Installing and loading CHNOSZ

After starting R, install CHNOSZ by selecting the “Install packages from CRAN” or similar menu item in the R GUI or by using the following command: ⊕Or, install the package from a package file, which you can download from CRAN or (for the development version) from R-Forge.

install.packages("CHNOSZ")

Then load the CHNOSZ package to make its functions available in your R session:

library(CHNOSZ)

## CHNOSZ version 1.1.3 (2017-11-13)

## Please run data(thermo) to create the "thermo" object

Then load the thermo object, which contains the thermodynamic database and system settings for CHNOSZ:

data(thermo)

## data(thermo): attached environment "CHNOSZ"

## thermo$obigt: 1911 aqueous, 3588 total species

Note: Throughout this document, syntax highlighting is applied to the input of the code chunks. Double hash marks (##) precede the output, where black text denotes results and blue text is used for messages.

Getting help

After CHNOSZ is installed, type help.start() to browse the R help documents, then choose “Packages” followed by “CHNOSZ”. That shows an index of the manual (help pages) for each function; many of the help pages include examples. There are also links to the demos (longer examples) and vignettes (more in-depth documentation; this document is a vignette).

Suggestions for accessing the documentation are indicated here with blue text. For example, read ?`CHNOSZ-package` to get an overview of the package and a list of features. ⊕“?” is a shortcut to R’s help() function. The command here is equivalent to help(“CHNOSZ-package”).

Organization of major functions

CHNOSZ is made up of a set of functions and supporting datasets. The major components of the package are shown in the figure below, which is an updated version of the diagram in Dick (⊕2008Dick JM. 2008. Calculation of the relative metastabilities of proteins using the CHNOSZ software package. Geochemical Transactions 9: 10. doi: 10.1186/1467-4866-9-10). Rectangles and ellipses represent functions and datasets; bold text indicates primary functions.

Structure of CHNOSZ.

Many functions in CHNOSZ have no side effects. That is, the function only returns a result; to use the result elsewhere, it can be assigned to a variable with <-. In this document, the names of these functions are set in green text (not applicable to the code chunks). ⊕When they are mentioned, names of functions in the base and recommended packages of R are said to belong to R. Example: Use R’s plot() to plot the data. Major functions without side effects in CHNOSZ are:

• info(): search for species in the thermodynamic database;
• subcrt(): calculate the thermodynamic properties of species and reactions;
• affinity(): calculate the affinities of formation reactions using given chemical activities;
• equilibrate(): calculate the equilibrium chemical activities of the species of interest;
• diagram(): plot the results.

Some functions in CHNOSZ do have side effects: they modify the thermo data object in the current R session. In this document, the names of these functions are set in red text (but not in the code chunks). Major functions with side effects are:

• basis(): set the basis species and their chemical activities;
• species(): set the species of interest and their (non-equilibrium) chemical activities;
• data(thermo): reset the database, restoring all settings to their default values.

The following pseudocode shows a common sequence of commands. In actual usage, the ... are replaced by arguments that define the chemical species and variables:

data(thermo)         ## initialize system settings
basis(...)
species(...)
a <- affinity(...)
e <- equilibrate(a)  ## optional
diagram(e)           ## or diagram(a)
data(thermo)         ## clear settings for next calculation

The info() function

info() provides an interface to the thermodynamic database packaged with CHNOSZ. Suppose you are interested in the thermodynamic properties of aqueous methane. You can search for the species by name:

info("methane")

## info.character: found methane(aq), also available in liq, gas

## [1] 859

Multiple entries exist for methane; the index of the aq (aqueous) species is returned by default. A second argument can be used to specify a different physical state:

info("methane", "gas")

## [1] 3307

## info.character: found methane(aq), also available in liq, gas

Knowing that aqueous methane is species number 859 in the database, you can again use info() to retrieve the set of standard molal thermodynamic properties and equations of state parameters:

info(859)

##        name abbrv formula state       ref1 ref2      date     G      H  S    Cp
## 859 methane  <NA>     CH4    aq PS01 [S07] <NA> 04.Oct.00 -8140 -20930 21 60.23
##      V    a1    a2     a3     a4    c1    c2  omega Z
## 859 36 1.769 -1530 -67.88 114700 40.87 64500 -40000 0

This number can be used as an argument (ispecies) for other functions in CHNOSZ to uniquely identify any species; some commonly used functions also accept the species names. Liquid water is species number 1; it has NA entries in the database because specialized functions are used to compute its properties:

info(info("water"))

##    name abbrv formula state ref1 ref2      date  G  H  S Cp  V  a  b  c  d  e
## 1 water  <NA>     H2O   liq <NA> <NA> 25.Oct.06 NA NA NA NA NA NA NA NA NA NA
##    f lambda  T
## 1 NA     NA NA

Fuzzy searches

Calling info() with a string that does not exactly match the name of any species invokes a fuzzy search of the database:

info("acid")

## info.approx: 'acid' is ambiguous; has approximate matches to 84 species:

##  [1] "a-aminobutyric acid"     "acetamide"               "formic acid"             "acetic acid"             "propanoic acid"          "n-butanoic acid"        
##  [7] "n-pentanoic acid"        "n-hexanoic acid"         "n-heptanoic acid"        "n-octanoic acid"         "n-nonanoic acid"         "n-decanoic acid"        
## [13] "n-undecanoic acid"       "n-dodecanoic acid"       "n-benzoic acid"          "o-toluic acid"           "m-toluic acid"           "p-toluic acid"          
## [19] "oxalic acid"             "malonic acid"            "succinic acid"           "glutaric acid"           "adipic acid"             "pimelic acid"           
## [25] "suberic acid"            "azelaic acid"            "sebacic acid"            "glycolic acid"           "lactic acid"             "2-hydroxybutanoic acid" 
## [31] "2-hydroxypentanoic acid" "2-hydroxyhexanoic acid"  "2-hydroxyheptanoic acid" "2-hydroxyoctanoic acid"  "2-hydroxynonanoic acid"  "2-hydroxydecanoic acid" 
## [37] "aspartic acid"           "glutamic acid"           "cis-aconitic acid"       "isocitric acid"          "a-ketoglutaric acid"     "fumaric acid"           
## [43] "malic acid"              "oxaloacetic acid"        "pyruvic acid"            "uracil"                  "citric acid"             "metacinnabar"           
## [49] "sanidine,high"           "acide fulvique"          "acide humique"           "aspartic acid"           "glutamic acid"           "2-iodobenzoic acid"     
## [55] "3-iodobenzoic acid"      "4-iodobenzoic acid"      "uracil"                  "phosphoric acid"         "citric acid"             "acetamide"              
## [61] "nicotinamide,red"        "nicotinamide,ox"         "acetic acid"             "propanoic acid"          "n-butanoic acid"         "n-pentanoic acid"       
## [67] "n-hexanoic acid"         "n-heptanoic acid"        "n-octanoic acid"         "n-nonanoic acid"         "n-decanoic acid"         "n-undecanoic acid"      
## [73] "n-dodecanoic acid"       "n-tridecanoic acid"      "n-tetradecanoic acid"    "n-pentadecanoic acid"    "n-hexadecanoic acid"     "n-heptadecanoic acid"   
## [79] "n-octadecanoic acid"     "n-nonadecanoic acid"     "n-eicosanoic acid"       "2-iodobenzoic acid"      "3-iodobenzoic acid"      "4-iodobenzoic acid"

## [1] NA

The message includes e.g. “uracil” and “metacinnabar” because their names have some similarity to the search term.

As “ribose” is the name of a species in the database, to find species with similar names, add an extra character to the search:

info(" ribose")

## info.approx: ' ribose' is ambiguous; has approximate matches to 8 species:

## [1] "ribose"               "deoxyribose"          "ribose-5-phosphate"  
## [4] "ribose-5-phosphate-1" "ribose-5-phosphate-2" "ribose"              
## [7] "deoxyribose"          "ribose-5-phosphate"

## [1] NA

The messages may be useful for browsing the database, but owing to their ambiguous results, these fuzzy searches return an NA value for the species index.

Counting elements, chemical formulas, ZC()

Continuing with the example of methane, let’s look at its chemical formula:

info(859)$formula

## [1] "CH4"

We can use makeup() to count the elements in the formula, followed by as.chemical.formula() to rewrite the formula on one line:

makeup(859)

## C H 
## 1 4

as.chemical.formula(makeup(859))

## [1] "CH4"

For organic species, a calculation of the average oxidation state of carbon (Z_C) is possible given the species index, chemical formula, or elemental count:

ZC(859)

## [1] -4

ZC(info(859)$formula)

## [1] -4

ZC(makeup(859))

## [1] -4

Changing units

The default units of temperature, pressure, and energy are °C, bar, and calories. The functions T.units(), P.units(), and E.units() can be used to change the units used by various functions in CHNOSZ. What is the Gibbs energy in J/mol of aqueous methane at 298.15 K and 0.1 MPa?

T.units("K")
P.units("MPa")
E.units("J")
subcrt("methane", T = 298.15, P = 0.1)$out$methane$G

## [1] -34057.8

A related function, convert(), can be used to convert given values between units. Let’s convert the standard Gibbs energy of aqueous methane listed in the database from cal/mol to J/mol:

convert(info(info("methane"))$G, "J")

## [1] -34057.8

As expected, we get the same result from both operations.

Use data(thermo) to restore the units and all other settings for CHNOSZ to their defaults:

data(thermo)

## thermo$obigt: 1911 aqueous, 3588 total species

Reaction definitions

To calculate the thermodynamic properties of reactions, give the names of species, the physical states (optional), and reaction coefficients as the arguments to subcrt(). Here we calculate properties for the dissolution of CO₂: ⊕Because of aqueous speciation, this doesn’t give the solubility of CO₂. For an example of a solubility calculation, see demo(solubility), which is based on a figure in Manning et al. (2013).

subcrt(c("CO2", "CO2"), c("gas", "aq"), c(-1, 1), T = seq(0, 250, 50))

## subcrt: 2 species at 6 values of T and P (wet)

## $reaction
##      coeff           name formula state ispecies
## 3308    -1 carbon dioxide     CO2   gas     3308
## 1576     1            CO2     CO2    aq     1576
## 
## $out
##        T        P      rho     logK       G         H         S       V      Cp
## 1   0.01  1.00000 0.999829 -1.10873 1385.80 -5907.949 -26.70130 23.6978 49.1458
## 2  50.00  1.00000 0.988030 -1.71865 2541.27 -3947.724 -20.08036 36.9843 34.6260
## 3 100.00  1.01322 0.958393 -2.00354 3420.89 -2271.995 -15.25623 41.5584 33.0446
## 4 150.00  4.75717 0.917058 -2.10770 4080.94  -596.111 -11.05290 44.5642 34.3421
## 5 200.00 15.53650 0.864743 -2.09920 4544.74  1228.815  -7.00814 47.9553 39.3857
## 6 250.00 39.73649 0.799072 -2.01184 4815.90  3513.101  -2.49026 54.3882 55.7180

In order to make a plot like Figure 18 of Manning et al. (⊕2013Manning CE, Shock EL, Sverjensky DA. 2013. The chemistry of carbon in aqueous fluids at crustal and upper-mantle conditions: Experimental and theoretical constraints. Reviews in Mineralogy and Geochemistry 75(1): 109–148. doi: 10.2138/rmg.2013.75.5), let’s run more calculations and store the results. In addition to the reaction definition, we specify a greater number of temperature points than the default:

Calculated equilibrium constants for dissolution of CO₂, CO, and CH₄.

T <- seq(0, 350, 10)
CO2 <- subcrt(c("CO2", "CO2"), c("gas", "aq"), c(-1, 1), T = T)$out$logK
CO <- subcrt(c("CO", "CO"), c("gas", "aq"), c(-1, 1), T = T)$out$logK
CH4 <- subcrt(c("CH4", "CH4"), c("gas", "aq"), c(-1, 1), T = T)$out$logK
logK <- data.frame(T, CO2, CO, CH4)

Now we can make the plot, using R’s matplot(). Here, axis.label() and expr.species() are used to create formatted axis labels and chemical formulas:

matplot(logK[, 1], logK[, -1], type = "l", col = 1, lty = 1,
        xlab = axis.label("T"), ylab = axis.label("logK"))
text(80, -1.7, expr.species("CO2"))
text(240, -2.37, expr.species("CO"))
text(300, -2.57, expr.species("CH4"))

Unbalanced reactions

A balanced chemical reaction conserves mass. subcrt() won’t stop you from running an unbalanced reaction, but it will give you a warning:

subcrt(c("CO2", "CH4"), c(-1, 1))

## info.character: found CO2(aq), also available in gas

## info.character: found CH4(aq), also available in liq, gas

## subcrt: 2 species at 15 values of T and P (wet)

## subcrt: reaction is not balanced; it is missing this composition:

##  H  O 
## -4  2

## Warning: reaction was unbalanced, missing H-4O2

In other words, to balance the reaction, we should add 4 H to the left and 2 O to the right. That could be done manually be redefining the reaction with the appropriate species. There is another option: balancing the reaction automatically using basis species.

