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Fig_SI_1_no_N_carb_syst.R
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Fig_SI_1_no_N_carb_syst.R
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#----------------------------------------------------------------------------#
# Fig. SI.1 Main program #
# Modeling the carbonate system in the algal boundary layer; no N source #
# #
# M. Lavoie 2018 #
#----------------------------------------------------------------------------#
rm(list=ls())
#------------------------------#
# Load packages #
#------------------------------#
library(ReacTran)
#-----------------------------------------------------------------------------#
# Function calculating the chemical species of the carbonate system #
# in the boundary layer. N uptake rate = 0 #
# This function computes the concentration (mol/m^3) of CO2, HCO3-, CO32-, H+ and OH- as a function of the distance
# from the cell surface. This also computes relative changes in the boundary layer.
# I : ionic strengh in mol/L
# R : radius (m)
# L : length of the layer surrounding the cell (m)
# pH : negative logarithm of H+ activity far from the cell
#--------------------------------------------------------------------------#
boundary_carb_syst_No_N<- function(I, R, L, pH) {
#------------------------------#
# MODEL #
#------------------------------#
#--- Partial derivative equations
boundary_layer <- function(t, state, parms) {
{ with( as.list( c(t, state, parms) ), {
# Reshape state variable as a 2D matrix
S <- matrix(nrow = X.grid$N, ncol = 5, data = state)
# Initialize dC/dt matrix
dCdt <- 0*S
# Rate of change for CO2
tran_1 <- tran.1D(C = S[,1], D = D, flux.up = F1, C.down = C[1], A = A.grid, dx = X.grid, full.output = T)
prod1 <- (kb[1] * S[,4] + kb[2]) * S[,2]
loss1 <- (kf[1] + kf[2] * S[,5]) * S[,1]
dCdt[,1] <- tran_1$dC + prod1 - loss1
# Rate of change for HCO3
tran_2 = tran.1D(C = S[,2], D = D, C.down = C[2], flux.up = 0, A = A.grid, dx = X.grid, full.output = T)
prod2 = (kf[1] * S[,1]) + (kf[2] * S[,1] * S[,5]) + (kf[4] * S[,3] * S[,4]) + (kb[5] * S[,3])
loss2 = (kb[1] * S[,4] * S[,2]) + (kb[2] * S[,2]) + (kb[4] * S[,2]) + (kf[5] * S[,2] * S[,5])
dCdt[,2] = tran_2$dC + prod2 - loss2
# Rate of change for CO3
tran_3 = tran.1D(C = S[,3], D = D, C.down = C[3], flux.up = 0, A = A.grid, dx = X.grid, full.output = T)
prod3 = (kb[4] * S[,2]) + (kf[5] * S[,2] * S[,5])
loss3 = (kf[4] * S[,4] * S[,3]) + (kb[5] * S[,3])
dCdt[,3] = tran_3$dC + prod3 - loss3
# flux.up = 0 since N uptake rate is set to 0.
# Rate of change for H
tran_4 = tran.1D(C = S[,4], D = D, C.down = C[4], flux.up = 0, A = A.grid, dx = X.grid, full.output = T)
prod4 = (kb[4] - kb[1] * S[,4]) * S[,2] + (kf[1] * S[,1]) + kb[6]
loss4 = (kf[4] * S[,4] * S[,3]) + (kf[6] * S[,4] * S[,5])
dCdt[,4] = tran_4$dC + prod4 - loss4
# Rate of change for OH
tran_5 = tran.1D(C = S[,5], D = D, C.down = C[5], flux.up = 0, A = A.grid, dx = X.grid, full.output = T)
prod5 = (kb[2] * S[,2]) + (kb[5] * S[,3]) + kb[6]
loss5 = (kf[2] * S[,1] * S[,5]) + (kf[5] * S[,2] * S[,5]) + (kf[6] * S[,4] * S[,5])
dCdt[,5] = tran_5$dC + prod5 - loss5
return(list(dCdt = dCdt))
} )
}
}
#------------------------------#
# Model grid definition #
#------------------------------#
# Number of grid layer
N <- 10000
# Radius of the cell (m)
R <- R
# Model grid setup
X.grid <- setup.grid.1D(x.up = R, L = L, N = N) # x.up = radius of the cell
# Interface area
A.grid <- setup.prop.1D(grid = X.grid, func = function(r) 4*pi*r^2)
#------------------------------#
# Model parameters #
#------------------------------#
options(digits=15)
