---
title: "Equilibrium in CHNOSZ"
author: "Jeffrey M. Dick"
output:
  html_vignette:
    mathjax: null
vignette: >
  %\VignetteIndexEntry{Equilibrium in CHNOSZ}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: vig.bib
csl: elementa.csl
---
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```{r setup, include=FALSE}
## use pngquant to optimize PNG images
library(knitr)
knit_hooks$set(pngquant = hook_pngquant)
pngquant <- "--speed=1 --quality=0-25"
if (!nzchar(Sys.which("pngquant"))) pngquant <- NULL
```

```{r CHNOSZ_reset, include=FALSE}
library(CHNOSZ)
reset()
```

This vignette was compiled on `r Sys.Date()` with CHNOSZ version `r sessionInfo()$otherPkgs$CHNOSZ$Version`.
This document defines the concepts and provides examples of equilibrium calculations in CHNOSZ.

First, some limitations.
CHNOSZ is not a general-purpose speciation code.
The types of equilibrium calculations it can handle are restricted to those systems that can be described as *balanced on an element*.
For example, we can predict the speciation of Au in complexes with chloride, hydroxide, and sulfide all with specified activities.
But we can not at the same time predict the formation of species such as NaCl(aq), which is required for a complete equilibrium model.

## Concepts

System
:   Chemical system defined by formed species (referred to simply as species) and basis species, which are analogous to thermodynamic components.
The possible species are any that are available in the [OBIGT thermodynamic database](OBIGT.html).

    Species
    :   Selection of possible chemical species for which you want to calculate relative stabilities.

    Basis species
    :   Any minimal set of possible species that can be used to balance the formation reactions of the species.
    Additionally, in CHNOSZ the number of basis species must be equal to the number of elements.

Formation reaction
:   Chemical reaction giving the stoichiometry of basis species combined to make 1 mole of a species.

Transformation reaction
:   Any combination of two formation reactions in opposite directions (1 mole of a species reacts to form 1 mole of another species).

Balancing coefficients
:   (*n*~balance~) For a system that is balanced on one of the basis species,
the number of moles of that basis species in the formation reaction of each of the species.
The number can be positive or negative but not zero.
Can also represent a virtual quantity, such as 1 (balance on moles of species) or number of amino acids in a protein sequence (balance on protein length).

Speciation diagram
:   Diagram showing the equilibrium activities of species as a function of one variable.

Predominance diagram
:   Diagram variables showing fields for species with highest equilibrium activity as a function of two variables.
Also known as an equal activity diagram for aqueous species or stability diagram for minerals.

Chemical affinity
:   Negative of the differential of Gibbs energy of a system with respect to reaction progress.
For a given reaction, chemical affinity is the negative of Gibbs energy of reaction: *A* = 2.303*RT*log(*K*/*Q*),
where *K* is the equilibrium constant and *Q* is the activity quotient of species in the reaction
(log here denotes base-10 logarithms, i.e. `log10` in R).

Maximum affinity method
:   Approach used to construct predominance diagrams not by finding the highest activity at equilibrium but by finding the highest affinity at a **reference activity**.
The reference activities are user-defined equal activities of species (e.g. unit activity for minerals and 10^-3^ for aqueous species).

Equilibration method
:   Calculations of the activities of species in equilibrium.

    Metastable equilibrium
    :   The condition that the affinities of all *transformation* reactions are zero.
    Implemented with the `equilibrate()` function in CHNOSZ, which is used below.

    Total equilibrium
    :   The condition that the affinities of all *formation* reactions are zero.
    Implemented with the `solubility()` function in CHNOSZ, which is not described here.

## Example 1: Amino acids

In this sytem the basis species are CO~2~, H~2~O, NH~3~, H~2~S, and O~2~ and the formed species are 20 amino acids.

```{r AAsetup, results = "hide", message = FALSE}
library(CHNOSZ)
reset()
basis("CHNOS")
species(aminoacids(""))
```

These functions are used for the different diagrams in the figure.