Setting the basis species

Basis species are a minimal number of chemical species that linearly combine to give the composition of any species in the system. The basis species are similar to thermodynamic components, but can include charged species. Basis species are used in CHNOSZ to automatically balance reactions; they are also required for making chemical activity diagrams.

Let’s start with an example that doesn’t work:

basis(c("CO2", "H2", "H2CO2"))

## Error in put.basis(ispecies, logact): singular stoichiometric matrix

That set of species has a singular (non-invertible) stoichiometric matrix. An error would also result from either an underdetermined or overdetermined system. A valid set of basis species has an invertible stoichiometric matrix and the same number of species as elements:

basis(c("CO2", "H2", "H2O"))

##     C H O ispecies logact state
## CO2 1 0 2     1576      0    aq
## H2  0 2 0       64      0    aq
## H2O 0 2 1        1      0   liq

The composition of any species made up of C, H, and O can be represented by a single linear combination of these basis species.

Automatically balancing reactions

Methanogenic metabolism in reducing environments may proceed by acetoclastic or hydrogenotrophic processes. To consider reactions involving a charged species (acetate), let’s define a basis with H⁺:

basis(c("CO2", "H2", "H2O", "H+"))

##     C H O Z ispecies logact state
## CO2 1 0 2 0     1576      0    aq
## H2  0 2 0 0       64      0    aq
## H2O 0 2 1 0        1      0   liq
## H+  0 1 0 1        3      0    aq

By identifying species other than the basis species, the reactions will be automatically balanced. This produces the balanced reaction for acetoclastic methanogenesis:

subcrt(c("acetate", "methane"), c(-1, 1))$reaction

##      coeff    name formula state ispecies
## 1070    -1 acetate C2H3O2-    aq     1070
## 859      1 methane     CH4    aq      859
## 1576     1     CO2     CO2    aq     1576
## 3       -1      H+      H+    aq        3

We can similarly consider reactions for hydrogenotrophic methanogenesis as well as acetate oxidation (without production of methane):

acetate_oxidation <- subcrt("acetate", -1)
hydrogenotrophic <- subcrt("methane", 1)
acetoclastic <- subcrt(c("acetate", "methane"), c(-1, 1))

Use describe.reaction() to write the reactions on a plot:

plot(0, 0, type = "n", axes = FALSE, ann=FALSE, xlim=c(0, 5), ylim=c(5.2, -0.2))
text(0, 0, "acetoclastic methanogenesis", adj = 0)
text(5, 1, describe.reaction(acetoclastic$reaction), adj = 1)
text(0, 2, "acetate oxidation", adj = 0)
text(5, 3, describe.reaction(acetate_oxidation$reaction), adj = 1)
text(0, 4, "hydrogenotrophic methanogenesis", adj = 0)
text(5, 5, describe.reaction(hydrogenotrophic$reaction), adj = 1)

Chemical affinity

Usually, subcrt() returns only standard state thermodynamic properties. ⊕The standard state adopted for H₂O is unit activity of the pure component at any T and P. The standard state for aqueous species is unit activity of a hypothetical one molal solution referenced to infinite dilution at any T and P. Thermodynamic models often consider a non-standard state (i.e. non-unit activity). The activities of basis species can be modified with basis(), and those of the other species using the logact argument in subcrt().

Let us calculate the chemical affinity of acetoclastic methanogenesis. ⊕The affinity is equal to the negative of the overall (non-standard) Gibbs energy change of the reaction. We begin by changing the energy units to Joules. Then, we change the state of H₂O and CO₂ in the basis from aq (aqueous) to gas, and set the logarithm of fugacity of gaseous H₂ and the pH, using values from Mayumi et al. (⊕2013Mayumi D, Dolfing J, Sakata S, Maeda H, Miyagawa Y, Ikarashi M, Tamaki H, Takeuchi M, Nakatsu CH, Kamagata Y. 2013. Carbon dioxide concentration dictates alternative methanogenic pathways in oil reservoirs. Nature Communications 4: 1998. doi: 10.1038/ncomms2998). The activity of acetate and fugacity of methane, as well as temperature and pressure, are set in the call to subcrt():

E.units("J")
basis(c("CO2", "H2", "H2O", "H+"))
basis(c("CO2", "H2"), "gas")
basis(c("H2", "pH"), c(-3.92, 7.3))

subcrt(c("acetate", "methane"), c(-1, 1),
       c("aq", "gas"), logact = c(-3.4, -0.18), T = 55, P = 50)$out

##    T  P      rho    logK        G       H       S        V      Cp  logQ
## 1 55 50 0.987821 13.5984 -85429.6 18706.9 317.685 -40.1339 39.2144 10.52
##         A
## 1 19339.4

The new A column shows the affinity; the other columns are unaffected and still show the standard-state properties. Let’s repeat the calculation for hydrogenotrophic methanogenesis.

subcrt("methane", 1, "gas", logact = -0.18, T = 55, P = 50)$out

##    T  P      rho   logK       G       H        S       V      Cp logQ       A
## 1 55 50 0.987821 18.828 -118284 -251807 -407.195 36.4746 33.4702 15.5 20907.8

Under the specified conditions, the affinities of hydrogenotrophic and acetoclastic methanogenesis are somewhat greater than and less than 20 kJ, respectively. This result matches Figure 4b in Mayumi et al. (2013) at unit fugacity of CO₂.

We can go even further and reproduce their plot. ⊕The reproduction is not identical, owing to differences of thermodynamic data and of calculations of the effects of temperature and pressure. To make the code neater, we write a function that can run any of the reactions:

rxnfun <- function(coeffs) {
  subcrt(c("acetate", "methane"), coeffs,
         c("aq", "gas"), logact = c(-3.4, -0.18), T = 55, P = 50)$out
}

Now we’re ready to calculate and plot the affinities. Here, we use R’s lapply() to list the results at two values of logarithm of fugacity of CO₂. We insert an empty reaction to get a line at zero affinity. R’s do.call() and rbind() are used to turn the list into a data frame that can be plotted with R’s matplot(). There, we plot the negative affinities, equal to Gibbs energy, as shown in the plot of Mayumi et al. (2013).

Gibbs energies of acetate oxidation and methanogenesis (after Mayumi et al., 2013).

Adat <- lapply(c(-3, 3), function(logfCO2) {
  basis("CO2", logfCO2)
  data.frame(logfCO2,
    rxnfun(c(0, 0))$A,
    rxnfun(c(-1, 0))$A,
    rxnfun(c(-1, 1))$A,
    rxnfun(c(0, 1))$A
  )
})
Adat <- do.call(rbind, Adat)
matplot(Adat[, 1], -Adat[, -1]/1000, type = "l", lty = 1, lwd = 2,
  xlab = axis.label("CO2"), ylab = axis.label("DG", prefix = "k"))
legend("topleft", c("acetate oxidation", "acetoclastic methanogenesis",
  "hydrogenotrophic methanogenesis"), lty = 1, col = 2:4)

Let’s not forget to clear the system settings, which were modified by basis() and E.units(), before running other calculations:

data(thermo)

Species of interest

By formation reaction is meant the stoichiometric requirements for formation of one mole of any species from the basis species. The species() function is used to set these species of interest. Let’s consider the stoichiometry of some aqueous sulfur-bearing species. Here we use basis() with a keyword to load a preset basis definition. ⊕Some available keywords are CHNOS (including CO₂, H₂O, NH₃, H₂S, and O₂), CHNOS+ (also including H⁺), and CHNOSe (including H⁺, and e^- instead of O₂). See ?basis for more options. ⊕What is SO42-? Is it 1 S, 4 O, and 2 negative charges, or 1 S, 42 O, and 1 negative charge? The ambiguity of a digit that could belong to the coefficient for the following charge or to that for the preceding element is why formulas in CHNOSZ are written with the number of charges after the + or - symbol. SO4-2 is unambiguously parsed as 1 S, 4 O and 2 negative charges.

basis("CHNOS+")

species(c("H2S", "HS-", "HSO4-", "SO4-2"))

##   CO2 H2O NH3 H2S O2 H+ ispecies logact state  name
## 1   0   0   0   1  0  0       67     -3    aq   H2S
## 2   0   0   0   1  0 -1       22     -3    aq   HS-
## 3   0   0   0   1  2 -1       25     -3    aq HSO4-
## 4   0   0   0   1  2 -2       24     -3    aq SO4-2

Aqueous species are assigned default activities of 10^-3 (logact is -3). Now, we can use affinity() to calculate the affinities of the formation reactions of each of the species. R’s unlist() is used here to turn the list of values of affinity into a numeric object that can be printed in a couple of lines (note that the names correspond to ispecies above): ⊕The values returned by affinity() are dimensionless, i.e. A/(2.303RT).

unlist(affinity()$values)

## energy.args: temperature is 25 C

## energy.args: pressure is Psat

## subcrt: 10 species at 298.15 K and 1 bar (wet)

##        67        22        25        24 
##  -4.00000  -3.98775 -29.48836 -24.46748

The same result (but expressed in units of cal/mol or J/mol) could be obtained using subcrt(); however, affinity() has the advantage of being able to perform calculations on a grid of T, P, or activities of basis species. Let’s choose a set of variables commonly used in aqueous speciation diagrams: Eh and pH. To use Eh as a variable, the electron (e^-) should be in the basis. To put the electron in there, we can use a different keyword (basis("CHNOSe")), or swap oxygen out of the existing basis:

swap.basis("O2", "e-")

## subcrt: 6 species at 298.15 K and 1 bar (wet)
## subcrt: 6 species at 298.15 K and 1 bar (wet)

##     C H N O S  Z ispecies   logact state
## CO2 1 0 0 2 0  0     1576 -3.00000    aq
## H2O 0 2 0 1 0  0        1  0.00000   liq
## NH3 0 3 1 0 0  0       66 -4.00000    aq
## H2S 0 2 0 0 1  0       67 -7.00000    aq
## e-  0 0 0 0 0 -1        2  6.22377    aq
## H+  0 1 0 0 0  1        3 -7.00000    aq

swap.basis() changed the basis species and recalculated their activities, but preserved the species of interest. ⊕That is, running affinity()$values again would give the same result.

Aqueous sulfur species at 25 °C.

Now we can calculate the affinities on an Eh-pH grid:

a <- affinity(pH = c(0, 12), Eh = c(-0.5, 1))

Potential diagrams

Given values of affinity, the diagram() function uses the maximum affinity method to make a potential diagram (i.e. a Pourbaix diagram). Areas corresponding to Eh-pH conditions beyond the stability limits of water are colored slate gray. Another function, water.lines(), is used to draw lines at the water stability limits:

diagram(a, fill = "heat")
water.lines(a)

Note that the calculation of affinity implies a non-equilibrium reference state of equal activities of species (see above). Generally, then, diagram() gives a potential diagram because it shows regions of maximum affinity. In systems where equilibrium is attainable, it makes sense to call this a predominance diagram, showing regions of maximum activity.

The same plot, with different colors and labels.

The names of species that can be parsed as chemical formulas are formatted with subscripts and superscripts; if this is not desired, set format.names = FALSE. The default colors for diagrams shown on the screen use R’s heat.colors() palette. Some arguments in diagram() can be used to control the color, labels, and lines, and title. The tplot argument turns off plot customizations used in CHNOSZ. Additional arguments are passed to R’s plotting functions; here, we use bty to remove the box around the plot.

diagram(a, fill = "terrain", lwd = 3, lty = 3,
        names = c("hydrogen sulfide", "bisulfide", "bisulfate", "sulfate"),
        tplot = FALSE, main = "sulfur species, 25 °C", bty = "n")

Mineral stability diagrams often depict activity ratios, e.g. log (aCa+2/aH+2), on one or both axes. The variables used for potential calculations in CHNOSZ include only a single chemical activity, e.g. log aCa+2. However, you can set pH = 0 to generate diagrams that are geometrically equivalent to those calculated using activity ratios, and use ratlab() to make the axes labels for the ratios. See demo(activity_ratios) for some examples.

