Bioenergetics

• Bioenergetics is the quantitative study of energy transformations in biological systems -- these concepts are the underpinnings for all of biochemistry.
  
  
• The change in Gibbs free energy (DG) for a reaction quantitates the energy available to do useful work. It is related to the change in enthalpy and the change in entropy:
  DG = DH - TDS
  
  
• The standard free energy change for a reaction (DG°) is the change in free energy for going from STANDARD CONDITIONS TO EQUILIBRIUM.  It is related to the equilibrium constant by the equation
  DG° = - RTlnK_eq
  
  
• The actual free energy change (DG) depends on 2 parameters:
  • the standard free energy change for that reaction (DG°) (and thus to K_eq, defining where equilibrium for this reaction lies), and
  • the actual mass action ratio, reflecting the actual starting conditions, the actual concentrations of reactants and products

    
    

    actual DG = DG° + RTln{actual mass action ratio}
    
    
• The sign of DG tells us in which DIRECTION the reaction would have to go to reach equilibrium (the "spontaneous" direction), but DG gives NO information about RATE at which reaction will go.
  
  
• Free energy changes are ADDITIVE: for coupled or sequential reactions, the overall free energy change for the process is the SUM of the DGs for the component reactions.  This permits FREE ENERGY COUPLING, in which an exergonic reaction can "drive" an endergonic reaction if they can be coupled.
  
  
• The essence of protein function/action is BINDING (recognition of and interaction with other molecules); biochemists generally use the equilibrium dissociation constant (K_d) for a bimolecular complex to describe the strength of the binding interaction.
  

General chemical reaction:

• aA + bB  <=>   cC + dD
  
• All reactions/processes proceed in direction required to go TOWARD EQUILIBRIUM.
  

• For this reaction, the mass action ratio is given by
  

• The mass action (m.a.) ratio at equilibrium is the equilibrium constant for the reaction, K_eq:
  
  

• Bioenergetics: the quantitative study of energy transformations in biological systems (part of thermodynamics)
  • essential for understanding
• how metabolic processes provide energy for the cell
• the structures of macromolecules
• how membrane transport processes occur
• all the fundamental processes that define biochemistry!
  
  
• bioenergetics useful for describing conditions under which processes occur spontaneously 
  All reactions/processes proceed spontaneously in whatever direction is required to achieve, or at least go toward, equilibrium; "spontaneous" direction is always toward equilibrium
  
  
• Although we can use bioenergetics to determine whether a process will occur spontaneously, bioenergetics gives us no information as to how fast the process will occur.

• Gibbs Free Energy, G
  (the thermodynamic function that is most useful for biochemistry)
  
  G a function of
  • Enthalpy, H, a measure of the energy (heat content) of the system at constant pressure, and
  • Entropy, S, a measure of the randomness (disorder) of the system
    

  G = H - TS
  

• For any process,
  • If DH is negative, then heat is released (a favorable enthalpy change)
    • making bonds:   DH < 0, favorable
    • breaking bonds:   DH > 0, unfavorable
      
      
• If DS is positive, then the randomness of the system increases (a favorable entropy change). 
  • increased disorder:   DS > 0, favorable
    
  • increased order:   DS < 0, unfavorable
    
    
• change in free energy for any process:   DG = DH - TDS. 
• If DG is negative (DG < 0): process goes in direction written (left to right)
• If DG = 0: process is at equilibrium
• If DG is positive (DG > 0): process goes in reverse (right to left)
• Value and sign of DG depend on interplay of enthalpy and entropy 
  (Neg. DH doesn't necessarily  -->  neg. DG, and positive DS doesn't necessarily  -->  neg. DG.)
  
  
• Example: Consider melting of ice and values of DG, DH, DS at various temperatures.
  
  • PROCESS: H₂O_solid --> H₂O_liquid
  • DH positive (unfavorable) because hydrogen bond are breaking
  • DS positive (favorable) because H₂O molecules more disordered in water than in ice

• More examples that illustrate how DH and DS interact to produce a favorable DG (DG < 0):

• critical role of temperature in determining sign of DG (DG = DH - TDS).

