# Cellular Metabolism

The term "basal metabolism" refers collectively to those physiological functions which your body performs while resting. The basal metabolic "rate" is the power required to sustain those functions. Many of the functions are macroscopic, such as the beating of your heart, your breathing and your digestion. But we will consider the work done by that energy at the cellular level. For instance, two thirds of your basal metabolism is required to move ions against concentration gradients (see Chapter 4). In fact, one third is used just to power the sodium pumps!

In order to deal with cellular metabolism quantitatively, we must work with several kinds of energy. The distinctions are necessary because we are dealing with a statistical system, in this case, the large numbers of chemical reactions necessary to keep you alive. We distinguish between the "internal" energy (U), which denotes thermal energy, and work (pressure times volume) done by or against the outside environment. We also distinguish between the free energy available to make chemical reactions occur (see Section B) and "entropic" energy (heat), which is a consequence of the degeneracy of the system. The latter contributes to the random nature of the system. We also define the "enthalpy" (H, an extensive state variable), which we will think of as the total energy:

H = U + P V

as well as

H = G + T S.

These are normally written as changes in energy (often for particular reactions):

D H = D U + P D V

= D G + T D S,

since most of the reactions we are interested take place at constant pressure ("isobarically") and constant temperature ("isothermally").

Note that the physicist's use of entropy in the last equation differs somewhat from the chemist's. For the chemist, that equation means

H P - H R = G P - G R + T (S P - S R),

where P and R denote products and reactants: specific species. The enthalpy is the "heat of formation," the free energy is the stored energy which can be retrieved, and the entropy is the entropy of a single molecule. Hence the chemist considers the system to be a molecular one. We, on the other hand, will think of the entropy of the statistical system consisting of a great many molecules in the cellular environment. Hence the meaning of free energy is essentially unchanged: if a reaction has a negative free energy change, it will occur spontaneously, while a reaction with a positive free energy change will not. But for us, the entropy reflects the energy "wasted" as thermal excitations in the cell. This is the energy which we were so concerned with losing in the last section. If the system is closed (as we will imagine), then the enthalpy is conserved, and

D S = | G P - G R | / T,

or, as we will use it,

D S = | G in - G stored for later use | / T.

Metabolic processes involve chemical reactions which release energy ("exergonic" reactions), and reactions which require energy to in order to occur ("endergonic" reactions). The former supply the needs of the latter. As indicated by the last equation, however, not all of the energy released by an exergonic reaction is free to be used by an endergonic one. There is always some "waste" entropic energy. We can quantify the efficiency of any system as the ratio of the work done by the system to the energy it requires to accomplish that work:

e = W done / E input ,

which is of course dimensionless. This is equivalent to, for our purposes,

e = G stored for later use / G in.

Typical system-wide physiological efficiencies are about 20 %.

We can now make some qualitative statements about life. Equilibrium conditions exist when the free energy G is at a minimum: mathematically, when

DG = 0.

This situation corresponds to death, since no free energy is available to drive endergonic reactions. Life requires a steady state relatively far from equilibrium, and in all cases that we know of, this state is maintained by sustaining non-equilibrium concentration gradients across membranes. This activity requires a constant influx of energy, and the consequent generation of entropy.

We can summarize the general features of metabolic reactions as follows.

1. All reactions must proceed spontaneously, either because they are exergonic or because they are coupled with a sufficiently exergonic reaction to provide the extra energy necessary to drive an otherwise non-spontaneous endergonic reaction.
2. Free energy and entropy changes are independent of path (they are state variables). Therefore changes in their values for a whole process are the sums of the changes for individual reactions.
3. The use of small steps between equilibrium states allows the biological system to approach "reversibility", which increases efficiency.
4. Efficiencies of "cyclic" processes (in which the final state is the same as the initial one) are generally better than non-cyclic ones.
5. As always, the energy where we select E = 0 is arbitrary.