Mosaic diagrams

If sulfur is in the basis species, then we should consider that its speciation is sensitive to Eh and pH, as shown in the preceding diagram. Mosaic diagrams, which are often shown for oxide, sulfide, and carbonate minerals, account for speciation of the basis species. These diagrams are made by constructing individual diagrams for the possible basis species. The individual diagrams are then combined, each one contributing to the final diagram only in the range of stability of the corresponding basis species.

Let’s use mosaic() to make a diagram for aqueous species and minerals in the Cu-S-Cl-H₂O system, similar to Figure 5a of Caporuscio et al. (⊕2017Caporuscio FA, Palaich SEM, Cheshire MC, Jové Colón CF. 2017. Corrosion of copper and authigenic sulfide mineral growth in hydrothermal bentonite experiments. Journal of Nuclear Materials 485: 137–146. doi: 10.1016/j.jnucmat.2016.12.036). To know what aqueous copper chloride complexes are available in the database, we can use a fuzzy search:

info(" CuCl")

## info.approx: ' CuCl' is ambiguous; has approximate matches to 7 species:

## [1] "CuCl"    "CuCl2-"  "CuCl3-2" "CuCl+"   "CuCl2"   "CuCl3-"  "CuCl4-2"

We wish to include chalcocite (Cu₂S) in the system. This mineral undergoes phase transitions; to find out the temperatures of the phase transitions, we can also use info():

info(info("chalcocite", c("cr", "cr2", "cr3")))$T

## [1]  376  717 1400

Those are temperatures in Kelvin (regardless of the T.units()); at 200 °C we should use the second phase.

Next we define the basis, and set the activities of the H₂S and Cl^- basis species. These represent the total activity of S and Cl in the system, which are distributed among the minerals and aqueous species. Three minerals and the aqueous copper chloride species are included:

basis(c("Cu", "H2S", "Cl-", "H2O", "H+", "e-"))
basis("H2S", -6)
basis("Cl-", -0.7)
species(c("copper", "tenorite"))
species("chalcocite", "cr2")
species(c("CuCl", "CuCl2-", "CuCl3-2", "CuCl+", "CuCl2", "CuCl3-", "CuCl4-2"))

We use mosaic() to generate and combine diagrams for each candidate basis species (H₂S, HS^-, HSO₄^-, or SO₄^-2) as a function of Eh and pH. The key argument is bases, which identifies the candidate basis species, starting with the one in the current basis. The other arguments, like those of affinity(), specify the ranges of the variables; res indicates the grid resolution to use for each variable (the default is 128). The first call to diagram() plots the species of interest; the second adds the predominance fields of the basis species. We turn off the gray coloring beyond the water stability limits (limit.water) but plot the red dotted lines using water.lines():

Copper minerals and aqueous complexes with chloride, 200 °C.

T <- 200
res <- 200
bases <- c("H2S", "HS-", "HSO4-", "SO4-2")
m1 <- mosaic(bases, blend = TRUE, pH = c(0, 12, res), Eh=c(-1.2, 0.75, res), T=T)
diagram(m1$A.species, lwd = 2, fill = NA, limit.water = FALSE)
diagram(m1$A.bases, add = TRUE, col = "blue", col.names = "blue", lty = 2,
        limit.water = FALSE)
water.lines(m1$A.species, col = "red", lwd = 2, lty = 3)

The argument blend = TRUE is used to combine the diagrams according to the equilibrium activities of the basis species by themselves (see below). The smooth transitions between basis species cause the appearance of curved lines on the plot. Without that argument, the diagrams would be combined using the dominant basis species, and all of the line segments would be straight.

We have seen the effects of speciation of S in the basis species. However, the choice of other basis species can also affect the diagram. For instance, we can use H₂ or O₂ in place of e^-. To do that, let’s write a function to swap those basis species and make a diagram. We use R’s do.call() to construct the argument list for mosaic(); this way, the name of the newvar argument to our function indicates the chosen variable.

mosaicfun <- function(newvar, T = 200) {
  swap.basis("e-", names(newvar))
  if (names(newvar) == "O2") basis("O2", "gas")
  mosaicargs <- c(list(bases), blend=TRUE, pH=list(c(-2, 12, res)), newvar, T=T)
  m1 <- do.call(mosaic, mosaicargs)
  diagram(m1$A.species, lwd = 2, fill = rev(topo.colors(10)),
          limit.water = FALSE)
  diagram(m1$A.bases, add = TRUE, col = "blue", col.names = "blue", lty = 3,
          limit.water = FALSE)
  water.lines(m1$A.species, col = "red", lwd = 2, lty = 3)
  swap.basis(names(newvar), "e-")
}
par(mfrow = c(1, 3))
mosaicfun(list(Eh = c(-1, 1, res)))
mosaicfun(list(H2 = c(-30, 10, res)))
mosaicfun(list(O2 = c(-70, 5, res)))

The same chemical system projected into different sets of basis species.

T, P, activity transects

Above, we used evenly-spaced grids of chemical activities of basis species; the ranges of variables were given by two or three values (minimum, maximum, and optionally resolution). affinity() can also perform calculations along a transect, i.e. a particular path along one or more variables. A transect is calculated when there are four or more values assigned to the variable(s). Let’s use this feature to calculate affinities (negative Gibbs energies) of methanogenesis and biosynthetic reactions in a hydrothermal system.

Some results of mixing calculations for seawater and vent fluid from the Rainbow hydrothermal field, calculated using EQ3/6 by Shock and Canovas (⊕2010Shock E, Canovas P. 2010. The potential for abiotic organic synthesis and biosynthesis at seafloor hydrothermal systems. Geofluids 10(1-2): 161–192. doi: 10.1111/j.1468-8123.2010.00277.x), are included in a data file in CHNOSZ. Reading the file with R’s read.csv(), we set check.names = FALSE to preserve the NH4+ column name (which is not a syntactically valid variable name):

file <- system.file("extdata/cpetc/SC10_Rainbow.csv", package = "CHNOSZ")
rb <- read.csv(file, check.names = FALSE)

We take a selection of the species from Shock and Canovas (2010) with activities equal to 10^-6; methane is assigned an activity of 10^-3. We will write the synthesis reactions of organic species in terms of these basis species: ⊕The constant activity of methane is a simplification of the calculation reported by Shock and Canovas (2010). The code here could be expanded to vary the activity of methane.

basis(c("CO2", "H2", "NH4+", "H2O", "H2S", "H+"))
species("CH4", -3)
species(c("adenine", "cytosine", "aspartic acid", "deoxyribose",
          "methane", "leucine", "tryptophan", "n-nonanoic acid"), -6)

Now we can calculate affinities along the transect of changing temperature and activities of five basis species. Each variable is given as a named argument; NH4+ must be quoted. ⊕A shorter expression would use R’s do.call() to construct the function call: do.call(affinity, as.list(rb)). ⊕The target of the conversion is G, or free energy, from logK. That conversion requires temperature in Kelvin, which is obtained by conversion from °C. We finish with a negation (affinity is negative Gibbs energy) and scaling from cal to kcal. Using R’s lapply() to run convert() for each species, we convert the affinity from dimensionless values (A/(2.303RT)) to cal/mol, then kcal/mol.

a <- affinity(T = rb$T, CO2 = rb$CO2, H2 = rb$H2,
              `NH4+` = rb$`NH4+`, H2S = rb$H2S, pH = rb$pH)
T <- convert(a$vals[[1]], "K")
a$values <- lapply(a$values, convert, "G", T)
a$values <- lapply(a$values, `*`, -0.001)

Affinities of organic synthesis in a hydrothermal system, after Shock and Canovas (2010).

Finally, we use diagram() to plot the results. Although only temperature is shown on the x axis, pH and the activities of CO₂, H₂, NH₄⁺, and H₂S are also varied according to the data in rb. By default, diagram() attempts to scale the affinities by dividing by the reaction coefficients of a shared basis species (in this case, CO₂). To override that behavior, we set balance = 1 to plot the affinities of the formation reactions as written (per mole of the product species).

diagram(a, balance = 1, ylim = c(-100, 100), ylab = axis.label("A", prefix="k"),
        col = rainbow(8), lwd = 2, bg = "slategray3")
abline(h = 0, lty = 2, lwd = 2)

When making line plots, diagram() automatically places the labels near the lines. The additional arguments adj and dy can be used to fine-tune the positions of the labels (they are used in a couple of examples below). If labeling of the lines is not desired, add e.g. legend.x = "topright" to make a legend instead, or names = NULL to prevent any plotting of the names.

Buffers

There is one other feature of affinity() that can be mentioned here. Can we go the other direction: calculate the activities of basis species from the activities of the species of interest? This question relates to the concept of chemical activity buffers. In CHNOSZ there are two ways to perform buffer calculations:

1. Assign the name of a buffer (listed in thermo$buffer) to the basis species:

• more versatile (multiple activities can be buffered, e.g. both S₂ and O₂ by pyrite-pyrrhotite-magnetite);
• the buffers are active in calculations of affinity of other species;
• use mod.buffer() to change or add buffers in thermo$buffer;
• demo(buffer) uses it for mineral buffers (solid lines).

2. Use the what argument of diagram() to solve for the activity of the indicated basis species:

• more convenient (the buffers come from the currently defined species of interest), but only a single basis species can be buffered, and it’s not used in the calculation of affinity;
• demo(buffer) uses it for aqueous organic species as buffers (dotted and dashed lines).

As an example of method 1, let’s look at the pyrite-pyrrhotite-magnetite (PPM) buffer at 300 °C. ⊕For other examples, see ?buffer and demo(protbuff) (hypothetical buffer made of proteins). Without the buffer, the basis species have default activities of zero. Under these conditions, the minerals are not in equilibrium, as shown by their different affinities of formation:

basis(c("FeS2", "H2S", "O2", "H2O"))
species(c("pyrite", "magnetite"))
species("pyrrhotite", "cr2")

⊕The affinity of formation of pyrite happens to be zero because it is identical to one of the selected basis species.

unlist(affinity(T = 300, P = 100)$values)

##     2113     2081     2118 
##   0.0000 -50.6501 -20.6134

We use mod.buffer() to choose the cr2 phase of pyrrhotite, which is stable at this temperature (see above for how to get this information for minerals with phase transitions). Then, we set up H₂S and O₂ to be buffered by PPM, and inspect their buffered activities:

mod.buffer("PPM", "pyrrhotite", "cr2")

## mod.buffer: changed state and/or logact of pyrrhotite in PPM buffer

basis(c("H2S", "O2"), c("PPM", "PPM"))

unlist(affinity(T = 300, P = 100, return.buffer = TRUE)[1:3])

##       H2S        O2      FeS2 
##  -2.35581 -36.51522   0.00000

Values of logf_H2 corresponding to mineral buffers or to given activities of aqueous species.

Et voilà! We have found loga_H2S and logf_O2 that are compatible with the coexistence of the three minerals. Under these conditions, the affinities of formation reactions of the minerals in the buffer are all equal to zero:

unlist(affinity(T = 300, P = 100)$values)
## 2031 1999 2036 
##    0    0    0

Another example, based on Figure 6 of Schulte and Shock (⊕1995Schulte MD, Shock EL. 1995. Thermodynamics of Strecker synthesis in hydrothermal systems. Origins of Life and Evolution of the Biosphere 25(1-3): 161–173. doi: 10.1007/BF01581580), is given in demo(buffer). Here, values of logf_H2 buffered by minerals or set by equilibrium with given activities of aqueous species are calculated using the two methods:

demo(buffer)

Getting from affinity to equilibrium

The equilibrate() function in CHNOSZ automatically chooses between two methods for calculating equilibrium. ⊕For more information, see the vignette Equilibrium in CHNOSZ. The method based on the Boltzmann equation is fast, but is applicable only to systems where the coefficient on the balanced basis species in each of the formation reactions is one. The reaction-matrix method is slower, but can be applied to systems were the balanced basis species has reaction coefficients other than one.

Three views of carbonate speciation: affinity, activity, degree of formation.