# Calculation of CO2 concentration at I = 0
pCO2 <- 10^-3.51 # CO2 partial presure (atm)
Kh <- 0.034 # Henry's constant (mol atm-1 L-1) at I =0 ; Correction of Kh at higher I can be neglected at I typical of freshwaters (See Weiss, 1974)
CO2 <- pCO2 * Kh * 1000 # CO2 concentration (mol m-3)
# Fixed H+ and OH- concentrations at a given pH (mol m-3)
source("func_activity_I.R")
coeff1 <- coeff_act(charge = 1, I = I)$coeff_act
H <- (1/coeff1) * 1000 * (10^-pH)
OH <- (1/coeff1) * 1000 * 1E-14 / (10^-pH)
# Calculations of HCO3- concentration
K1 <- 10^6.35 # Thermodynamic equilibrium constant of H+ + HCO3- = CO2
source("func_K_corr2_I.R") # Loading the function for ionic strength correction
K1_corr <- K_corr2(K0=K1, I=I, chargeA=1, chargeB=1, chargeC=0)
K1_corr
HCO3 <- (CO2 * 1000 * (1/K1_corr$K1))/H
# Calculations of CO32- concentration
K2 <- 10^10.33 # Thermodynamic equilibrium constant of H+ + CO32- = HCO3-
K2_corr <- K_corr2(K0=K2, I=I, chargeA=1, chargeB=2, chargeC=1)
K2_corr
CO3 <- (HCO3 * 1000 * (1/K2_corr$K1)) / H
# Concentration of CO2,HCO3,CO3,H,OH in bulk solution (mol m^-3) at I=0 and T=25 C and a given pH
C <- c(CO2, HCO3, CO3, H, OH)
# Growth rate (u in d-1)
u <- 1.12 * ((R*1E+06)^-0.75)
# Uptake flux of CO2 by the cell (mol m-^2 s^-1)
F1 <- -(23230/3) * R * u / 86400
# Diffusion coefficient of CO2,... (m^2 s^-1)
D <- 1.18E-09
# Rate constants at I=0
kb_I0 <- c(7.88029840776055E+01, 1.8E-04, NA, 2.33867570643599, 6.54216399387684E+05, 1.4) # m^3 mol^-1 s^-1; s^-1; ; s^-1; s^-1; mol m^-3 s^-1
kf_I0 <- c(3.52E-02, 8.04030465871734, NA, 5E+07, 3.06E+06, 1.4E+08) # s-^1; m^3 mol^-1 s^-1; ; m^3 mol^-1 s^-1; m^3 mol^-1 s^-1; m^3 mol^-1 s^-1
source("func_k_rate_cst_corr2_I.R") # Load the function converting rate constant at a given I
kwat_I0 <- 1E-14 # Ion product of water at 25 ?C
kwat <- kwat_I0 / (coeff1 * coeff1) # Conditional ion product of water
kb1 <- k_corr2(k0 = kb_I0[1], I = I, chargeA = 1, chargeB = 1, chargeC = 0)$k1 # HCO3- + H+ -> CO2
kb1
kf1 <- 1000* kb1/K1_corr$K1 # CO2 + H2O -> HCO3- + H+
kf1
kf2 <- k_corr2(k0 = kf_I0[2], I = I, chargeA = 0, chargeB = 1, chargeC = 1)$k1 # CO2 + OH- -> HCO3-
kf2
kb2 <- kf2 * K1_corr$K1 * kwat * 1000 # HCO3- -> CO2 + OH-
kb2
kf4 <- k_corr2(k0 = kf_I0[4], I = I, chargeA = 2, chargeB = 1, chargeC = 1)$k1 # CO32- + H+ -> HCO3-
kf4
kb4 <- kf4 * (1/(K2_corr$K1 / 1000)) # HCO3- -> CO32- + H+
kb4
kf5 <- k_corr2(k0 = kf_I0[5], I = I, chargeA = 1, chargeB = 1, chargeC = 2)$k1 # HCO3- + OH- <-> CO32- + H2O
kf5
kb5 <- kf5 * kwat * 1000 * 1000 * K2_corr$K1 / 1000 # CO32- + H2O -> HCO3- + OH-
kb5
kf6 <- k_corr2(k0 = kf_I0[6], I = I, chargeA = 1, chargeB = 1, chargeC = 0)$k1 # H+ + OH- -> H2O
kf6
kb6 <- kf6 * kwat * 1000 * 1000
kb6
# Rate constants at I = I
kb <- c(kb1, kb2, NA, kb4, kb5, kb6)
kf <- c(kf1, kf2, NA, kf4, kf5, kf6)
# Define parameters + grid definition vector
parms <- list(C=C, F1=F1, D=D, kb=kb, kf=kf, X.grid=X.grid, A.grid=A.grid)
#------------------------------#
# Model solution #
#------------------------------#
# Numerical solution at steady state
Cini <- matrix(C, nrow=N, ncol=5, byrow=T)
# I added pos = TRUE
boundary <- steady.1D(y = Cini, func = boundary_layer, pos = TRUE, atol=1E-8, parms = parms, nspec = 5, names = c('CO2','HCO3','CO3','H','OH'))
#------------------------------#
# Plotting output #
#------------------------------#
# Using S3 plot method of package rootSolve'
plotmult <- plot(boundary, grid = X.grid$x.mid, xlab = 'distance from centre, m', ylab = 'mol/m3', main = c('CO2', 'HCO3', 'CO3', 'H', 'OH'), ask = F, mfrow = c(1,1))
# Storing X-axis label
xlabel <- X.grid$x.mid
