Units and conversions

I just viva'd for an SPR based PhD and got some corrections (this site and the corresponding book we're a huge help, thanks).

The biggest criticism my external made was of the way units are used in chemical equations and the related models, and I wasn't sure how to respond, or quite how to correct this.

Normally a chemical equation will be of the form:

[A]+[L]⇌_{k_a}^k_d[AL]

and each of [A], [L] and [AL] will be a quantity or concentration of a chemical species, and their concentrations can be represented with ODEs:

d[AL]/dt=k_a[A][L]-k_d[AL] eq1
d[L]/dt =-k_a[A][L]+k_d[AL] eq2
d[A]/dt =-k_a[A][L]+k_d[AL] eq3

with each of the concentrations being measured in M, k_a and k_d in per M per s, and per s respectively. However in SPR, if [L] represents the ligand bound to a chip it should be in Mol per unit area; however, its measured in RU (which are equivalent to pg per mm^2).

Shouldn't this change the units of the kinetic rate constants? And shouldn't there be a conversion factor?

Hi,
Thank you for your kind words.

Let's take eq 1 for the association of the complex: d[LA]/dt = k_a [L][A]-k_d [LA].
Expressed as a chemical equation, Ligand, analyte and complex formed are concentrations.

But let's look what is happening when the analyte binds to the (immobilized)ligand in an SPR instruments. The flow will add more and more analyte to the system and is effectively constant (-> C). The amount of free ligand [L] at any time is the starting amount [Lo] minus the formed complex (->[L]=[Lo]-[LA]). Thus eq 2:

Now the crucial part: In SPR the signal generated is proportional to the formed [LA]-complex. We can substitute [LA] = Rt (response). When all the ligand [Lo] is occupied by the analyte the maximal signal is reached, thus [Lo] = Rmax. If this is substituted in the equation we get eq 3:
To have the response in Mol per volume we have to somehow calculate the concentration (Mol L^-1) of formed complex. In general we assume that 1000 RU is equivalent to 1 ng/mm², is equivalent to 10 mg/ml.

The conversion from response to Mol per liter is (eq 4):
This relation was established in the early days of SPR for the CM5 sensor chip with a dextran layer of approximately 100 nm thickness.

However, many types of sensor chips are different in construction and the conversion should be established for each sensor chip type independently. In addition, we don't know if the conversion relation holds now for the CM5 sensor chip because the manufacturing can be changed.
Luckily, equation 3 holds in all situations because the exact concentration-response conversion is the same on both sides of the equation. Thus the rate constants and the analyte concentration are in the appropriate units and the ligand concentration is removed from the equation.

I hope it is clear that there is a very practical reason to use the response (units) (and different instruments use pixels, milli degrees, etc.) and not to try to convert these to actual complex concentrations.

Kind regards
Arnoud