The distribution of aqueous carbonate species as a function of pH (a type of Bjerrum plot) is a classic example of an equilibrium calculation. We can begin by plotting the affinities of formation, for equal activities of the species, calculated at 25 °C and 150 °C. Here, CO₂ is in the basis, so it has zero affinity, which is greater than the affinities of HCO₃^- and CO₃^-2 at low pH. To avoid overplotting the lines, we offset the species labels in the y direction using the dy argument:

par(mfrow = c(3, 1))
basis("CHNOS+")
species(c("CO2", "HCO3-", "CO3-2"))
a25 <- affinity(pH = c(4, 13))
a150 <- affinity(pH = c(4, 13), T = 150)
diagram(a25, dy = 0.4)
diagram(a150, add = TRUE, col = "red")

Now we use equilibrate() to calculate the activities of species. Our balancing constraint is that the total activity of C is 10^-3. This shows a hypothetical metastable equilibrium; we know that for true equilibrium the total activity of C is affected by pH.

e25 <- equilibrate(a25, loga.balance = -3)
e150 <- equilibrate(a150, loga.balance = -3)
diagram(e25, ylim = c(-6, 0), dy = 0.15)
diagram(e150, add = TRUE, col = "red")

To display the species distribution, or degree of formation, add alpha = TRUE to the argument list:

diagram(e25, alpha = TRUE, dy = -0.25)
diagram(e150, alpha = TRUE, add = TRUE, col = "red")

The possible reactions between species are all balanced on 1 C. Therefore, although pH alters the total activity of C, in a system with ideal mixing the total activity of C doesn’t affect the relative activities of these species. ⊕See demo(solubility) for calculations of the total activity of C in this ideal system; uncomment a line in the demo to run calculations for CO₂ instead of calcite.

Groups of species

Sometimes it is helpful to look at the summed activities of species as groups on species distribution diagrams. The groups argument of diagram() can be used to sum the activities of species.

To demonstrate this feature, let’s consider the distribution of carbon among organic and inorganic species in the hydrothermal mixing scenario described by Shock and Schulte (⊕1998Shock EL, Schulte MD. 1998. Organic synthesis during fluid mixing in hydrothermal systems. Journal of Geophysical Research 103(E12): 28513–28527. doi: 10.1029/98JE02142). First we define the basis and add two inorganic species. The index.return = TRUE argument tells info() to return the index (number) of the species in the current species definition; these indices are saved for use below:

basis("CHNOS+")
ii <- species(c("CO2", "HCO3-"), index.return = TRUE)

Next, we add each group of organic species: C₁–C₈ alcohols, C₃–C₈ ketones, C₂–C₁₂ carboxylic acids and their corresponding anions, and C₂–C₈ alkenes. To do this, we provide info() with a set of ispecies values that identify these species. In the database, the species in each group are ordered by carbon number, so we construct a sequence from the starting to ending ispecies for each group using R’s seq() function, wrapped by the seq2() function we write here to make the code shorter:

seq2 <- function(x) seq(x[1], x[2])
ia <- species(seq2(info(c("methanol", "octanol"))), index.return = TRUE)
ik <- species(seq2(info(c("acetone", "2-octanone"))), index.return = TRUE)
ic <- species(seq2(info(c("acetic acid","n-dodecanoic acid"))),index.return=TRUE)
ica <- species(seq2(info(c("acetate", "n-dodecanoate"))), index.return = TRUE)
ie <- species(seq2(info(c("ethylene", "octene"))), index.return = TRUE)

Now we read two data files that contain values of logf_O2 and pH as a function of temperature, digitized from Figure 5 of Shock and Schulte (1998). ⊕The specific values are for calculations with vent fluids initially set by the fayalite-magnetite-quartz buffer minus 1/2 logf_O2 (FMQ - 1/2). These values were calculated using a speciation and mixing model that is not available in CHNOSZ; however, we can use these intermediate values as input to the “downstream” calculations that are available in CHNOSZ. Because of the noise introduced by digitization of the figure, we smooth the data using R’s smooth.spline(); the lower T limit reflects the absence of data below this temperature in the figure for logf_O2.

O2dat <- read.csv(system.file(
  "extdata/cpetc/SS98_Fig5a.csv", package = "CHNOSZ"))
pHdat <- read.csv(system.file(
  "extdata/cpetc/SS98_Fig5b.csv", package = "CHNOSZ"))
T <- seq(8, 350)
O2 <- predict(smooth.spline(O2dat$T, O2dat$logfO2), T)$y
pH <- predict(smooth.spline(pHdat$T, pHdat$pH), T)$y

We are ready to calculate affinities and equilibrium activities of the species. This calculation utilizes the transect mode of affinity() (see above). The call to equilibrate() runs with the default balance (in this case, CO₂), with a log activity set to -2.5. ⊕Actually, the total concentration of carbon depends on the mixing ratio, ranging from about 10^-2.2 (seawater) to 10^-2.6 (vent fluid). A setting to vary the activity of the balanced basis species is not yet implemented in CHNOSZ, so a single value is used here.

a <- affinity(T = T, O2 = O2, pH = pH)
e <- equilibrate(a, loga.balance = -2.5)

At last we come to the diagram. The groups are identified by the current species numbers in a list; the elements of the list are given names that will appear on the diagram. When summing the activities of species in the groups, each activity is multiplied first by the balance coefficient on that species. Therefore, the total activity is that of a basis species (or of an element that is present only in that basis species, like carbon in this example).

par(mfrow = c(1, 3))
groups <- list(inorganic = ii, alcohols = ia, ketones = ik,
               `carboxylic acids` = c(ic, ica), alkenes = ie)
diagram(e, alpha = TRUE, groups = groups, col = 1:5)

That makes a diagram that is similar to Figure 6b of Shock and Schulte (1998). ⊕Some differences from the original diagrams could be caused by the sensitivity of the calculations to logf_O2, for which the values we use here may carry artifacts introduced by digitization of their plot. It is also possible to plot the distribution of species within individual groups, such as alcohols and ketones. We do this by setting the names and line types for the other species to values that prevent them from being plotted:

# plot only alcohols
names <- within(species(), name[-ia] <- "")$name
lty <- ifelse(names == "", 0, 1)
diagram(e, alpha = TRUE, ylim = c(0, 0.32), lty = lty, names = names)
# plot only ketones
names <- within(species(), name[-ik] <- "")$name
lty <- ifelse(names == "", 0, 1)
diagram(e, alpha = TRUE, ylim = c(0, 0.16), lty = lty, names = names)

Distribution of inorganic and groups of organic species (left plot) and of alcohols and ketones (middle and right plots) as a function of T, pH, and logf_O2.

Balancing differently

How about the choice between balancing constraints? Be default, equilibrate() and diagram() balance reactions on the first basis species that is present in each of the species of interest. Let’s look at some amino acids in a hypothetical metastable equilibrium. This calculation is loosely based on one described by Shock (⊕1990Shock EL. 1990. Do amino acids equilibrate in hydrothermal fluids? Geochimica et Cosmochimica Acta 54(4): 1185–1189. doi: 10.1016/0016-7037(90)90450-Y) for five amino acids. Here we include 20 proteinogenic amino acids, whose names are returned by aminoacids(""). We use ZC.col() to generate colors based on the average oxidation state of carbon of the amino acids (red and blue for relatively reduced and oxidized).

basis("CHNOS")
basis("CO2", "gas")
swap.basis("NH3", "N2")
species(aminoacids(""))
a <- affinity(O2 = c(-50, -25, 200), CO2 = c(-10, 15, 200), T = 250, P = 265)
aa.ZC <- ZC(info(aminoacids("")))
col <- ZC.col(aa.ZC)

To make plots using different balance constraints, let’s write a function that sets the balance argument of diagram() and adds a title to the plot. The first plot is the most similar to Figure 4 of Shock (1990), except for the absence of alanine (probably due to different thermodynamic data) and the presence of some other amino acids. There, we set balance = 1, which indicates that moles of species are conserved; this is equivalent to balancing on the amino acid backbone. In the remaining plots, the balance is set to each of the basis species in turn (except for O₂), then on volume. expr.species() together with R’s substitute() is used to make titles that include formatted chemical formulas:

aafun <- function(balance) {
  diagram(a, balance = balance, fill = col)
  blab <- expr.species(balance)
  title(main = substitute("balanced on" ~ b, list(b = blab)))
}
par(mfrow = c(1, 5))
lapply(c("1", "CO2", "H2O", "N2", "volume"), aafun)

Plots of maximum affinity at 250 °C and 265 bar using different reaction balances for 20 amino acids.

There are some broad similarities—increasing logf_O2 favors more oxidized amino acids—but also substantial differences. It is interesting that there is more “going on” in the middle part of the diagram showing volume conservation.

Caveat lector. These plots demonstrate some possibilities in CHNOSZ and are not necessarily realistic portrayals of this system. It does seem odd to balance on a fugacious component like O₂ or H₂O. Unlike different choices of basis species, which are thermodynamically equivalent (see above), the choice of balance reflects extra-thermodynamic factors. For instance, the widespread rule of thumb for balancing mineral reactions on a chemical component is unrealistic for processes where volume is conserved (Merino and Dewers, ⊕1998Merino E, Dewers T. 1998. Implications of replacement for reaction–transport modeling. Journal of Hydrology 209(1–4): 137–146. doi: 10.1016/S0022-1694(98)00150-4). While choosing an inappropriate balance leads to infeasible models, consideration of the different possibilities might give insight into the conditions affecting the dynamics of some systems.

Transformation of variables

Following the main workflow of CHNOSZ, nonideal() normally does not need to be used directly. Intead, invoke the calculations by setting the IS argument in subcrt() or affinity(). There are a few things to remember when using activity coefficients:

• H⁺ is assumed to behave ideally, so its activity coefficient is 1 for any ionic strength. You can calculate activity coefficients of H⁺ by setting thermo$opt$ideal.H <<- FALSE.

• Using subcrt() with IS not equal to zero, calculated values of G are the adjusted Gibbs energy at specified ionic strength (denoted by ΔG°(I); Alberty, ⊕1996Alberty RA. 1996. Recommendations for nomenclature and tables in biochemical thermodynamics. European Journal of Biochemistry 240(1): 1–14. Blackwell Science Ltd. doi: 10.1111/j.1432-1033.1996.0001h.x).

• Using subcrt() with IS not equal to zero, values in the logact argument stand for log molality of aqueous species in affinity calculations.

• Using affinity() with IS not equal to zero, the following values stand for log molality of aqueous species:
  • values of logact set by basis();
  • values of logact set by species().
• Using equilibrate() on the output of affinity calculated with IS not equal to zero, the following values stand for log molality of aqueous species:
  • the value of loga.balance used by equilibrate() (i.e., logarithm of total molality of the balancing basis species);
  • values of loga.equil returned by equilibrate().

In other words, the activation of activity coefficients effects a transformation from activity to molality in the main workflow. A simple but comprehensive series of calculations demonstrating these tranformations is in tests/testthat/test-logmolality.R.

Because it is not possible to dynamically change the names of arguments in the functions, the user should be aware of the transformations mentioned above. However, the labels on diagrams can be automatically adjusted; accordingly, activities of aqueous species are relabeled as molalities by diagram() when IS is used in the calculation of affinity().

Biochemical example

For the following calculations, we change the nonideality method to Alberty; this is a simpler formulation with parameters that are suitable for biochemical species at relatively low temperatures:

oldnon <- nonideal("Alberty")

## nonideal: setting nonideal option to use Alberty method

Let’s take a look at calculated activity coefficients at two temperatures and their effect on the standard Gibbs energies of formation (ΔG°_f) of species with different charge:

subcrt(c("MgATP-2", "MgHATP-", "MgH2ATP"),
       T = c(25, 100), IS = c(0, 0.25), property = "G")$out

## subcrt: 3 species at 2 values of T and P (wet)

## nonideal: calculated activity coefficients for 2 species (Alberty method)

## $`MgATP-2`
##     T       P       G    loggam   IS
## 1  25 1.00000 -773609  0.000000 0.00
## 2 100 1.01322 -780416 -0.656185 0.25
## 
## $`MgHATP-`
##     T       P       G    loggam   IS
## 1  25 1.00000 -780994  0.000000 0.00
## 2 100 1.01322 -789617 -0.164046 0.25
## 
## $MgH2ATP
##     T       P       G
## 1  25 1.00000 -786200
## 2 100 1.01322 -795012

The logarithms of the activity coefficients (loggam) are more negative for the higher-charged species, as well as at higher temperature, and have a stabilizing effect. That is, the adjusted Gibbs energies at I > 0 are less than the standard Gibbs energies at I = 0.

We can use these calculations to make some speciation plots, similar to Figures 1.2–1.5 in Alberty (2003). These figures show the distribution of differently charged species of adenosine triphosphate (ATP) as a function of pH, and the average number of H⁺ and Mg⁺² bound to ATP in solution as a function of pH or pMg (-loga_Mg⁺²).

