%% Output mapping
% _Xout_ contains a row for each state variable. It can be mapped to the
% data. If you need to transform the model values to match the data, do it
% here. 

% Xout(:,1) = Xout(:,1).^3; % e.g., do something on first column, like cube it ...

% % To obtain the output of the derivatives at each time point. The values in
% % dXout might be used to replace values in Xout, if the data to be fitted
% % are the changes (rates) instead of the state variable itself.
% % dXout = zeros(size(Xout)); % initialise with zeros
% for i = 1:length(t) % run through all time points
%     dXout(i,:) = derivatives(t(i),Xout(i,:),par,c); 
%     % derivatives for each stage at each time
% end

%% Events function
% Modify this part of the code if _eventson_=1. This subfunction catches
% the 'events': in this case, it looks for the scaled internal
% concentration where the NEC is exceeded, and the start of investment in
% reproduction (L=Lp). This function should be adapted to the problem you
% are modelling (this one matches the byom_debtox_... files). You can
% catch more events by making a vector out of _values_.
%
% Note that the eventsfun has the same inputs, in the same sequence, as
% <derivatives.html derivatives.m>.

function [value,isterminal,direction] = eventsfun(t,X,par,c)

Lp  = par.Lp(1);  % length at puberty (mm)
c0  = par.c0(1);  % no-effect concentration sub-lethal
c0s = par.c0s(1); % no-effect concentration survival

nevents = 3;         % number of events that we try to catch

value       = zeros(nevents,1); % initialise with zeros
value(1)    = X(1) - c0;        % follow when scaled internal conc exceed the NEC
value(2)    = X(3) - Lp;        % follow when body length exceeds length at puberty
value(3)    = X(1) - c0s;       % follow when scaled internal conc exceed the NEC
isterminal  = zeros(nevents,1); % do NOT stop the solver at an event
direction   = zeros(nevents,1); % catch ALL zero crossing when function is increasing or decreasing