3.9 Biological Information Processing

27

can be understood through consideration of (a) transport phenomena and (b) reaction

kinetics, and coupling between them.

The general expression of flux density upper JJ for the iith molecular entity is

upper J Subscript i Baseline equals minus upper D Subscript i Baseline StartFraction partial differential c Subscript i Baseline Over partial differential x EndFraction plus upper D Subscript i Baseline StartFraction upper Z Subscript i Baseline upper F Over upper R upper T EndFraction c Subscript i Baseline upper EJi = −Di

∂ci

∂x ^{+ Di}

Zi F

RT ^{ci E}

(3.4)

where upper DD is ii’s diffusivity, cc its concentration, xx a spatial coördinate, upper ZZ ii’s charge,

upper FF the Faraday constant, and upper EE the electric field. Mass balance is expressed by

StartFraction partial differential c Subscript i Baseline Over partial differential t EndFraction equals StartFraction partial differential upper J Subscript i Baseline Over partial differential x EndFraction plus v Subscript i^∂ci

∂t ^{= ∂Ji}

∂x ^{+ vi}

(3.5)

wherevv accounts for all chemical reactions. One could write a similar expression for

charge balance but it may be assumed that in practice sufficient supporting electrolyte

is present for that to become unimportant. Combining (3.4) with (3.5) yields

StartFraction partial differential c Subscript i Baseline Over partial differential t EndFraction equals upper D Subscript i Baseline StartFraction partial differential squared c Subscript i Baseline Over partial differential x squared EndFraction minus upper D Subscript i Baseline StartFraction z Subscript i Baseline upper F Over upper R upper T EndFraction upper E StartFraction partial differential c Subscript i Baseline Over partial differential x EndFraction plus v Subscript i Baseline period^∂ci

∂t ^{= Di}

∂²ci

∂x^{2 −Di}

zi F

RT ^{E ∂ci}

∂x ^{+ vi .}

(3.6)

This is the fundamental equation of biophysicochemical information processing. ⁵

Systems constructed on this basis can have fully integrated functions, unlike biosen-

sors, in which the biosensing element is coupled to a physical transducer. Some

examples of the information processing achievable in such systems are active trans-

port (against a concentration gradient), clocks, mathematical operations (addition,

multiplication, etc.), control (e.g., stopping a function), storage and amplification.

As a more detailed example (See footnote 4), consider a membrane in which

enzyme ESubscript 11 is distributed homogeneously in the left half of a membrane separating two

compartments containing a substance S, the concentration of S being significantly

higher in the right hand compartment than in the left hand one, and enzyme ESubscript 22

distributed homogeneously in the right half of a membrane. Applying Eq. (3.6) in

the absence of an electric field (upper E equals 0E = 0) yields:

StartLayout 1st Row StartFraction partial differential s Over partial differential t EndFraction equals upper D Subscript normal upper S Baseline StartFraction partial differential squared s Over partial differential x squared EndFraction plus v 2 minus v 1 semicolon 2nd Row Blank 3rd Row StartFraction partial differential p Over partial differential t EndFraction equals upper D Subscript normal upper P Baseline StartFraction partial differential squared p Over partial differential x squared EndFraction plus v 1 minus v 2 EndLayout^∂s

∂t ^{= DS}

∂²s

∂x^{2 + v2 −v1 ;}

(3.7)

∂ p

∂t ^{= DP}

∂² p

∂x^{2 + v1 − v2}

(3.8)

wheress andpp are respectively the concentrations of substrate S and product P. Some

of the solutions of this two-equation set have asymmetrical concentration profiles

s left parenthesis x right parenthesiss(x) andp left parenthesis x right parenthesisp(x): for example, depletion of S in the left-hand side where ESubscript 11 operates, and

accumulation in the right-hand side where ESubscript 22 operates. Such a profile corresponds

to active transport against a concentration gradient.

5 Mostly, systems of these equations (one for eachii) have to be solved numerically.