Use info() to see what ATP species are available. The sources of high-temperature thermodynamic data for these species are two papers by LaRowe and Helgeson (⊕2006aLaRowe DE, Helgeson HC. 2006a. Biomolecules in hydrothermal systems: Calculation of the standard molal thermodynamic properties of nucleic-acid bases, nucleosides, and nucleotides at elevated temperatures and pressures. Geochimica et Cosmochimica Acta 70(18): 4680–4724. doi: 10.1016/j.gca.2006.04.010; ⊕2006bLaRowe DE, Helgeson HC. 2006b. The energetics of metabolism in hydrothermal systems: Calculation of the standard molal thermodynamic properties of magnesium-complexed adenosine nucleotides and NAD and NADP at elevated temperatures and pressures. Thermochimica Acta 448(2): 82–106. doi: 10.1016/j.tca.2006.06.008).

info(" ATP")

## info.approx: ' ATP' is ambiguous; has approximate matches to 14 species:

##  [1] "ATP-4"    "HATP-3"   "H2ATP-2"  "H3ATP-"   "H4ATP"    "dATP-4"  
##  [7] "dHATP-3"  "dH2ATP-2" "dH3ATP-"  "dH4ATP"   "MgATP-2"  "MgHATP-" 
## [13] "MgH2ATP"  "Mg2ATP"

The plots for this system in Alberty’s book were made for I = 0.25 M and T = 25 °C. As a demonstration of CHNOSZ’s capabilities, we can assign a temperature of 100 °C.

T <- 100

Use the following commands to set the basis species, add the variously protonated ATP species, calculate the affinities of the formation reactions, equilibrate the system, and make a degree of formation (α) or mole fraction diagram. This is similar to Figure 1.3 of Alberty (2003), but is calculated for I = 0 M and T = 100 °C: ⊕To make the code more readable, commands for plotting titles and legends are not shown. All of the commands are available in the source of this document.

basis("MgCHNOPS+")
species(c("ATP-4", "HATP-3", "H2ATP-2", "H3ATP-", "H4ATP"))
a <- affinity(pH = c(3, 9), T = T)
e <- equilibrate(a)
d <- diagram(e, alpha = TRUE, tplot = FALSE)

Note that we have saved the numeric results of diagram(), i.e. the degrees of formation of the species (α). With that, we can calculate and plot the average number of protons bound per ATP molecule. To do so, we use R’s rbind() and do.call() to turn alpha into a matrix, then multiply by the number of protons bound to each species, and sum the columns to get the total (i.e. average proton number, N_H⁺):

alphas <- do.call(rbind, d$plotvals)
nH <- alphas * 0:4
Hlab <- substitute(italic(N)[H^`+`])
plot(a$vals[[1]], colSums(nH), type = "l", xlab = "pH", ylab=Hlab, lty=2, col=2)

Adding the IS argument to affinity(), we can now plot N_H⁺ at the given ionic strength. Here we set plot.it = FALSE in diagram() because we use the computed α to make our own plot. This is similar to Figure 1.3 of Alberty (2003), but at higher temperature:

a <- affinity(pH = c(3, 9), IS = 0.25, T = T)
e <- equilibrate(a)
d <- diagram(e, alpha = TRUE, plot.it = FALSE)
alphas <- do.call(rbind, d$plotvals)
nH <- alphas * 0:4
lines(a$vals[[1]], colSums(nH))

Next, we add the Mg⁺²-complexed ATP species:

species(c("MgATP-2", "MgHATP-", "MgH2ATP", "Mg2ATP"))

Here is a function to calculate and plot N_H⁺ for a given pMg:

Hplot <- function(pMg, IS = 0.25) {
  basis("Mg+2", -pMg)
  a <- affinity(pH = c(3, 9), IS = IS, T = T)
  e <- equilibrate(a)
  d <- diagram(e, alpha = TRUE, plot.it = FALSE)
  alphas <- do.call(rbind, d$plotvals)
  NH <- alphas * c(0:4, 0, 1, 2, 0)
  lines(a$vals[[1]], colSums(NH), lty = 7 - pMg, col = 7 - pMg)
}

With that function in hand, we plot the lines corresponding to pMg = 2 to 6. This is similar to Figure 1.4 of Alberty (2003):

plot(c(3, 9), c(0, 2), type = "n", xlab = "pH", ylab = Hlab)
lapply(2:6, Hplot)

The next function calculates and plots the average number of Mg⁺² bound to ATP (N_Mg⁺²) for a given pH. Here we multiply alpha by the number of Mg⁺² in each species, and negate loga_Mg⁺² (the variable used in affinity()) to get pMg:

Mgplot <- function(pH, IS = 0.25) {
  basis("pH", pH)
  a <- affinity(`Mg+2` = c(-2, -7), IS = IS, T = T)
  e <- equilibrate(a)
  d <- diagram(e, alpha = TRUE, plot.it = FALSE)
  alphas <- do.call(rbind, d$plotvals)
  NMg <- alphas * species()$`Mg+`
  lines(-a$vals[[1]], colSums(NMg), lty = 10 - pH, col = 10 - pH)
}

Using that function, we plot the lines corresponding to pH = 3 to 9. This is similar to Figure 1.5 of Alberty (2003):

Mglab <- substitute(italic(N)[Mg^`+2`])
plot(c(2, 7), c(0, 1.2), type = "n", xlab = "pMg", ylab = Mglab)
lapply(3:9, Mgplot)

Binding of H⁺ and Mg⁺² to ATP at 100 °C and I = 0 M (first plot) or I = 0.25 M (third and fourth plots).

We have calculated the distribution of ATP species and average binding number of H⁺ and Mg⁺² for given pH, pMg, ionic strength, and temperature. Accounting for the distribution of chemical species lends itself to thermodynamic models for reactions between reactants that have multiple ionized and complexed states. In contrast, Alberty (2003) and others propose models for biochemical reactions where the ionized and complexed species are combined into a single representation. Those models invoke Legendre-transformed thermodynamic properties, such as transformed Gibbs energies that are tabulated for specified pH, pMg, and ionic strength. Although the conceptual pathways are different, the two approaches lead to equivalent results concerning the energetics of the overall reactions and the conditions for equilibrium (Sabatini et al., ⊕2012Sabatini A, Vacca A, Iotti S. 2012. Balanced biochemical reactions: A new approach to unify chemical and biochemical thermodynamics. PLoS ONE 7(1): e29529. doi: 10.1371/journal.pone.0029529). The example here shows how the required calculations can be performed at the species level using conventional standard Gibbs energies for species referenced to infinite dilution (zero ionic strength). The effects of ionic strength are modeled “on the fly” in CHNOSZ by setting the IS argument in subcrt() or affinity() to invoke the nonideality model on top of the standard Gibbs energies of species.

Now that we’re finished, we can reset the nonideality method to the default. (This really isn’t needed here, because there aren’t any nonideality calculations below):

nonideal(oldnon)

## nonideal: setting nonideal option to use Helgeson method

Group additivity and ionization

The group additivity calculations for proteins are based on equations and data from Amend and Helgeson (⊕2000Amend JP, Helgeson HC. 2000. Calculation of the standard molal thermodynamic properties of aqueous biomolecules at elevated temperatures and pressures. II. Unfolded proteins. Biophysical Chemistry 84(2): 105–136. doi: 10.1016/S0301-4622(00)00116-2), Dick et al. (⊕2006Dick JM, LaRowe DE, Helgeson HC. 2006. Temperature, pressure, and electrochemical constraints on protein speciation: Group additivity calculation of the standard molal thermodynamic properties of ionized unfolded proteins. Biogeosciences 3(3): 311–336. doi: 10.5194/bg-3-311-2006), and LaRowe and Dick (⊕2012LaRowe DE, Dick JM. 2012. Calculation of the standard molal thermodynamic properties of crystalline peptides. Geochimica et Cosmochimica Acta 80: 70–91. doi: 10.1016/j.gca.2011.11.041). There are two major options for the calculations: whether to calculate properties for crystalline or aqueous groups, and, for the latter, whether to model the ionization of the sidechain and terminal groups as a function of pH (and T and P). By default, additivity of aqueous groups is used, but the ionization calculations are not available in subcrt():

subcrt("LYSC_CHICK")$out[[1]][1:6, ]

##        T       P      rho    logK        G         H       S       V      Cp
## 1   0.01 1.00000 0.999829 3217.68 -4021765 -10423733 3685.29 10049.2 4409.32
## 2  25.00 1.00000 0.997061 3019.80 -4119738 -10283083 4176.74 10421.0 6415.52
## 3  50.00 1.00000 0.988030 2861.40 -4230972 -10113250 4723.39 10600.2 7073.98
## 4  75.00 1.00000 0.974864 2734.31 -4355838  -9932209 5262.87 10708.1 7376.58
## 5 100.00 1.01322 0.958393 2632.00 -4493930  -9745475 5780.77 10782.9 7548.44
## 6 125.00 2.32014 0.939073 2549.29 -4644330  -9554920 6274.14 10840.9 7665.20

Let’s compare experimental values of heat capacity of four proteins, from Privalov and Makhatadze (⊕1990Privalov PL, Makhatadze GI. 1990. Heat capacity of proteins. II. Partial molar heat capacity of the unfolded polypeptide chain of proteins: Protein unfolding effects. Journal of Molecular Biology 213(2): 385–391. doi: 10.1016/S0022-2836(05)80198-6), with those calculated using group additivity. After dividing Privalov and Makhatadze’s experimental values by the lengths of the proteins to get per-residue values, we convert those to calories, then plot them.

The heat capacity calculated by group additivity closely approximates experimental values for aqueous proteins. For a related figure showing the effects of ionization in the calculations, see ?ionize.aa.

PM90 <- read.csv(system.file("extdata/cpetc/PM90.csv", package = "CHNOSZ"))
plength <- protein.length(colnames(PM90)[2:5])
Cp_expt <- t(t(PM90[, 2:5]) / plength)
matplot(PM90[, 1], convert(Cp_expt, "cal"), type = "p", pch = 19,
        xlab = axis.label("T"), ylab = axis.label("Cp0"), ylim = c(28, 65))

The loop calculates the properties of each protein using group additivity with aqueous or crystalline groups, then plots the per-residue values.

for(i in 1:4) {
  pname <- colnames(Cp_expt)[i]
  aq <- subcrt(pname, "aq", T = seq(0, 150))$out[[1]]
  cr <- subcrt(pname, "cr", T = seq(0, 150))$out[[1]]
  lines(aq$T, aq$Cp / plength[i], col = i)
  lines(cr$T, cr$Cp / plength[i], col = i, lty = 2)
}
legend("right", legend = colnames(Cp_expt),
       col = 1:4, pch = 19, lty = 1, bty = "n", cex = 0.9)
legend("bottomright", legend = c("experimental", "calculated (aq)",
       "calculated (cr)"), lty = c(NA, 1, 2), pch = c(19, NA, NA), bty = "n")

Although subcrt() has no provision for protein ionization, the properties of ionization can be calculated via affinity(), which calls ionize.aa() if a charged species is in the basis. ⊕Whether to calculate properties using aqueous or crystalline groups is determined by the value of thermo$opt$state; if it is changed from its default of aq to cr, no ionization is possible. The following plot shows the calculated affinity of reaction between nonionized proteins and their ionized forms as a function of pH. Charged and uncharged sets of basis species are used to to activate and suppress the ionization calculations. The calculation of affinity for the ionized proteins returns multiple values (as a function of pH), but there is only one value of affinity returned for the nonionized proteins, so we need to use R’s as.numeric() to avoid subtracting nonconformable arrays:

Affinity of ionization of proteins. See demo(ionize) for ionization properties calculated as a function of temperature and pH.

ip <- pinfo(c("CYC_BOVIN", "LYSC_CHICK", "MYG_PHYCA", "RNAS1_BOVIN"))
basis("CHNOS+")
a_ion <- affinity(pH = c(0, 14), iprotein = ip)
basis("CHNOS")
a_nonion <- affinity(iprotein = ip)
plot(c(0, 14), c(50, 300), xlab = "pH", ylab = axis.label("A"), type = "n")
for(i in 1:4) {
  A_ion <- as.numeric(a_ion$values[[i]])
  A_nonion <- as.numeric(a_nonion$values[[i]])
  lines(a_ion$vals[[1]], A_ion - A_nonion, col=i)
}
legend("topright", legend = a_ion$species$name,
       col = 1:4, lty = 1, bty = "n", cex = 0.9)

The affinity is always positive, showing the strong energetic drive for ionization of proteins in aqueous solution. The degrees of ionization of amino and carboxyl groups increase at low and high pH, respectively, giving rise to the U-shaped lines.