# Relative enrichment calculations
bmat <- unlist(boundary$y)
bmat1 <- as.matrix(bmat)
CO2_res <- bmat1[,1]
CO2enrich <- CO2_res / CO2
HCO3_res <- bmat1[,2]
HCO3enrich <- HCO3_res / HCO3
CO3_res <- bmat1[,3]
CO3enrich <- CO3_res / CO3
H_res <- bmat1[,4]
Henrich <- H_res / H
OH_res <- bmat1[,5]
OHenrich <- OH_res / OH
return(list(plotmult = plotmult, CO2enrich = CO2enrich, HCO3enrich = HCO3enrich, CO3enrich = CO3enrich, Henrich = Henrich, OHenrich = OHenrich, xlabel = xlabel))
}
#--------------------#
# Plots of FigSI.1 #
#--------------------#
#------------------------------------------------#
# Calculations at different pH at R = 5 um #
#------------------------------------------------#
carb_pH7 <- boundary_carb_syst_No_N(I = 0.001, R = 5E-06 , L = 150E-06, pH = 7)
carb_pH5 <- boundary_carb_syst_No_N(I = 0.001, R = 5E-06 , L = 150E-06, pH = 5)
carb_pH8 <- boundary_carb_syst_No_N(I = 0.001, R = 5E-06 , L = 150E-06, pH = 8)
# Plot of relative enrichment (C/Co) of each chemical species at different pHs.
tiff( "FigS1_AE_no_N.tiff", res = 100)
oldpar <- par(mfrow=c(3,2), mar=c(0,0,0,3), oma = c(4,4.1,4,0.4), las=1)
# Panel A : CO2 concentrations (r = 5 um)
plot(carb_pH7$xlabel*1E+06-5, carb_pH7$CO2enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 60), ylim = c(0.9, 1))
lines(carb_pH5$xlabel*1E+06-5, carb_pH5$CO2enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-5, carb_pH8$CO2enrich, type = "l", lty = 3)
legend("bottomright", legend= c("pH7", "pH5", "pH8"), lty = c(1,2,3))
mtext(text = "A", side = 3, adj = 0.05, line = -1.4, font = 2)
mtext(text = "relative change", side=2, line = 2.7, outer=TRUE, las=0)
mtext(text = expression(paste(" distance from cell surface (", mu, "m)")), side = 1, line = 3, font = 2, outer=TRUE, las=1)
# Panel B : HCO3- concentrations (r = 5 um)
plot(carb_pH7$xlabel*1E+06-5, carb_pH7$HCO3enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 60), ylim = c(0.95, 1.01))
lines(carb_pH5$xlabel*1E+06-5, carb_pH5$HCO3enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-5, carb_pH8$HCO3enrich, type = "l", lty = 3)
mtext(text = "B", side = 3, adj = 0.05, line = -1.4, font = 2)
# Panel C : CO32- concentrations (r = 5 um)
plot(carb_pH7$xlabel*1E+06-5, carb_pH7$CO3enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 60), ylim = c(0.95, 1.03))
lines(carb_pH5$xlabel*1E+06-5, carb_pH5$CO3enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-5, carb_pH8$CO3enrich, type = "l", lty = 3)
mtext(text = "C", side = 3, adj = 0.05, line = -2.1, font = 2)
# Panel D : H+ concentrations (r = 5 um)
plot(carb_pH7$xlabel*1E+06-5, carb_pH7$Henrich, type = "l", lty = 1, xlim = c(0, 60), ylim = c(0.95, 1.01))
lines(carb_pH5$xlabel*1E+06-5, carb_pH5$Henrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-5, carb_pH8$Henrich, type = "l", lty = 3)
mtext(text = "D", side = 3, adj = 0.05, line = -1.4, font = 2)
# Panel E : OH- concentrations (r = 5 um)
plot(carb_pH7$xlabel*1E+06-5, carb_pH7$OHenrich, type = "l", lty = 1, xlim = c(0, 60), ylim = c(0.99, 1.03))
lines(carb_pH5$xlabel*1E+06-5, carb_pH5$OHenrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-5, carb_pH8$OHenrich, type = "l", lty = 3)
mtext(text = "E", side = 3, adj = 0.05, line = -2.1, font = 2)
par(oldpar)
dev.off()
#------------------------------------------------#
# Calculations at different pH at R = 30 um #
#------------------------------------------------#
carb_pH7 <- boundary_carb_syst_No_N(I = 0.001, R = 30E-06 , L = 900E-06, pH = 7)
carb_pH5 <- boundary_carb_syst_No_N(I = 0.001, R = 30E-06 , L = 900E-06, pH = 5)
carb_pH8 <- boundary_carb_syst_No_N(I = 0.001, R = 30E-06 , L = 900E-06, pH = 8)