FREE ENERGY AND CHEMICAL REACTIONS

• Standard conditions ("standard state"):  1 M in each reactant and product (or 1 atm for gaseous reactants or products), with temperature = 25 °C = 298 K.
  
  
• The chemical potential of compound A,, in solution:

 
where is the free energy of A in its standard state and [A] is the actual molar concentration of A in the solution.  A similar equation can be written for each component of the solution

• For the reaction



the overall free energy change involved in going from the actual starting chemical potentials of all components to equilibrium is given by  

• After substituting the equations for the chemical potential of each component, we get:

.

• This important general equation gives the free energy change for any reaction to go to equilibrium from ANY starting conditions.

• DG^o is the standard free energy change, the free energy change for the reaction when going from standard state chemical potentials of all components to equilibrium.

• at equilibrium, DG = 0
  thus, DG^o = - RTlnK_eq
  K_eq and DG^o are thus different ways to express the same information, and are interconvertible.
  

SOME PRACTICAL CONSIDERATIONS:

How do we obtain values for DH, DG and DS?

One way is to study the temperature dependence of the equilibrium constant for a reaction:

• van't Hoff Equation:

  .

• plot of RlnK_eq vs. 1/T (or 1000/T to keep the numbers greater than 1) should be a straight line with
  slope = - DH^o, the enthalpy change for reaction going from standard state to equilibrium.

These data show the temperature dependence for the thermal denaturation of a protein, a process which can be described as the conversion from the native state N to the denatured state D . This plot has a slope of -586 = -DHo, so DHo = 586 kJ/mol (unfavorable). At 54.5oC (327.5oK), Keq = 0.27, so DGo = 3.56 kJ/mol -- unfavorable at this temperature. Recall that DG = DH - TDS, so DSo = (DHo - DGo)/T = 1,778 J/mol*K (favorable). DSo is favorable because the denatured state is a large collection of different states, whereas the native state is a single state.

EQUILIBRIUM CONSTANTS AND FREE ENERGY CHANGES:

One of the important reactions in biochemistry is the hydrolysis of ATP:

How do we calculate free energy changes for this reaction?

• The free energy change for this reaction is given by
  
  
  
  
• In biochemistry, we make two simplifying assumptions (justifiable under physiological conditions):

• The concentration of water [H₂O] does not change during the reaction, i.e. [H₂O] = 55.5 M.
• The pH = 7.0 and does not change during the reaction, i.e. [H⁺] = 10^–7 M.

• Under these conditions the (constants) [H₂O] and [H⁺] are incorporated into DG^o to give a new "biochemical" standard free energy change, DG^o'.  The "biochemical" equilibrium constant, related to DG^o', is designated K_eq'.    NOTE:  "Biochemical" free energy changes are always supposed to be designated with "prime" symbols.  However, in this course we'll ALWAYS be dealing with biochemical free energy changes, so if the primes are left out sometimes, assume that they're supposed to be there!

• Using these assumptions, and for simplicity, using
• ADP instead of ADP^-3
• ATP, instead of ATP^-4
• P_i, instead of HPO₄^-2, the free energy equation becomes:

 

Sample Calculations

Now we can do some calculations ( step-by step answer; complete answers (PDF format).

COUPLED REACTIONS:

• Many biological processes are endergonic -- they require the input of free energy.
  endergonic reaction:  unfavorable (positive) free energy change for going in direction of equilibrium
  

• How can a reaction that "wants" to go BACKWARDS be "driven" forwards?
  by coupling it to an exergonic reaction (one with a negative, favorable, free energy change)
  

Fig. 1-9a:  Mechanical example
Downward motion of an object releases potential energy (pink side, exergonic) that can be used to do mechanical work, moving another object upward (blue side, endergonic).