There, we used the indices returned by pinfo() in the iprotein argument of affinity() to specify the proteins in the calculation. That approach utilizes some optimizations that can be realized due to group additivity, and is useful for calculations involving many proteins. An alternative, but slower, approach is to identify the proteins to species(); this produces results that are equivalent to using the iprotein argument:

species(c("CYC_BOVIN", "LYSC_CHICK", "MYG_PHYCA", "RNAS1_BOVIN"))
a_nonion_species <- affinity()

unlist(a_nonion_species$values)

##      3590      3589      3593      3595 
## -122.5026 -700.5444  -50.6508 -731.8601

⊕The ispecies index (top) refers to the rownumber of thermo$species; that is distinct from the iprotein index (bottom, printed as negative integers), which refers to the rownumber of thermo$protein.

unlist(a_nonion$values)

##        -5        -6        -7        -9 
## -122.5026 -700.5444  -50.6508 -731.8601

Compositional analysis

Functions in CHNOSZ make it easy to get the chemical formulas of proteins from their amino acid compositions. Calculations based on the formulas, such as the average oxidation state of carbon (Z_C), and the coefficients of basis species in formation reactions, are also available.

Let’s compare the Z_C of Rubisco with optimal growth temperature of organisms, as shown in Figure 6a of Dick (⊕2014Dick JM. 2014. Average oxidation state of carbon in proteins. Journal of the Royal Society Interface 11: 20131095. doi: 10.1098/rsif.2013.1095). First we read a CSV file with the IDs of the proteins and the optimal growth temperatures (T_opt); the midpoint of the range of T_opt is used for plotting. Then we use read.fasta() to read a FASTA file holding the amino acid sequences of the proteins; the function returns a data frame with the amino acid counts. To put the proteins in the right order, the IDs in the CSV file are matched to the names of the proteins in the FASTA file. Then, we calculate Z_C from the formulas of the proteins. Next, point symbols are assigned according to domain (Archaea, Bacteria, Eukaryota); numbers inside the symbols show the ordering of T_opt in three temperature ranges (0–35 °C, 37.5–60 °C, and 65–100 °C).

Average oxidation state of carbon in Rubisco compared with optimal growth temperature of organisms. This is an interactive image. Move the mouse over the points to show the names of the organisms, and click to open a reference in a new window. (Made with RSVGTipsDevice using code that can be found in the source of this document.)

datfile <- system.file("extdata/cpetc/rubisco.csv", package = "CHNOSZ")
fastafile <- system.file("extdata/fasta/rubisco.fasta", package = "CHNOSZ")
dat <- read.csv(datfile)
aa <- read.fasta(fastafile)
Topt <- (dat$T1 + dat$T2) / 2
idat <- match(dat$ID, substr(aa$protein, 4, 9))
aa <- aa[idat, ]
ZC <- ZC(protein.formula(aa))
pch <- match(dat$domain, c("E", "B", "A")) - 1
col <- match(dat$domain, c("A", "B", "E")) + 1
plot(Topt, ZC, pch = pch, cex = 2, col = col,
     xlab = expression(list(italic(T)[opt], degree*C)),
     ylab = expression(italic(Z)[C]))
text(Topt, ZC, rep(1:9, 3), cex = 0.8)
abline(v = c(36, 63), lty = 2, col = "grey")
legend("topright", legend = c("Archaea", "Bacteria", "Eukaryota"),
       pch = c(2, 1, 0), col = 2:4, pt.cex = 2)

Let’s look at the different ways of representing the chemical compositions of the proteins. protein.basis() returns the stoichiometry for the formation reaction of each proteins. Dividing by protein.length() gives the per-residue reaction coefficients (n̅). Using the set of basis species we have seen before (CO₂, NH₃, H₂S, H₂O, O₂) there is a noticeable correlation between n̅_O2 and Z_C, but even more so between n̅_H2O and Z_C (shown in the plots on the left). ⊕The calculation of Z_C, which sums elemental ratios, is not affected by the choice of basis species. The QEC keyword to basis() loads basis species including three amino acids (glutamine, glutamic acid, cysteine, H₂O, O₂). This basis strengthens the relationship between Z_C and n̅_O2, but weakens that between Z_C and n̅_H2O (shown in the plots on the right).

Compositions of proteins projected into different sets of basis species.

layout(matrix(1:4, nrow = 2))
par(mgp = c(1.8, 0.5, 0))
pl <- protein.length(aa)
ZClab <- axis.label("ZC")
nO2lab <- expression(bar(italic(n))[O[2]])
nH2Olab <- expression(bar(italic(n))[H[2]*O])
lapply(c("CHNOS", "QEC"), function(thisbasis) {
  basis(thisbasis)
  pb <- protein.basis(aa)
  nO2 <- pb[, "O2"] / pl
  plot(ZC, nO2, pch = pch, col = col, xlab = ZClab, ylab = nO2lab)
  nH2O <- pb[, "H2O"] / pl
  plot(ZC, nH2O, pch = pch, col = col, xlab = ZClab, ylab = nH2Olab)
  mtext(thisbasis, font = 2)
})

By projecting the compositions of proteins into the QEC set of basis species, n̅_O2 emerges as a strong indicator of oxidation state, while n̅_H2O is a relatively uncorrelated (i.e. independent) compositional variable. These independent variables make it easier to distinguish the effects of oxidation and hydration state in proteomic datasets (Dick, ⊕2017Dick JM. 2017. Chemical composition and the potential for proteomic transformation in cancer, hypoxia, and hyperosmotic stress. PeerJ 5: e3421. doi: 10.7717/peerj.3421).

Normalization to residues

As with other systems, a balance must be chosen for calculations of the metastable equilibrium distribution for proteins. Balancing on the number of backbone units (the sequence length) seems a reasonable choice given the polymeric structure of proteins. ⊕Balancing on one of the basis species remains a possibility, using the balance argument in equilibrate() or diagram(). However, there is an additional consideration: owing to the large size of proteins compared to the basis species, the distribution of proteins in metastable equilibrium has many orders of magnitude separation between the activities of the dominant and less-dominant proteins. The metastable coexistence of the residues (i.e. per-residue formulas, or residue equivalents) of the same proteins spans a much smaller range of chemical activities. In CHNOSZ, the calculation of metastable equilibrium activities of the residue equivalents is referred to as normalization. ⊕See the vignette Equilibrium in CHNOSZ for other examples using normalization.

To take an example, let’s look at the metastable equilibrium distribution of selected proteins in the ER-to-Golgi location of S. cerevisiae (yeast). This example brings in a few other functions we haven’t seen yet: yeastgfp(), unitize(), and revisit(). yeastgfp() queries a data file in CHNOSZ for the names and relative abundances of proteins taken from the YeastGFP study of Ghaemmaghami et al. (⊕2003Ghaemmaghami S, Huh W-K, Bower K, Howson RW, Belle A, Dephoure N, O’Shea EK, Weissman JS. 2003. Global analysis of protein expression in yeast. Nature 425(6959): 737–741. doi: 10.1038/nature02046). There are six proteins identified in the ER-to-Golgi location; one has NA abundance, so it is excluded from the comparisons:

y <- yeastgfp("ER.to.Golgi")
ina <- is.na(y$abundance)

Next, we get the amino acid compositions of the proteins and add them to thermo$protein:

aa <- yeast.aa(y$protein[!ina])
ip <- add.protein(aa)

The YeastGFP study reported absolute abundances of molecules, but the thermodynamic calculations give relative chemical activities of the proteins. In order to make a comparison between them, we use unitize() to scale the abundances or activities of proteins (in logarithmic units) such that the total abundance or activity of residue equivalents is unity. To do that, we must have the lengths of the proteins. Here, the first call to unitize() generates equal logarithms of activities of proteins for unit total activity of residues; this is used as the reference state for affinity(). The second call to unitize() scales the logarithms of experimental abundances for unit total activity of residues; this is used for comparison with the theoretical results:

pl <- protein.length(ip)
logact <- unitize(numeric(5), pl)
logabundance <- unitize(log10(y$abundance[!ina]), pl)

Now we can load the proteins and calculate their activities in metastable equilibrium as a function of logf_O2: ⊕The commented line uses mod.obigt() to revert the parameters of the methionine sidechain group to those present in older versions of CHNOSZ (Dick et al., 2006). The current database, with parameters set by data(thermo) and used here, contains updated group additivity parameters for methionine (LaRowe and Dick, 2012).

par(mfrow = c(1, 3))
basis("CHNOS+")
#mod.obigt("[Met]", G = -35245, H = -59310)
a <- affinity(O2 = c(-80, -73), iprotein = ip, loga.protein = logact)
e <- equilibrate(a)
diagram(e, ylim = c(-5, -2), col = 1:5, lwd = 2)

Whoa! The proteins look very non-coexistent in metastable equilibrium. We get a different view by considering per-residue rather than per-protein reactions, through the normalize argument for equilibrate(): ⊕The normalization step is followed by conversion of activities of residues to activities of proteins; that conversion can be skipped using the as.residue argument in equilibrate().

e <- equilibrate(a, normalize = TRUE)
diagram(e, ylim = c(-5, -2.5), col = 1:5, lwd = 2)
abline(h = logabundance, lty = 1:5, col = 1:5)

The experimental relative abundances are plotted as thin horizontal lines with the same style and color as the thicker curved lines calculated for metastable equilibrium. With the exception of YNL049C, the overlap between the calculations and experiments looks to be greatest near the middle-left part of the figure. The revisit() function (see below) offers some statistical and thermodynamic measures that can quantify this comparison. Here, we plot the information-theoretic free energy ΔG_inf (calculated from the relative entropy or Kullback–Leibler divergence):

revisit(e, "DGinf", logabundance)

ER-to-Golgi proteins: calculations without and with length normalization, and free energy difference between experimental and calculated abundances in metastable equilibrium with normalization.

The minimum free energy difference occurs near logf_O2 = -78. This agrees with the assessment shown in Figure 4 of Dick (⊕2009Dick JM. 2009. Calculation of the relative metastabilities of proteins in subcellular compartments of Saccharomyces cerevisiae. BMC Systems Biology 3: 75. doi: 10.1186/1752-0509-3-75) (note that the old parameters for the methionine sidechain group were used in that study).

An affinity baseline

Because affinities of proteins often vary strongly with oxygen fugacity and other variables, it can be helpful to express the values as differences from a baseline. The following example compares the affinities for formation of transcription factors involved in embryonic dorsal-ventral patterning with that of the morphogen, Sonic hedgehog (Shh), as a function of logf_O2 and loga_H2O (Dick, ⊕2015Dick JM. 2015. Chemical integration of proteins in signaling and development. bioRxiv. doi: 10.1101/015826). We first list the UniProt names of Shh and 10 transcription factors, and get the iprotein indices (rownumbers of thermo$protein):

pname <- c("SHH", "OLIG2", "NKX22", "FOXA2", "IRX3",
  "PAX6", "NKX62", "DBX1", "DBX2", "NKX61", "PAX7")
ip <- pinfo(pname, "HUMAN")

Next, set up the basis species:

basis("CHNOS")
basis("NH3", -7)

Now calculate the affinities of formation of the proteins from the basis species. The values chosen for logf_O2 and loga_H2O covary, so we are using the transect mode of affinity():

O2 <- seq(-70, -106, length.out = 50)
H2O <- seq(0.5, -5.5, length.out = 50)
a <- affinity(H2O = H2O, O2 = O2, iprotein = ip)

We would like to compare the affinities of the proteins on a per-residue scale. R’s lapply() could be used here, but to show the operation more clearly we use a for() loop:

pl <- protein.length(ip)
for(i in seq_along(a$values)) a$values[[i]] <- a$values[[i]] / pl[i]

Then, we calculate the relative affinities, using Shh as the baseline:

a.Shh <- a$values[[1]]
for(i in 1:length(a$values)) a$values[[i]] <- a$values[[i]] - a.Shh

Next we use diagram() to plot the affinities. We set balance = 1 to plot the values as they are—without that, diagram() divides the values by protein length, which we have already done! ⊕See demo(Shh) for a plot with more interpretive labels and comments. For this plot, we highlight and label the proteins with the highest relative affinity at some combination of logf_O2 and loga_H2O along the transect. Those proteins are Olig2, Irx3, Nkx6.2, Dbx1, and Shh (numbers 2, 5, 7, 8, 1 in the set we have identified). Here, adj = 0 makes the labels left-aligned, dy = 0.1 adds a y offset to the labels, and format.names = FALSE prevents formatting of the names as if they were chemical formulas (that causes subscripted numbers to appear). The last few lines are used to make a second x axis, using a label generated with axis.label():

Per-residue affinities for formation of transcription factors relative to Shh.

