A physicist’s overview of biological free energy

The 2nd law of thermodynamics is well known, and can be summarised thus: entropy always increases. Life appears to buck this trend, spontaneously generating extremely highly ordered structures. Living things decreases their own entropy at the expense of the rest of the universe. The free energy of a system (for example the Gibbs energy G = H – TS) is a measure of the entropy change of the entire universe corresponding to changes in the system, and therefore is an indicator of the spontaneous direction of any process of change.

The processes of life can therefore be considered as the gathering, storage and manipulation of free energy. Most globally, life is driven by the free energy of photons and converts it ultimately into heat. But there is a lot going on in between. This lecture introduces the main types of biological free energy and the ways in which they are interconverted. Bioenergetics is a whole lecture course in itself. Here we will only give a very brief overview of the processes shown in Slide 1. (Pmf is protonmotive force, which will be introduced and discussed in this lecture.)


1.1 Living things and free energy.

Free energy can be thought of as the ‘energy currency’ of life: living things gather, store and manipulate sources of free energy. Any reaction that requires free energy input (eg. making DNA from nucleic acids, doing mechanical work, building a protonmotive force) must be ‘paid for’ by coupling to a reaction that releases free energy.

Free energy is handled by chains of coupled chemical reactions or processes, each of which passes free energy from one form to another. Many of the reactions are called electron transport reactions, because they can be described in terms of the transfer of electrons between molecules with different electron affinity. The traditional biologist’s overview divides bioenergetics into two: respiration and photosynthesis (Slide 2).


Respiration

Overall, uses conversion of food + oxygen to CO2 and H2O to transfer electrons from molecules where they have low free energy (high affinity) onto molecules where they have high free energy.

Photosynthesis

Overall, uses energy form sunlight to transfer electrons from molecules where they have low free energy (high affinity) onto molecules where they have high free energy.

Slide 3 shows a physicist’s overview of bioenergetics. This is a very much simplified overview: each arrow represents a highly organised and controlled process that is achieved by one or more molecular machines.


Almost all biological energy begins with light. Phototrophes harvest sunlight, and organotrophes live off the organic chemicals that phototrophes produce. Lithotrophes, which derive energy from reactions between available chemicals that have no biological origin (e.g. microbes that live near hydrothermal vents) are the minority exception.

Light harvesting (see Lecture 3 in this topic) is the first part of photosynthesis – it occurs in membranes within specialist organelles in plants (chloroplasts) and in the cell membranes of photosynthetic bacteria. Energy from light is stored chemically and physically. Absorbed quanta are used to synthesise NADH (nicotinamide adenine dinucleotide (NAD+) plus hydrogen, see Lecture 2 in this topic), an energy-storage molecule, and to pump hydrogen ions (protons) across a membrane, which acts as a capacitor. This charge separation across a membrane is called the protonmotive force (pmf).

One of the most important processes that is directly driven by the pmf is ATP synthesis. ATP (adenosine triphosphate) is a high-energy molecule that is often called the ‘free energy currency’ of the cell. Energy stored in ATP is released by ATP hydrolysis, which is a reaction that is coupled to most important metabolic processes that require free energy input – e.g. chemical synthesis, movement.

The second part of photosynthesis is carbon fixation: energy stored in NADH and ATP is used to reduce CO2 to form carbon-carbon bonds in carbohydrates.

The chemical energy stored in molecules generated by this process is the source of all energy for non-photosynthetic organisms – this is food, and it is broken down to provide energy in a process called respiration. Respiration also generates a pmf, and ATP as before is generated from the pmf.

Some processes are driven by other high-energy chemicals (eg: NADH, GTP – guanosine triphosphate) or directly by the pmf (if they are membrane-bound processes).

1.2 Chemical potential and electrochemical potential

Before proceeding further it is useful to recap some key ideas from other Biological Physics lectures.

1.2.1 Thermodynamics

Key ideas from thermodynamics are summarised below and in Slide 4. Also see the topic ‘Elements of Statistical Mechanics and Soft Condensed Matter’, Lecture 1 ‘Statistical mechanics’.


For a system of N components, comprising ni molecules of component i, with Gibbs free energy G (T, p, n1, n2 ...nm):

Equation 1
Equation 1


where the chemical potential, μ, of the ith species (per molecule) is defined as

Equation 2
Equation 2


It is also sometimes convenient to define and use the molar chemical potential, i.e. per Avogadro’s number of molecules. (Note that µ is an intensive quantity.) For an ideal gas mixture (and, to a reasonable approximation, for a mixture of molecules or ions in dilute solution):

Equation 3
Equation 3


where µi0 is the ‘standard’ chemical potential and ci the concentration of the ith component; c0 is a standard reference concentration (which is normally 1M i.e. 1 mole per dm3). The concentration-dependent term is derived from the term RTlnpi, where pi = ci p is the partial pressure of the ith component, in the expression for the molar free energy of the system.