There are two primary sources of energy for endergonic reactions. These are the exergonic reactions

 ATP + H 2 O -> ADP + P DG = - 7.3 kcal / mol NADH + H+ + 1/2 O 2 -> NAD+ + H 2 O DG = - 52.4 kcal / mol

The negative signs on the free energy indicate that the reactants lose energy: the reactions are exergonic. ATP (adenosine triphosphate) is used throughout the body to store energy that would otherwise be released as heat. Similarly, the "reduction" (gain of one or more electrons by a molecule, raising its internal energy) reaction for nicotinamide adenine dinucleotide

stores energy which is released by the "oxidation" (loss of an electron, freeing energy) of NADH. Living organisms also use the reduction of flavin adenine dinucleotide

FAD + H 2 O -> FADH 2 + 1/2 O 2 ,

which stores 45.9 kcal / mol of energy. Its associated oxidation reaction,

FADH 2 + 1/2 O 2 -> FAD + H 2 O

releases it. We will assume that there is plenty of water, ADP, NAD+, FAD and hydrogen ions available in the cell, and will leave those reactants out of the reactions below.

We are now in a position to discuss cellular respiration: the "burning" of glucose with oxygen, producing carbon dioxide and water:

C 6 H 12 O 6 + 6O 2 -> 6CO 2 + 6H 2 O,

DG = -686 kcal / mol.

This reaction can take place inorganically, and essentially all of the energy released is entropic: heat. Cellular respiration breaks the process up into myriad steps which involve small changes in energy, and therefore small releases of entropy. The steps are reversible, given sufficient variations in concentrations, and so the overall process gains a great deal in efficiency.

We will ignore (until the problems!) much of the details and list the four major parts of cellular respiration. Note that water is necessary to supply hydrogen ions and oxygen for these processes; it is also a product of many of them. It is left out of most equations to make the counting easier.

1. Glycolysis:

(DG of complete aerobic oxidation of pyruvate is -273 kcal / mol)

Glycolysis is very efficient because most of the reactions involved take place under nearly reversible conditions.

2. Intermediate step:

(assume DG = 0)

3. TriCarboxylic Acid (TCA) Cycle:

4. Electron Transport:

FADH 2 + 1/2 O 2 -> FAD + 3H 2 O + 2ATP

This is a nearly reversible series of oxidations to O 2; this pathway includes the:

5. Proton pump: oxidation is used to increase the proton concentration in mitochondria; that gradient is used to drive the generation of ATP. The proton potential is the sum of the membrane potential and the concentration gradient

D V = .16V + k T ln (H+out / H+in) / e

(note that log10(H+) = -pH, and pHin - pHout = 1). Therefore 2 protons provide enough energy to generate one ATP.

As an example of the thermodynamics of these processes, consider the TCA cycle (also known as the "Krebs" cycle). The change in free energy for the complete oxidation of AcetylCoA is 220.6 kcal / mol. By subtracting from this the energy stored in the ATP molecule and the four reduced molecules (NADH and FADH 2), we find the free energy change for the TCA cycle to be -10.2. Since the entropy change is assumed to be at constant body temperature, we simply divide the change in free energy (which we assume is wasted heat) by the temperature to find a generation of .03 kcal / mol K of entropy. The ratio of the energy stored in the various molecules to the free energy of AcetylCoA is the efficiency, which is about 95 %. This high efficiency is characteristic of cyclic processes.

We now summarize aerobic cellular respiration. For "eukaryotic" cells (those with nucleii, as are most of ours), 2 ATP are used to transport cytoplasmic NADH to the mitochondria, so the final reaction is

C 6 H 12 O 6 + 6O 2 -> 6CO 2 +6H 2 O +36 ATP.

The change in free energy for this reaction is - 423.2 kcal / mol; the associated entropy generated is 1.36 kcal / mol K, and the efficiency is about 38 %. This is a relatively efficient process for biological systems.

The next section is about photosynthesis and entropy generation in the biome.

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