Setting power variable values in kinetic microbial reaction

[John Lisle]

Good afternoon, I've been trying to figure out the rational behind assigning values to the "power", "powerA" and "powerD" options in the kinetic microbial reaction page of the Reactants window in GWB11. I've read through the GWB Reaction Modeling Guide (rel. 11), the Microbial Kinetics chapter in G&BRM book and the various videos and tutorials on your website. Its not clear to me when to use these options and what determines my decision to enter nothing, enter 1 or a value greater than 1 (which I've seen in some of the examples). My interest in using this option of GWB is I'm trying to model microbial processes (at this point just in solution and not on the aquifer matrix) in an anaerobic and reduced (-350 mV) groundwater system. I'm currently using a spreadsheet method to generate data from the kinetic model described by Jin & Bethke's series of papers and Chp 18 in G&BRM but would like to move these calculations into GWB due to the graphs and other data generated and to compliment the geochemical analyses. If you could provide some guidance on how these variables and the values entered influence the outputs from the analyses I would be truly grateful. If there are specific papers or other resources which can guide me through this please send those along and I'll work through those.

[Brian Farrell]

Hi John,

 

The first step is to determine the microbial rate law that’s applicable to your situation. After that, one can specify the rate law within the GWB using the "power", "powerA" and "powerD" settings, which I’ll explain below. To get started, let’s look at half-cell reactions in general:

 

Donating half-reaction: ΣD = ΣD⁺ + ne^-

Accepting half-reaction: ΣA + ne^- = ΣA^-

 

where D and D⁺ are all the species on left and right sides, respectively, of the e- donating half-reaction, and A and A^- are the species on left and right sides, respectively, of the e- accepting half-reaction.

 

To keep this simple, let’s look at the generalized kinetic terms for electron donation (F_D) and acceptance (F_A), assuming for now that the exponents applied to the various species (the beta terms in the GBRM textbook) are 1.

 

F_D = (Πm_D)/( Πm_D + K_D Πm_D+)

F_A = (Πm_A)/( Πm_A + K_A Πm_A-)

 

Here Π is the product function, m refers to the molal concentration of the species in the reactions, and K_D and K_A are constants.

 

Let’s look at microbial sulfate reduction as an example: CH3COO- + SO4-- = 2 HCO3- + HS-

 

Donating half-reaction: CH3COO- + 4 H2O = 2 HCO3- + 9 H+ + 8 e-

Accepting half-reaction: SO4-- + 9 H+ + 8e- = HS- + 4 H2O

 

We can write out the kinetic terms in general form by taking species from the left and right sides of the half-cell reactions and inserting them into the equations:

 

F_D = (mCH3COO-)/((mCH3COO-) + K_D(mHCO3-*mH+))

F_A = (mSO4--*mH+)/((mSO4--*mH+) + K_A(mHS-))

 

If we wanted to set this rate law, we’d use the “power” keyword for species in the numerator of either kinetic factor (I think this is straightforward enough), “powerA” for species in the denominator of the accepting half-reaction kinetic factor, and “powerD” for species in the denominator of the donating half-reaction kinetic factor.

 

Looking at the donating half-reaction first, you would set:

 

mpower(CH3COO-) = 1, mpowerD(CH3COO-) = 1

mpowerD(HCO3-) = 1

mpowerD(H+) = 1

 

And now, the accepting half-reaction:

 

mpower(SO4--) = 1, mpowerA(SO4--) = 1

mpower(H+) = 1, mpowerA(H+) = 1

mpowerA(HS-) = 1

 

The “1” settings simply mean the concentration of the species are used in each part of the rate law (a “power” of 1), rather than being squared, as an example (mpower = 2) or raised to some non-integer value (mpower = 1.2). You don’t have to “pick” the species to go on the left or right side of the denominator for either term (i.e. whether they appear by themselves or are multiplied by the kinetic constant). They’re assigned to one side or the other based on what side of the donating half-reaction they’re on. All you have to specify is that they’re in the denominator of the kinetic term for donation (mpowerD) or acceptance (mpowerA).

 

Now, species whose concentrations don’t change much don’t need to be carried in the rate law. Perhaps the pH is buffered, and reaction of the microbially produced sulfide with ferrous iron in solution precipitates FeS of some sort, so the amount of HS- in solution doesn’t build up. In that case, only the CH3COO-, SO4—, and HCO3- change concentrations much. We could simplify our rate law to this:

 

F_D = mCH3COO-/(mCH3COO- + KD*mHCO3-)

F_A = mSO4--/(mSO4-- + KA)

 

In this case, we just set

 

mpower(CH3COO-) = 1, mpowerD(CH3COO-) = 1

mpowerD(HCO3-) = 1

 

and

 

mpower(SO4--) = 1

mpowerA(SO4--) = 1

 

The rate law might be even simpler. The acetate might be supplied in excess, and there might be quite a bit of HCO3- in solution so that its concentration doesn’t change much. In that case, the rate law might look like a traditional Monod equation, written for the electron acceptor. In that case, you just set mpower for SO4—to appear in the numerator and mpowerA for SO4—in the denominator.

 

As for why they’re all set to 1? The beta values are determined empirically through careful experiments, but they’re commonly taken to be 1. This is written in the paragraph after equation 18.24 in the GBRM text and is explained in a bit more detail in Jin and Bethke 2005 (the text in section 3.1 and their Figure 1). The reference also discusses how the kinetic terms might be written in a general form accounting for all of the species in the donating and accepting half-reactions, and where simplifications are commonly made.

 

As far as I’m aware, all of the microbial example in the GBRM textbook have “mpower”, “mpowerD”, and “mpowerA” set to 1. You should know that the “mpower” variable, for species appearing in the numerator, is also used in the various other kinetic reactions built into the GWB (mineral dissolution, complex dissociation, gas transfer, redox transformation). Perhaps the example you found with “mpower” set to greater than 1 is one of these reaction types. A complexation reaction A + A -> B, for example, might proceed at a rate r = k * (mA)^2. In that case, mpower(A) is set to 2 (note that’s “mpower()”, not “mpowerA()”).

 

Hope this helps,

 

Brian Farrell

Aqueous Solutions LLC