If species i is an ion of charge q with a position-dependent electrostatic potential energy qV(r) then this can be included in a generalised µ which becomes the electrochemical potential µ*:

Equation 4
Equation 4


If a single ion of charge qwere to move from a region of potential V1 and concentration c1 to a region of potential V 2 and concentration c2 then the Gibbs energy of the system would change by:

Equation 5
Equation 5


1.2.2 Chemical kinetics

Key ideas from chemical kinetics are summarised below and on Slide 5. Also see the supplementary Boxes ‘Chemical reaction kinetics and equilibrium’.


Consider a chemical reaction:

aA + bB + .....xX + yY + .....

Equation 6

If the reaction were to proceed from left to right (at constant temperature and pressure) by one formula’s worth of reactant molecules or ions, then the Gibbs energy of the system would change by:

Equation 7a
Equation 7a


Equation 7b
Equation 7b


where

Equation 8
Equation 8


is the standard free energy change of the reaction.

If we were following more usual conventions in chemistry, the intensive quantity µ would be defined per mole, not per molecule, kB would be replaced by the molar gas constant R and ΔG and ΔG0 in the formulae above would be, respectively, the molar free energy change and standard molar free energy change of the reaction.

In equilibrium, G is minimum so ΔG = 0. This is consistent with the law of mass action, that in equilibrium:

Equation 9
Equation 9


where K is the equilibrium constant:

Equation 10
Equation 10


and

Equation 11
Equation 11


This relationship is generally true. For single-step reactions (elementary reactions) the rates of the forward and backward reactions are proportional to the probabilities that the reactants collide, so (for any set of concentrations):

Equation 12a
Equation 12a


Equation 12b
Equation 12b


where k+,k- are rate constants. In equilibrium:

Equation 13
Equation 13


1.3 Biological membranes

Slides 6-8 are reminders of some important properties of biological membranes. Also see the topic ‘Essential elements of statistical mechanics’: Lecture 3 ‘Amphiphile aggregation: critical micelle concentration’, Lecture 4 ‘Micelle geometry’ and Lecture 5 ‘Lipid bilayers’.

Phospholipids (Slide 6) are important constituents of biological membranes. They are amphiphiles – they have two non-polar, hydrophobic tails and polar, hydrophilic heads (negatively charged or zwitterionic – i.e. one each of positive and negative charges). The tails typically contain 14-24 carbon atoms. One usually contains one or more double bond, which introduces kinks.


Above a critical concentration, amphiphiles in water spontaneously aggregate into stable structures driven by the hydrophobic interaction as explained in Lecture 3 of the statistical mechanics topic: ‘Amphiphile aggregation: critical micelle concentration’– the hydrophobic tails pack together, leaving only the polar head groups exposed to the solvent. The structures formed depend on the structure of the amphiphilic molecules. Single-tailed lipids can form stable spherical micelles in which the tails pack together to form a hydrophobic core. Phospholipids are the wrong shape to do this – they are roughly cylindrical, and can pack much more readily to form bilayers (Slide 7). A bilayer can avoid the high energy cost of a strained capping region by forming a closed shell – for example a vesicle, which is a small, aqueous compartment surrounded by a lipid bilayer.


A biological membrane – for example the plasma membrane that surrounds each cell – is a lipid bilayer about 5 nm thick consisting of amphiphilic molecules – mainly phospholipids. Embedded within the membrane are other molecules, e.g. transmembrane proteins which are responsible for most of the dynamic processes carried out by the membrane. The membrane is a 2D fluid – its fluidity is increased by the double bonds in the hydrocarbon tails of the lipids and by included molecules such as cholesterol which disrupt stable packing.

The plasma membrane (Slide 8) defines the boundary of cell. It keeps molecules generated within the cell from leaking out, and prevents unwanted molecules from diffusing in. It is selectively permeable – there are transport mechanisms that allow specific ions and molecules to be taken up from the surroundings, and others to be deliberately removed from the cell. Concentration gradients across the plasma membrane are used for energy storage – energy obtained from oxidation of chemical fuels or from absorption of light during photosynthesis is used to pump hydrogen ions across a membrane to create a pH gradient and a capacitive voltage – energy recovered by allowing hydrogen ions back across the membrane is used to generate ATP.


1.4 Electrical properties of lipid bilayers

1.4.1 Lipid bilayers as insulators

Lipid bilayers are insulators – they are very effective barriers to charged molecules, in particular, small ions (Slide 9).


The ‘self energy’ of a spherical ion is

Equation 14
Equation 14


The energy required to take an ion from water (dielectric constant 80) to the middle of a bilayer (dielectric constant ~2) can be estimated by integrating the energy stored in its electric field:

Equation 15
Equation 15


For a potassium ion with radius ~0.2nm this is ~2.6×10-19 J (1.6 eV, 65 kBT). The associated Boltzmann factor is e-65 ~10-28, so there is essentially no chance of spontaneous partitioning of the ion into the membrane. Screening of the ion’s charge in water is by orientation of water molecules (see Slide 9), which are electric dipoles. A more accurate calculation would account for the microscopic structure of the water molecules close to the ion – most of the energy associated with screening can be attributed to the first hydration shell.