# line type, width, and color
twc <- lapply(c(3, 1, 1), rep, length(pname))
ihigh <- c(2, 5, 7, 8, 1)
twc[[1]][ihigh] <- 1
twc[[2]][ihigh] <- 3
col <- c("#f9a330", "#63c54e", "#f24e33", "#d4e94e", "#0f0f0f")
twc[[3]][ihigh] <- col
names <- rep("", length(pname))
names[ihigh] <- c("Olig2", "Irx3", "Nkx6.2", "Dbx1", "Shh")
ylab <- substitute(italic(A) / (2.303 * italic(RT)) * " relative to Shh")
diagram(a, balance = 1, ylim = c(-0.5, 5), xlim = c(0.5, -5.5),
  lty = twc[[1]], lwd=twc[[2]], col = twc[[3]], ylab = ylab,
  names = names, adj = 0, dy = 0.1, format.names = FALSE)
par(usr = c(-70, -106, -0.5, 5), tcl = -0.3)
axis(3, at = seq(-70, -106, by = -10))
mtext(axis.label("O2"), line = 1.2)

Among these proteins, Olig2 and Shh have the greatest affinities for formation at the extremes of oxidation and hydration state. This thermodynamic description highlights possible links between the chemical compositions of the proteins and their patterns of expression along with chemical changes in developing embryos (Dick, 2015).

Getting amino acid compositions

In the Rubisco example above, we saw the use of read.fasta() to read amino acid sequences from a FASTA file. There are several other methods for inputting amino acid compositions.

R’s read.csv() can be used to read amino acid compositions from a CSV file with the same columns that are present in thermo$protein. Note the use of as.is = TRUE to prevent reading character data as factors. The nrows argument can be added to read that number of rows:

file <- system.file("extdata/protein/DS11.csv", package = "CHNOSZ")
aa_bison <- read.csv(file, as.is = TRUE, nrows = 5)

yeast.aa() retrieves amino acid composition of proteins in Saccharomyces cerevisiae from a data file that is included with CHNOSZ:

aa_YML020W <- yeast.aa("YML020W")

read.fasta() reads a FASTA file and returns the amino acid compositions of the sequences. The iseq argument can be used to read those sequences from the file:

file <- system.file("extdata/fasta/EF-Tu.aln", package = "CHNOSZ")
aa_Ef <- read.fasta(file, iseq = 1:2)

seq2aa() counts the amino acids in a user-supplied sequence and generates a data frame of the amino acid composition: ⊕See also ?count.aa, which can process both protein and nucleic acid sequences.

aa_PRIO <- seq2aa("PRIO_HUMAN", "
MANLGCWMLVLFVATWSDLGLCKKRPKPGGWNTGGSRYPGQGSPGGNRYPPQGGGGWGQP
HGGGWGQPHGGGWGQPHGGGWGQPHGGGWGQGGGTHSQWNKPSKPKTNMKHMAGAAAAGA
VVGGLGGYMLGSAMSRPIIHFGSDYEDRYYRENMHRYPNQVYYRPMDEYSNQNNFVHDCV
NITIKQHTVTTTTKGENFTETDVKMMERVVEQMCITQYERESQAYYQRGSSMVLFSSPPV
ILLISFLIFLIVG
")

uniprot.aa() returns the amino acid composition of a single amino acid sequence downloaded from UniProt. To get sequences for many proteins, use R’s lapply(), do.call(), and rbind():

IDs <- c("ALAT1_HUMAN", "P02452")
aa <- lapply(IDs, uniprot.aa)
## uniprot.aa: trying http://www.uniprot.org/uniprot/ALAT1_HUMAN ... accession P24298 ...
## >sp|P24298|ALAT1_HUMAN Alanine aminotransferase 1 OS=Homo sapiens GN=GPT PE=1 SV=3 (length 496)
## uniprot.aa: trying http://www.uniprot.org/uniprot/P02452 ... accession P02452 ...
## >sp|P02452|CO1A1_HUMAN Collagen alpha-1(I) chain OS=Homo sapiens GN=COL1A1 PE=1 SV=5 (length 1464)
aa_UniProt <- do.call(rbind, aa)

aa_UniProt

##      protein organism ref abbrv chains Ala Cys Asp Glu Phe Gly His Ile Lys Leu
## 1  sp|P24298    HUMAN  NA ALAT1      1  51  10  20  34  21  38   9  18  16  55
## 11 sp|P02452    HUMAN  NA CO1A1      1 139  18  66  75  27 391   9  24  57  48
##    Met Asn Pro Gln Arg Ser Thr Val Trp Tyr
## 1   12  12  33  30  37  25  18  41   1  15
## 11  13  28 278  49  71  60  45  47   6  13

These amino acid compositions can be processed using functions such as protein.length() and protein.formula():

myaa <- rbind(aa_YML020W, aa_Ef, aa_PRIO)
protein.length(myaa)

## [1] 664 394 394 253

Adding proteins and using iprotein

Once the amino acid compositions have been obtained, use add.protein() to add these proteins to thermo$protein:

add.protein(myaa)

## add.protein: added 4 new protein(s) to thermo$protein

## [1] 512 513 514 515

Then, subcrt() can be used to calculate the standard thermodynamic properties of any of these proteins:

subcrt("PRIO_HUMAN", T = 25)

## $species
##            name               formula state ispecies
## 3618 PRIO_HUMAN C1227H1834N352O352S16    aq     3618
## 
## $out
## $out$PRIO_HUMAN
##    T P      rho    logK        G         H       S       V      Cp
## 1 25 1 0.997061 5486.45 -7484855 -19116027 8124.78 20084.3 12174.4

Or we can add any of these proteins to the species list with species() and calculate the affinity:

basis("CHNOS")

species("YML020W_Sce")

##    CO2  H2O NH3 H2S    O2 ispecies logact state        name
## 1 3406 1247 932  13 -3517     3619     -3    aq YML020W_Sce

a <- affinity()

Or we can calculate the affinities of formation reactions of the proteins without adding them as species:

ip <- add.protein(aa_UniProt)
a <- affinity(iprotein = ip)

As shown there, the iprotein argument of affinity() can be used to calculate the affinities of reactions to form the indicated proteins, bypassing the species() step. Let’s see this in action using amino acid compositions deduced from metagenomic sequences in the Bison Pool hot spring in Yellowstone (Dick and Shock, ⊕2011Dick JM, Shock EL. 2011. Calculation of the relative chemical stabilities of proteins as a function of temperature and redox chemistry in a hot spring. PLoS ONE 6(8): e22782. doi: 10.1371/journal.pone.0022782). We read a data file of amino acid compositions produced in that study, taking those labeled “transferase”. Then we add the proteins and get their indices using add.protein(), set the basis, calculate the affinities, and make a potential diagram with temperature and activity of dissolved hydrogen as variables:

Potential diagram for metagenomically identified sequences of transferases in Bison Pool hot spring. See also the vignette Hot-spring proteins in CHNOSZ.

file <- system.file("extdata/protein/DS11.csv", package = "CHNOSZ")
aa <- read.csv(file, as.is = TRUE)
aa <- aa[grep("transferase", aa$protein), ]
ip <- add.protein(aa)
basis(c("HCO3-", "H2O", "NH3", "HS-", "H2", "H+"))
basis(c("HCO3-", "NH3", "HS-", "H+"), c(-3, -4, -7, -7.9))
T <- c(50, 100)
res <- 300
a <- affinity(T = c(T, res), H2 = c(-8, -3, res), iprotein = ip)
fill <- ZC.col(ZC(protein.formula(ip)))
diagram(a, normalize = TRUE, fill = fill, names = 1:5)

Site numbers 1–5 correspond to a cooling gradient along the outflow channel of the hot spring. The colors represent the relative Z_C of the proteins (red is more reduced). The points indicate the T and loga_H2 that optimize a thermodynamic model for relative abundances of phyla that were estimated by taxonomic classification of metagenomic sequences (Dick and Shock, ⊕2013Dick JM, Shock EL. 2013. A metastable equilibrium model for the relative abundances of microbial phyla in a hot spring. PLoS ONE 8(9): e72395. doi: 10.1371/journal.pone.0072395):

T <- c(93.3, 79.4, 67.5, 65.3, 57.1)
logaH2 <- c(-3.38, -4.14, -5.66, -7.47, -10.02)
lines(T, logaH2, lty = 2, lwd = 2)
points(T, logaH2, pch = 21, bg = "white", cex = 1.5)

Viewing data sources: thermo.refs()

The database in CHNOSZ lists one or two sources for each entry, and citation information for the sources is listed in thermo$refs. You can locate and view the references with thermo.refs(). Running the function without any arguments opens a browser window with the complete table of references. ⊕See the vignette, Thermodynamic data in CHNOSZ, for a more nicely formatted presentation of the sources of thermodynamic data, along with notes and additional comments. Where available, links to the web page for the articles and books are displayed:

thermo.refs()  ## shows table in a browser

A numeric argument to thermo.refs() gives one or more species indices for which to get the references:

iATP <- info("ATP-4")
iMgATP <- info("MgATP-2")
thermo.refs(c(iATP, iMgATP))

##       key                          author year                               citation                                                    note
## 99  LH06a D. E. LaRowe and H. C. Helgeson 2006 Geochim. Cosmochim. Acta 70, 4680-4724        nucleic-acid bases, nucleosides, and nucleotides
## 101 LH06b D. E. LaRowe and H. C. Helgeson 2006           Thermochim. Acta 448, 82-106 Mg-complexed adenosine nucleotides (ATP), NAD, and NADP
##                                           URL
## 99  https://doi.org/10.1016/j.gca.2006.04.010
## 101 https://doi.org/10.1016/j.tca.2006.06.008

A character argument gives the source key(s):

thermo.refs(c("HDNB78", "MGN03"))

##       key                                    author year                 citation                                              note                                   URL
## 8  HDNB78       H. C. Helgeson, J. M. Delany et al. 1978  Am. J. Sci. 278A, 1-229 data for minerals (n = 167) and phase transitions http://www.worldcat.org/oclc/13594862
## 90  MGN03 J. Majzlan, K.-D. Grevel and A. Navrotsky 2003 Am. Mineral. 88, 855-859        goethite, lepidocrocite, and maghemite GHS https://doi.org/10.2138/am-2003-5-614

If the argument holds the result of subcrt(), references for all species in the reaction are returned: ⊕The exception is H₂O. With the default settings, thermodynamic properties for H₂O are derived from SUPCRT92 (Johnson et al., 1992).

substuff <- subcrt(c("C2H5OH", "O2", "CO2", "H2O"), c(-1, -3, 2, 3))
thermo.refs(substuff)

##      key                                           author year                               citation                      note                                           URL
## 84  PS01                  A. V. Plyasunov and E. L. Shock 2001 Geochim. Cosmochim. Acta 65, 3879-3900   aqueous nonelectrolytes https://doi.org/10.1016/S0016-7037(01)00678-0
## 23 SHS89 E. L. Shock, H. C. Helgeson and D. A. Sverjensky 1989 Geochim. Cosmochim. Acta 53, 2157-2183 inorganic neutral species  https://doi.org/10.1016/0016-7037(89)90341-4

The URLs of the references can be copied to a browser, or opened using R’s browseURL():

iFo <- info("forsterite")
ref <- thermo.refs(iFo)
browseURL(ref$URL)  ## opens a link to worldcat.org

Optional data

Thermodynamic properties of minerals in the default database are mostly taken from Helgeson et al. (⊕1978Helgeson HC, Delany JM, Nesbitt HW, Bird DK. 1978. Summary and critique of the thermodynamic properties of rock-forming minerals. American Journal of Science 278A: 1–229. Available at http://www.worldcat.org/oclc/13594862. ). These are identified by the state cr, and cr2, cr3, etc. for higher-temperature polymorphs. Optionally, thermodynamic properties of minerals can be calculated using the equations and dataset of Berman (⊕1988Berman RG. 1988. Internally-consistent thermodynamic data for minerals in the system Na₂O–K₂O–CaO–MgO–FeO–Fe₂O₃–Al₂O₃–SiO₂–TiO₂–H₂O–CO₂. Journal of Petrology 29(2): 445–522. doi: 10.1093/petrology/29.2.445); these minerals are identified in the database by having the state cr_Berman.

