Lecture Notes

Click here for related assignment.

Use your Back button to return to the main course page.

Lecture Number 1

Date: 9/8/98

1. The importance of tracing molecules in biological systems:

Are there enough molecules for a stoichiometric task? At what rate are these reactant molecules transformed into products? (These tasks include providing required energy, producing chemicals useful to the organism, and providing the structural material of the organism.)

How are signaling molecules produced, transported, and recognized? What role do these molecules play in the regulation and coordination of an organism?

2. Only two things can happen to molecules

They can move.

They can react.

3. The concept of concentration

Amount (mass or molar concentrations) per unit volume.

Compare with mole fraction, molality, partial pressure

Mass and molar density as

4. Control volume: arbitrary region in space characterized by an enclosing surface, e.g. S(x,y,z) and a the volume interior to that surface.
5. "Arbitrary" control volume
6. Specific examples: rectangular parallelpiped, cylinder, sphere, hollow cylinder, hollow sphere.
7. Concept of flux, n (mass), N (molar):

Net flow of designated species, reckoned at a point. Flux is a vector. Elements of a surface, e.g. dS, can also be represented as a vector. Integral of dot product of flux and dS, over all or part of a surface gives the flow through that surface.

Flux of "a" is velocity of "a" (vector) times concentration of "a". (All molecules at a perpendicular distance between 0 and v from a surface will pass that surface in unit time.)

8. Concept of ensemble average of molecular velocities:

Zero for quiescent ensemble

Finite for moving ensemble

Concentration relationships in a binary (two-component) system (after Bird, et al.):

Mass density relationships	Molar density relationships
Mole and weight fractions	Interconversion of mole and weight fractions

 

9.  Conventions in dealing with mathematical descriptions of physical systems. (Physical systems will "do what they do" irrespective of the physical frameworks we impose on them, but we will describe the same system differently according to the coordinates, the units of measurement and the definitions we overlay on it.) Vectorial quantities are positive when their components point in the positive direction defined by the coordinate axes.
10. The concept of flux: the net flow rate through a unit area of a designated species, reckoned at a point (e.g. x,y,z). (We do not use the physicist's flux which is the total flow from a source or into a sink.) The units of flux are thus m/(l²t). Flux is a vector. A differential area can also be represented as a vector, the direction representing a normal to the surface and the magnitude representing the extent of the surface. A flowrate is the integral of the dot product of a flux vector by a differential vector representing the components of the corresponding area:



11. Flux can be related to the velocity of an ensemble of the species represented by the flux. Molecules are in constant motion. If the concentrations in a problem do not change much in a region that nonetheless contains many molecules, one can calculate an ensemble average velocity. If that average is zero, there is no macroscopic flux in the region. If the average is a vector, it represents the uncancelled, net movement of the center of mass of the ensemble. This velocity, multiplied by the concentration of the species is the flux of the species in this region. (Think of a unit area held perpendicular to the flux vector. In one unit of time a volume v that has a concentration c_A will pass through this area. Note that the units of velocity (l/t) when multiplied by the units of concentration (m/l³) are equal to the units of flux (m/(l²t)).
12. We use v (no subscripts) to represent the mass average velocity and v^* to represent the molar average velocity. We use n_A to represent the mass flux of A and N_A to represent the molar flux of A. Then:

Definitions of mass, molar average velocities:	Flux-velocity relationships:

Note that the average velocities are weighted averages of the species velocities, in one case weighted by the mass concentration and in the other by the molar concentration.

13. The concept of reaction rate (homogeneous reactions -- those that occur throughout a volume and not at a boundary or surface): We use the symbol r_A to represent the rate of formation of A per unit volume. The rate is generally a function of concentrations and temperature as well as the concentration of any catalyst (enzyme) that may be present. Not that the units of r_A are m/(l³t). As with other quantities the upper case R represents a molar rate of reaction per unit volume while the lower case r represents the mass rate.
14. The concept of catalysis. The marvel of biological systems might be the reactions that do not occur. If all feasible reactions occurred life could not exist. Biological systems operate at low temperatures where most reactions are held in check. Just a small, selected set of reactions occur and these are largely controlled by enzymes, biological catalysts. It is important to note that enzymes, like all other catalysts, are not consumed or made by the reactions they catalyze (although they are made and degraded by other reactions). Any catalyst that accelerates a reaction proceeding in one direction also accelerates the reverse reaction. Thus the equilibrium (which can be seen as a balance between forward and reverse reactions) is not affected by a catalyst. This is a fundamental, unviolated fact. Don't forget it and don't try to reason out an answer that contradicts the fact.