The membrane is an insulating plane between two conducting solutions, i.e. a capacitor (Slide 10). The specific capacitance of the membrane is relatively high because it is only two molecules thick.

Capacitance per unit area = ε0εr / t

Equation 16


1.4.2 Nernst equation and diffusion potential

If one type of ion is allowed to cross the membrane freely – for example through an ion channel specific to that type of ion – ions will cross until equilibrium is reached. At equilibrium the free energy of ions is the same on both sides of the membrane. This is equivalent to stating that the electrochemical potential is the same on both sides of the membrane (see Slide 4).


The voltage at this equilibrium is called a diffusion potential or Nernst potential and the equation that relates the diffusion potential ΔV to the ion concentrations either side of the membrane is called the Nernst Equation:

Equation 17a
Equation 17a


Equation 17b
Equation 17b


Despite the relatively high membrane capacitance, the number of charges that need to move to establish a typical membrane potential is very small compared to the number present in a typical solution.

(Biological membranes usually have a negative surface potential due to a negative surface charge. The balance between electrostatic and entropic forces on dissolved ions near charged surfaces is treated by Nelson, chapter 7.4, 11.1. Lecturers may wish to extend this topic to include treatments of the Poisson-Boltzmann equation, charge screening and the Debye length.)

1.4.3 Ion motive force and proton motive force

If ions are pumped across a membrane away from equilibrium by coupling this process to an external source of free energy (Slide 12), the electrochemical potential is non-zero. This means that each ion gives up a free energy equal to the electrochemical potential Δµ* when it crosses the membrane back towards equilibrium. Δµ*/q is called the ion-motive force and has units of volts.


In general, discontinuity in electrochemical potential drives current:

Equation 5
Equation 5


In equilibrium: Δµ* = 0, leading to the Nernst equation

qΔV + kBT Δln(c) = 0

Equation 18

Each ion type has its own Δµ* – though all are subject to the same electrostatic potential.

If we are dealing with H+ ions (protons) the ion-motive force is called the protonmotive force, pmf (Slide 13). Note, however, that H+ ions are not free protons dissolved in water, but are always attached to a parent water molecule making a hydronium ion, H3O+.


pmf is a measure of the deviation from proton equilibrium across the membrane – the ‘driving force’ for proton transport across the membrane. It is generated by active transport of protons across the membrane, using for example absorption of photons or chemical break-down of food as a free energy source.

pmf is the difference across the membrane in the electro-chemical potential of H+:

pmf = Δµ* = Vm + Δµ/e

Equation 19a

= Vm + ( kBT/e ) ln (cin / cout)

Equation 19b

Here Vm is the electrical potential (defined as free energy per unit charge) and Δµ is the chemical potential (defined as free energy).

pmf is the extra free energy per unit charge inside the cell relative to that outside. The sign convention is ‘inside minus outside’ (e.g. cin – cout). Usually both components are negative, but inside has the lower voltage and lower [H+]. Typically, pmf is in the range -150 mV to -200 mV.

1.5 ATP – Adenosine triphosphate

One of the most important processes driven by pmf is ATP synthesis (Slide 14). ATP – adenosine triphosphate – is a building block of RNA. It is also a high-energy molecule that is often called the ‘free energy currency’ of the cell. Most biological ATP is synthesised from ADP (adenosine diphosphate) and inorganic phosphate in a reaction driven by the pmf, and is thus coupled to the energy-yielding oxidation of foodstuffs (in animal cells, fungi, and some bacteria) or to the capture of light energy (in plant cells and some bacteria) – these are the processes that generate pmf. The hydrolysis of ATP back to ADP and inorganic phosphate in turn provides the energy to drive many cellular reactions, including synthesis of molecules with a positive ΔG of formation, and to operate molecular motors – e.g. kinesin and myosin.


A human body contains about 250 g of ATP, but each molecule is continuously recycled (ATP to ADP + Pi and back to ATP; Pi is inorganic phosphate) so that we synthesise and hydrolyse of the order of our body weight of ATP each day.

The biochemical standard free energy of ATP hydrolysis, ΔG0 is 30 kJ mol-1 which is about 12 kBT per molecule (under standard conditions: 25 °C, in water at pH 7, 1 atmosphere pressure, all chemical species at 1 M concentration).

The actual free energy ΔG in living cells is nearer 20-30 kBT, but it depends upon the concentrations of ATP, ADP, Pi etc. The concentration of Mg2+ is also important. ATP has charge 4-, ADP has charge 2-, usually they are electrostatically bound to one magnesium ion (Mg2+).

Some processes use GTP instead of ATP as an energy source (guanine instead of adenine as the base.)

Phosphoanhydride bonds between phosphate groups are ‘high-energy’ bonds for the following reasons (in order of increasing importance):

  • free phosphate ions are stabilised by resonance, which increases their entropy, i.e. electrons associated with P-O bonds are more delocalised than in ATP;
  • electrostatic repulsion between negative phosphate groups in ATP;
  • hydration free energy is larger for the products, partly as a result of better hydrogen bonding to water.