Two other optional datasets can be activated by using add.obigt():

add.obigt("SUPCRTBL") – This loads updated thermodynamic parameters for aqueous SiO₂, aluminum and arsenic species, and minerals, as compiled in the SUPCRTBL package (Zimmer et al., ⊕2016Zimmer K, Zhang Y, Lu P, Chen Y, Zhang G, Dalkilic M, Zhu C. 2016. SUPCRTBL: A revised and extended thermodynamic dataset and software package of SUPCRT92. Computers & Geosciences 90: 97–111. doi: 10.1016/j.cageo.2016.02.013). SUPCRTBL also includes modifications to SUPCRT92 for calculating thermodynamic properties of minerals using the Holland and Powell (⊕2011Holland TJB, Powell R. 2011. An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids. Journal of Metamorphic Geology 29(3): 333–383. doi: 10.1111/j.1525-1314.2010.00923.x) dataset, but that is not available in CHNOSZ. Instead, the Berman mineral data can be used with similar results; see demo(go-IU).

add.obigt("DEW") – These are aqueous species, with modified parameters, that are intended for use with the Deep Earth Water (DEW) model (Sverjensky et al., ⊕2014Sverjensky DA, Harrison B, Azzolini D. 2014. Water in the deep Earth: The dielectric constant and the solubilities of quartz and corundum to 60 kb and 1,200 ^∘C. Geochimica et Cosmochimica Acta 129: 125–145. doi: 10.1016/j.gca.2013.12.019). You should also run water("DEW") to activate the equations in the model; then, they will be used by subcrt() and affinity(). Examples are in demo(DEW).

Detailed lists of sources for these Optional Data are in the vignette Thermodynamic data in CHNOSZ (look under Solids / Berman, Optional Data / SUPCRTBL and Optional Data / DEW).

Adding data

You can also use add.obigt() to add data from a user-specified file to the database in the current session. The file must be a CSV (comma separated value) file with column headers that match those in the main database. To show the required format, take a look at BZA10.csv, a supplemental data file provided in CHNOSZ. Missing values are indicated by NA: ⊕R’s read.csv() has a useful option: as.is = TRUE. This prevents columns with character data from being read as factors (i.e. categorical data). Passing factors to functions that are designed for character data can give unexpected results or errors.

file <- system.file("extdata/thermo/BZA10.csv", package = "CHNOSZ")
read.csv(file, as.is = TRUE)

##      name abbrv formula state  ref1 ref2      date       G  H     S     Cp
## 1   CdCl+    NA   CdCl+    aq BZA10   NA 03.Jul.10  -52629 NA  7.06  11.12
## 2   CdCl2    NA   CdCl2    aq BZA10   NA 03.Jul.10  -84883 NA 25.72 116.01
## 3  CdCl3-    NA  CdCl3-    aq BZA10   NA 03.Jul.10 -115399 NA 45.15  97.78
## 4 CdCl4-2    NA CdCl4-2    aq BZA10   NA 03.Jul.10 -145583 NA 50.61  42.52
##       V    a1.a    a2.b    a3.c    a4.d    c1.e    c2.f omega.lambda z.T
## 1  2.20  2.2303 -2.3357  6.6681 -2.6824 16.6723 -0.7693       0.4372   1
## 2 42.21  7.5221 10.5852  1.5895 -3.2166 73.7023 20.5956      -0.0495   0
## 3 63.47 10.8045 18.5994 -1.5605 -3.5479 72.0244 16.8832       0.9378  -1
## 4 81.35 13.8329 25.9938 -4.4669 -3.8536 53.6766  5.6267       2.4766  -2

The column names with a dot (.) refer to different sets of equations for calculating standard thermodynamic properties. Aqueous species use the revised Helgeson-Kirkham-Flowers (HKF) equations (symbols before the dot), and crystalline, gas and liquid species other than H₂O use a general heat capacity equation. See ?thermo for details about what’s in the data table, and ?hkf and ?cgl for information about the thermodynamic equations.

Another use of add.obigt() is to add data from the file CHNOSZ_aq.csv, which holds provisional updates that aren’t loaded by default (i.e., using data(thermo)). In this mode, only the name of the species is used as the argument. Currently available species are adenine-old and pseudo-H4SiO4 (see demo(adenine) and the vignette, Regressing thermodynamic data).

Modifying data

Use mod.obigt() to add or modify the database in the current session. The function requires the name of a species and one or more properties to change. Let’s add data for the S₃^- ion from Pokrovski and Dubessy (⊕2015Pokrovski GS, Dubessy J. 2015. Stability and abundance of the trisulfur radical ion in hydrothermal fluids. Earth and Planetary Science Letters 411: 298–309. doi: 10.1016/j.epsl.2014.11.035). Although we could choose anything for the name (e.g. “trisulfur radical anion”), here we make it the same as the formula. The date for the entry is today() (i.e. today’s date using the date format inherited from SUPCRT92):

mod.obigt("S3-", formula = "S3-", state = "aq", ref1 = "PD15", date = today(),
          G = 13160, H = 10840, S = 28.6, Cp = 62.3, V = 37.7)

## mod.obigt: added S3-(aq)

## [1] 3620

The function prints a message saying that the species was added, and returns the species index of the new species. Now let’s modify the same species by adding the HKF coefficients:

mod.obigt("S3-", state = "aq", a1 = 2.5, a2 = 19.9, a3 = 9.2, a4 = -3.6,
          c1 = 50.2, c2 = 9.6, omega = 0.8, z = -1)

## mod.obigt: updated S3-(aq)

## [1] 3620

Let’s define basis species including S₃^- to automatically balance the reaction of SO₄^-2 and 2 H₂S:

basis(c("S3-", "O2", "H2O", "H+"), c("aq", "gas", "liq", "aq"))

##     H O S  Z ispecies logact state
## S3- 0 0 3 -1     3620      0    aq
## O2  0 2 0  0     3316      0   gas
## H2O 2 1 0  0        1      0   liq
## H+  1 0 0  1        3      0    aq

subcrt(c("H2S", "SO4-2"), c(-2, -1), T = c(25, 500), P = c(1, 700))

## $reaction
##      coeff   name formula state ispecies
## 67   -2.00    H2S     H2S    aq       67
## 24   -1.00  SO4-2   SO4-2    aq       24
## 3620  1.00    S3-     S3-    aq     3620
## 3316  0.75 oxygen      O2   gas     3316
## 1     2.50  water     H2O   liq        1
## 3    -1.00     H+      H+    aq        3
## 
## $out
##     T   P      rho      logK       G        H        S          V       Cp
## 1  25   1 0.997061 -45.97177 62716.7  75450.1  42.6525  -0.166515  90.6566
## 2 500 700 0.405762  -5.39411 19082.7 154614.4 175.2765 641.393901 714.1609

The reaction coefficients are identical to Reaction 4 of Pokrovski and Dubessy (2015), and the calculated values of logK match those shown in their Figure 5.

Cross-checking data entries

info() automatically performs some cross-checks of the thermodynamic data. ⊕This only checks the parameters for individual species, not the internal consistency of the database itself. Let’s do this for the new S₃^- species:

iS3 <- info("S3-")
info(iS3)

## checkGHS: G of S3- aq (3620) differs by 661 cal mol-1 from tabulated value

##      name abbrv formula state ref1 ref2      date     G     H    S   Cp    V
## 3620  S3-  <NA>     S3-    aq PD15 <NA> 13.Nov.17 13160 10840 28.6 62.3 37.7
##        a1   a2  a3     a4   c1    c2 omega  Z
## 3620 0.25 1990 9.2 -36000 50.2 96000 80000 -1

There checkGHS() calculated the value of ΔG°_f from those of ΔH°_f and S° and from the entropy of the elements (Cox et al., ⊕1989Cox JD, Wagman DD, Medvedev VA, editors. 1989. CODATA Key Values for Thermodynamics. New York: Hemisphere Publishing Corporation. Available at http://www.worldcat.org/oclc/18559968. ) in the chemical formula of the species. If the difference between the entered and calculated values of ΔG°_f is greater than 100 cal mol^-1, checkGHS() prints a message. The calculated value of ΔG°_f was found to be 661 cal mol^-1 higher than the entered value. After checking for any typographical errors in the entries for ΔG°_f, ΔH°_f, S°, and the chemical formula, the source should be consulted for clarification.

Some species in the main and secondary databases are known to have inconsistent entries. For example, the entry for cyclohexane has C_P° and V° that differ from those calculated from the HKF parameters. checkEOS() is a function that cross-checks these parameters and reports the differences (greater than 1 cm³ mol^-1 or 1 cal K^-1 mol^-1) when info() is run:

info(info("cyclohexane"))
## checkEOS: Cp of cyclohexane aq (1762) differs by 9.35 cal K-1 mol-1 from tabulated value
## checkEOS: V of cyclohexane aq (1762) differs by 6.64 cm3 mol-1 from tabulated value

The species with detected inconsistencies in OBIGT are listed in the file obigt_check.csv.

file <- system.file("extdata/thermo/obigt_check.csv", package = "CHNOSZ")
dat <- read.csv(file, as.is = TRUE)
nrow(dat)

## [1] 275

Without additional information, there is often no clear strategy for “fixing” these entries, and they are provided as is. ⊕See the Numerical Tools from GEOPIG for SUPCRT updates (slop files), including corrections that have not yet been ported to CHNOSZ.

Water: SUPCRT92 or IAPWS-95 or DEW

For calculations of the thermodynamic and dielectric properties of liquid and supercritical H₂O, CHNOSZ uses a Fortran subroutine (H2O92) from SUPCRT92 (Johnson et al., 1992). Alternatively, the IAPWS-95 formulation for thermodynamic properties (Wagner and Pruß, ⊕2002Wagner W, Pruß A. 2002. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of Physical and Chemical Reference Data 31(2): 387–535. doi: 10.1063/1.1461829) can be utilized. In part because of intrinsic thermodynamic differences between SUPCRT92 and IAPWS-95, as well as different equations used in CHNOSZ for calculating the dielectric constant when the IAPWS-95 option is active, this option could introduce inconsistencies with the data for aqueous species in the database and is not recommended for general use in CHNOSZ. However, the IAPWS-95 equations are useful for other applications, and may be extrapolated to a greater range of T and P than SUPCRT. See ?water for more information, as well as the last example in ?subcrt, where uncommenting the line for the IAPWS95 option allows extrapolation to lower temperatures for supercooled water.

More recently (late 2017), an implementation of the Deep Earth Water (DEW) model was added; see Optional data for more information.

Regressing thermodynamic data

EOSregress() and related functions can be used to regress “equation of state” parameters (e.g. coefficients in the HKF equations) from heat capacity and volumetric data. See ?EOSregress and the vignette, Regressing thermodynamic data.

Gibbs energy minimization

wjd() implements a Gibbs energy minimization using the method of steepest descent described by White et al. (⊕1958White WB, Johnson SM, Dantzig GB. 1958. Chemical equilibrium in complex mixtures. Journal of Chemical Physics 28(5): 751–755. doi: 10.1063/1.1744264). See ?wjd as well as the vignette, Winding journey down (in Gibbs energy).

Group additivity and EQ3/6 output

RH2obigt() implements a group additivity calculation of standard molal thermodynamic properties and equations of state parameters of crystalline and liquid organic molecules from Richard and Helgeson (⊕1998Richard L, Helgeson HC. 1998. Calculation of the thermodynamic properties at elevated temperatures and pressures of saturated and aromatic high molecular weight solid and liquid hydrocarbons in kerogen, bitumen, petroleum, and other organic matter of biogeochemical interest. Geochimica et Cosmochimica Acta 62(23-24): 3591–3636. doi: 10.1016/S0016-7037(97)00345-1).

eqdata() reads data, including concentrations of aqueous species, numbers of moles of solid phases, and mineral saturation states (affinities), from an EQ6 output file (Wolery, ⊕1992Wolery TJ. 1992. Eq3/6, A Software Package for Geochemical Modeling of Aqueous Systems: Package Overview and Installation Guide (Version 7.0). Lawrence Livermore National Laboratory. Report No.: UCRL-MA-110662 PT I. doi: 10.2172/138894).

NCBI taxonomy files

Some functions are available (see ?taxonomy) to read data from NCBI taxonomy files, traverse taxonomic ranks, and get scientific names of taxonomic nodes.