```{r AAfunctions}
aaA <- function() {
  a <- affinity(O2 = c(-90, -70), H2O = c(-20, 10))
  diagram(a, balance = 1, names = aa)
}

aaB <- function() {
  a <- affinity(O2 = c(-90, -70), H2O = c(-20, 10))
  e <- equilibrate(a, balance = 1)
  diagram(e, names = aa)
}

aaC <- function() {
  a <- affinity(O2 = c(-71, -66), H2O = c(-8, 4))
  diagram(a, balance = "CO2", names = aa)
}

aaD <- function() {
  a <- affinity(O2 = c(-71, -66), H2O = c(-8, 4))
  e <- equilibrate(a, balance = "CO2")
  diagram(e, names = aa)
}

aaE <- function() {
  basis("O2", -66)
  a <- affinity(H2O = c(-8, 4))
  e <- equilibrate(a, balance = "CO2")
  diagram(e, ylim = c(-5, -1), names = aa)
}

aaF <- function() {
  species(1:20, -7)
  a <- affinity(H2O = c(-8, 4))
  e <- equilibrate(a, balance = "CO2")
  diagram(e, ylim = c(-8, -4), names = aa)
}
```

The labels used for the diagrams are the one-letter abbreviations for the amino acids.

```{r AAabbrv}
aa <- aminoacids()
aa
```

Press the button to show all of the code for making the figure.

<button id="B-AAplot" onclick="ToggleDiv('AAplot')">Show code</button>
<div id="D-AAplot" style="display: none">
```{r AAplot, eval = FALSE}
showtime <- function(st) {
  # plot time in lower-right of figure region
  f <- usrfig()
  par(xpd=TRUE)
  if(st[3] > 2) col <- "red" else col <- "black"
  text(f$x[2], f$y[1], paste(round(st[3], 1), "s\n"), adj=1, col=col)
  par(xpd=FALSE)
}

layout(t(matrix(c(1:7, 11, 8:10, 12), nrow=4)), widths=c(1, 4, 4, 4), heights=c(0.7, 4, 4))

## row 0 (column titles)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
plot.new()
text(0.58, 0.5, "maximum affinity", cex=1.4)
plot.new()
text(0.58, 0.5, "equilibration", cex=1.4)
plot.new()
text(0.58, 0.5, "equilibration", cex=1.4)
par(opar)

## row 1 (balance = 1)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
text(0.5, 0.5, "balance = 1", srt=90, cex=1.4)
par(opar)
# figure A
st <- system.time(dA <- aaA())
showtime(st)
title(main="loga(species) = -3", cex.main=1)
label.figure("A", yfrac=0.92, xfrac=0.1, font = 2)
# figure B
st <- system.time(dB <- aaB())
showtime(st)
title(main=paste("loga(total species) =", round(dB$loga.balance[1], 2)), cex.main=1)
label.figure("C", yfrac=0.92, xfrac=0.1, font = 2)

## row 2 (balance = nCO2)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
text(0.5, 0.5, 'balance = "CO2"', srt=90, cex=1.4)
par(opar)
# figure C
st <- system.time(dC <- aaC())
showtime(st)
title(main="loga(species) = -3", cex.main=1)
label.figure("B", yfrac=0.92, xfrac=0.1, font = 2)
# figure D
st <- system.time(dD <- aaD())
showtime(st)
title(main=paste("loga(total CO2) =", round(dD$loga.balance[1], 2)), cex.main=1)
label.figure("D", yfrac=0.92, xfrac=0.1, font = 2)

## right (speciation at different total activity of CO2)
par(xpd=NA)
lines(c(-66, -64.5), c(4, 9), lty=2)
lines(c(-66, -64.5), c(-8, -8.5), lty=2)
par(xpd=FALSE)
# figure E
st <- system.time(dE <- aaE())
showtime(st)
title(main=paste("loga(total CO2) =", round(dE$loga.balance[1], 2)), cex.main=1)
label.figure("E", yfrac=0.92, xfrac=0.1, font = 2)
# figure F
st <- system.time(dF <- aaF())
showtime(st)
title(main=paste("loga(total CO2) =", round(dF$loga.balance[1], 2)), cex.main=1)
label.figure("F", yfrac=0.92, xfrac=0.1, font = 2)
```
</div>