# Plot of relative enrichment (C/Co) of each chemical species at different pHs.
tiff( "FigS1_F_no_N.tiff", res = 100)
oldpar <- par(mfrow=c(3,2), mar=c(0,0,0,3), oma = c(4.1,4,4,0.4), las=1)
# Panel F : CO2 concentrations (r = 30 um)
plot(carb_pH7$xlabel*1E+06-30, carb_pH7$CO2enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 100), ylim = c(0, 1))
lines(carb_pH5$xlabel*1E+06-30, carb_pH5$CO2enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-30, carb_pH8$CO2enrich, type = "l", lty = 3)
legend("bottomright", legend= c("pH7", "pH5", "pH8"), lty = c(1,2,3))
mtext(text = "F", side = 3, adj = 0.05, line = -1.4, font = 2)
mtext(text = "relative change", side=2, line = 2.6, outer=TRUE, las=0)
mtext(text = expression(paste(" distance from cell surface (", mu, "m)")), side = 1, line = 3, font = 2, outer=TRUE, las=1)
# Panel G : HCO3- concentrations (r = 30 um)
plot(carb_pH7$xlabel*1E+06-30, carb_pH7$HCO3enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 100), ylim = c(0.5, 1.01))
lines(carb_pH5$xlabel*1E+06-30, carb_pH5$HCO3enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-30, carb_pH8$HCO3enrich, type = "l", lty = 3)
mtext(text = "G", side = 3, adj = 0.03, line = -2.1, font = 2)
# Panel H : CO32- concentrations (r = 30 um)
plot(carb_pH7$xlabel*1E+06-30, carb_pH7$CO3enrich, type = "l", lty = 1, xaxt = "n", xlim = c(0, 100), ylim = c(0.85, 5.1))
lines(carb_pH5$xlabel*1E+06-30, carb_pH5$CO3enrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-30, carb_pH8$CO3enrich, type = "l", lty = 3)
mtext(text = "H", side = 3, adj = 0.03, line = -1.4, font = 2)
# Panel I : H+ concentrations (r = 30 um)
plot(carb_pH7$xlabel*1E+06-30, carb_pH7$Henrich, type = "l", lty = 1, xlim = c(0, 100), ylim = c(0, 1.1))
lines(carb_pH5$xlabel*1E+06-30, carb_pH5$Henrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-30, carb_pH8$Henrich, type = "l", lty = 3)
mtext(text = "I", side = 3, adj = 0.03, line = -1.4, font = 2)
# Panel J : OH- concentrations (r = 30 um)
plot(carb_pH7$xlabel*1E+06-30, carb_pH7$OHenrich, type = "l", lty = 1, xlim = c(0, 100), ylim = c(1, 5))
lines(carb_pH5$xlabel*1E+06-30, carb_pH5$OHenrich, type = "l", lty = 2)
lines(carb_pH8$xlabel*1E+06-30, carb_pH8$OHenrich, type = "l", lty = 3)
mtext(text = "J", side = 3, adj = 0.03, line = -1.4, font = 2)
par(oldpar)
dev.off()
#-----------------------------#
# References #
#-----------------------------#
# Martell, A. E., Smith, R. M., Motekaitis, R. J. 2004. NIST critical stability constants of metal complexes, version 8. National Institute of Standards and Technology. Gaithersburg, MD. In Gaithersburg, MD, 2004.
# Weiss, R.F. 1974. Carbon dioxide in water and seawater : The solubility of a non-ideal gas. Marine Chemistry 2 : 203-215