```{r AAplot, echo = FALSE, results = "hide", message = FALSE, fig.width = 13/2, fig.height = 8.7/2, out.width = "100%", pngquant = pngquant}
```

Diagrams **A** and **B** use the *maximum affinity method*, where the reference
activities of species are set to a single value.  Diagrams **C** and **D** use
the *equilibration method*, where the activities of species change across the
diagram and the lines are drawn at equal activity.  Other comments on the
figure:

* The equal-activity lines in Diagrams **A** and **C** are the same. That is,
  with the setting `balance = 1`, the conditions for equal activity of two
  species do not depend on the actual value of that activity.

* Diagrams **B** and **D** are different. Because balance ≠ 1 (in this case,
  the reactions are balanced on CO~2~), the conditions for equal activity of
  species depend on the actual value of that activity. In particular, with the
  equilibration method, the lines are curved due to the distribution of more
  than two species.

* Diagrams **E** and **F** both use the equilibration method to calculate
  activities of species as a function of log*a*~H2O~ at log*f*~O2~ = -66.
  Diagram **F** shows the default setting, where *a*~CO2~ is taken from the sum
  of initial activities of species (each 10^-3^). The positions of equal
  activities (where the lines cross) are the same as in Diagram **D** at
  log*f*~O2~ = -66.

* Diagram **F** is drawn for a lower total activity of *a*~CO2~. Not only are
  the activities of all amino acids decreased, but glycine starts to become
  predominant at some conditions. This is because, compared to other amino
  acids, it is smaller and has a lower coefficient on CO~2~ in its formation
  reaction.

## Example 2: Proteins

In this sytem the basis species are CO~2~, H~2~O, NH~3~, H~2~S, O~2~, and H^+^ and the
formed species are 6 proteins from the set of archaeal and bacterial surface
layer proteins considered by @Dic08.

The inclusion of charge in the basis species (with H^+^) allows ionization of
proteins to be calculated [@DLH06].

```{r PRsetup, results = "hide", message = FALSE}
basis("CHNOS+")
organisms <- c("METJA", "HALJP", "METVO", "ACEKI", "GEOSE", "BACLI")
proteins <- c(rep("CSG", 3), rep("SLAP", 3))
species(proteins, organisms)
```

Here are the lengths of the proteins.

```{r PRlength}
protein.length(species()$name)
```

These functions are used for the different diagrams in the figure.

```{r PRfunctions}
prA <- function() {
  a <- affinity(O2 = c(-90, -70), H2O = c(-20, 0))
  diagram(a, balance = "length", names = organisms)
}

prB <- function() {
  a <- affinity(O2 = c(-90, -70))
  e <- equilibrate(a, balance = "length")
  ylab <- quote(log~italic(a)~protein)
  diagram(e, names = organisms, ylim = c(-5, -1), ylab = ylab)
}

prC <- function() {
  a <- affinity(O2 = c(-90, -70), H2O = c(-20, 0))
  diagram(a, normalize = TRUE, names = organisms)
}

prD <- function() {
  a <- affinity(O2 = c(-90, -70))
  e <- equilibrate(a, normalize = TRUE)
  ylab <- quote(log~italic(a)~protein)
  diagram(e, names = organisms, ylim = c(-5, -1), ylab = ylab)
}

prE <- function() {
  a <- affinity(O2 = c(-90, -70), H2O = c(-20, 0))
  diagram(a, as.residue = TRUE, names = organisms)
}

prF <- function() {
  a <- affinity(O2 = c(-90, -70))
  e <- equilibrate(a, as.residue = TRUE, loga.balance = 0)
  ylab <- quote(log~italic(a)~residue)
  diagram(e, names = organisms, ylim = c(-3, 1), ylab = ylab)
}
```

<button id="B-PRplot" onclick="ToggleDiv('PRplot')">Show code</button>
<div id="D-PRplot" style="display: none">
```{r PRplot, eval = FALSE}
```
</div>

```{r PRplot, echo = FALSE, results = "hide", message = FALSE, fig.width = 13/2, fig.height = 8.7/2, out.width = "100%", pngquant = pngquant}
layout(t(matrix(1:12, nrow=4)), widths=c(1, 4, 4, 4), heights=c(0.7, 4, 4))

## row 0 (column titles)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
plot.new()
text(0.58, 0.5, 'balance = "length"', cex=1.4)
plot.new()
text(0.58, 0.5, "normalize = TRUE\n(balance = 1)", cex=1.4)
plot.new()
text(0.58, 0.5, "as.residue = TRUE\n(balance = 1)", cex=1.4)
par(opar)

## row 1 (maximum affinity 2D)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
text(0.5, 0.5, "maximum affinity", srt=90, cex=1.4)
par(opar)
# figure A (balance = "length")
st <- system.time(dA <- prA())
showtime(st)
label.figure("A", yfrac=0.9, xfrac=0.1, font = 2)
# figure C (normalize = TRUE)
st <- system.time(dC <- prC())
showtime(st)
label.figure("C", yfrac=0.9, xfrac=0.1, font = 2)
# figure E (as.residue = TRUE)
st <- system.time(dE <- prE())
showtime(st)
label.figure("E", yfrac=0.9, xfrac=0.1, font = 2)

## row 2 (equilibrate 1D)
opar <- par(mar=c(0, 0, 0, 0))
plot.new()
text(0.5, 0.5, "equilibration", srt=90, cex=1.4)
par(opar)
# figure B (balance = "length")
st <- system.time(prB())
showtime(st)
label.figure("B", yfrac=0.9, xfrac=0.1, font = 2)
# figure D (normalize = TRUE)
st <- system.time(prD())
showtime(st)
label.figure("D", yfrac=0.9, xfrac=0.1, font = 2)
# figure F (as.residue = TRUE)
st <- system.time(prF())
showtime(st)
label.figure("F", yfrac=0.9, xfrac=0.1, font = 2)
```

Diagrams **A**, **C**, and **E** are predominance (equal-activity) diagrams
made with different balancing constraints.  Diagrams **B**, **D**, and **F**
show a cross-section of the same system at log*a*~H2O~ = 0.

* In Diagrams **A** and **B**, the reactions are balanced on protein length.
  The drastic transitions between proteins in **B** result from the large
  balancing coefficients, which become exponents on activities in the
  expression for the activity quotient (*Q*).

* In Diagrams **C** and **D**, the chemical formulas of the proteins are
  *normalized*, or divided by the protein length, before the equilibration or
  maximum affinity calculation.  However, the final diagram is not drawn for
  activities of these "residue equivalents", but for rescaled activities of
  whole proteins. Because of the rescaling, the fields of the shorter METJA and
  METVO proteins grow at the expense of the longer BACLI protein.

* Diagrams **E** and **F** are like the normalization calculation, except that
  the final diagram is drawn for the residue equivalents and not the whole
  proteins.  The predominance diagram is the same as that for non-normalized
  reactions (**A**), but the equilibrium activities of residues are higher than
  those of the proteins.

## Example 3: Normalization

Here is another figure showing the effects of normalization.
This is like Figure 5 of @Dic08, extended to more extreme conditions.
If you wish to reproduce the diagram from the 2008 paper more closely, uncomment the `add.OBIGT()` command.

```{r ProteinSpeciation, results = "hide", message = FALSE, fig.width = 8, fig.height = 5.5, out.width = "100%", pngquant = pngquant}
organisms <- c("METSC", "METJA", "METFE",  "METVO", "METBU",
               "HALJP", "ACEKI", "GEOSE", "BACLI", "AERSA")
proteins <- c(rep("CSG", 6), rep("SLAP", 4))
# use red for Methano* genera
col <- c(rep(2, 5), rep(1, 5))
basis("CHNOS+")
species(proteins, organisms)
a <- affinity(O2 = c(-100, -65))
layout(matrix(1:2), heights = c(1, 2))
e <- equilibrate(a)
diagram(e, ylim = c(-2.8, -1.6), names = organisms, col = col)
water.lines(e, col = 4)
title(main="Equilibrium activities of proteins, normalize = FALSE")
e <- equilibrate(a, normalize = TRUE)
diagram(e, ylim = c(-4, -2), names = organisms, col = col)
water.lines(e, col = 4)
title(main = "Equilibrium activities of proteins, normalize = TRUE")
```

Although it is below the stability limit of H~2~O (vertical dashed line), there
is an interesting convergence of the metastable equilibrium activities of some
proteins at low log *f*~O2~.

## Document history

* 2009-11-29 Initial version with CSG example, titled "Calculating relative abundances of proteins"
* 2012-09-30 Renamed to "Equilibrium in CHNOSZ"; remove activity comparisons and add maximum affinity method.
* 2015-11-08 Move previous material to Appendix and add sections on concepts, organization, examples, and applications; use knitr vignette engine.
* 2020-07-10 Simplify to concepts and examples (amino acids, proteins, normalization); convert document from LaTeX (Rnw) to R Markdown (Rmd).

## References