• coupling mechanism (rope in this example) required to enable exergonic process to drive endergonic one

Fig. 1-9b:  Chemical example
phosphorylation of glucose to produce glucose-6-phosphate

• very important reaction in the cell
• first reaction in metabolism of glucose that enters a cell from the blood

• Reaction 1:  condensation of glucose (alcohol) with inorganic phosphate ion (acid) to make glucose-6-phosphate (an ester)
  

Glucose + P_i <=> glucose-6-phosphate + H₂O   (DG^o' = + 13.8 kJ/mol, endergonic)

• Reaction 2:  hydrolysis of ATP, a phosphoanhydride, to generate ADP and inorganic phosphate
  

  ATP + H₂O <=> ADP + P_i                        (DG^o' = - 30.5 kJ/mol, exergonic)

• To couple the 2 reactions (which requires some chemical mechanism, of course),
  add reactants on left, add products on right, and add DG^o' values to get DG^o' for coupled reaction:

  Glucose + ATP <=>  glucose-6-phosphate + ADP     (DG^o' = - 16.7 kJ/mol)

   

   

• The coupled reaction is exergonic; it will go spontaneously (forward, left to right) in the cell, but will it proceed at a rate consistent with cellular needs?

• There's NO information about rates in the value of a DG -- we can't answer this question from bioenergetics.
  
  
• Most biological reactions would proceed at a very slow rate indeed if they're not catalyzed.  The biological catalyst enabling the coupled reaction above to proceed on a biological timescale (as opposed to a geological timescale!) is an enzyme, hexokinase.
  
  
• Free energy coupling, with enzymes as catalysts, is the strategy used in metabolic pathways.

"HIGH-ENERGY" COMPOUNDS:

• energy to drive endergonic reactions comes from oxidation of fuels (exergonic)
• energy released during oxidation of nutrients trapped in the form of a few energy-rich or "high energy" compounds, such as ATP.
• "high-energy" compound (operational definition, not a very precise term but commonly used in biochemistry):
  a compound with a functional group, often a phosphoryl group, whose free energy of transfer to another compound proceeds with a large negative DG.
  • transfer to water = hydrolysis (DG^o' = -30.5 kJ/mol for hydrolysis of the terminal phosphoanhydride bond of ATP)
  • for comparison, hydrolysis of an ordinary phosphate ester bond (e.g. glucose-6-phosphate) has DG^o' = -14 kJ/mol.
• The somewhat arbitrary cut-off for calling a compound "high energy" is a free energy of hydrolysis more negative than about -25 kJ/mol.

Examples of "high-energy" and "low-energy" compounds:

• Free energy of hydrolysis (the numerical value of which is also referred to as a group transfer potential) represents potential energy, "stored" in the compound, that can be used in coupled reactions to drive unfavorable reactions like phosphorylation of glucose to form glucose-6-phosphate and to do other kinds of biological work.

  • A compound with a higher group transfer potential than ATP, e.g. phosphoenolpyruvate, can be used to transfer a phosphoryl group to ADP, making ATP.
  • Phosphoryl groups are sometimes shown in shorthand as a P in a circle, without specifying the oxygen atoms.
  • There are other groups in biomolecules besides phosphoryl groups that can have high group transfer potentials, e.g. acyl groups of thioesters of carboxylic acids (acetyl-CoA in the table above). (What's an "acyl group"?)

   

Fig. 14-9: Ranking of some biological molecules by standard free energies of hydrolysis (group transfer potential)

PEP as phosphoryl donor to generate ATP

• Addition of inorganic phosphate to ADP (phosphate anhydride formation) unfavorable under biochemical standard conditions:  DG^o' = + 30.5 kJ/mol
• Cells use other exergonic processes to drive ATP synthesis, e.g.,
  • exergonic transport processes (ion gradients)
  • direct phosphorylation by phosphoryl group donors with higher phosphoryl group transfer potentials than ATP
• Example of direct phosphorylation of ADP: step in glycolysis, phosphorylation of ADP to form ATP, with PEP (an enol phosphate) as phosphoryl donor:
  • PEP + H₂O <=> pyruvate + P_i (DG^o' = - 61.9 kJ/mol)
  • ADP + P_i <=> ATP + H₂O (DG^o' = + 30.5 kJ/mol)
    SUM:
    
  • PEP + ADP <=> pyruvate + ATP (DG^o' = - 31.4 kJ/mol)
• Reaction works because PEP has a higher group transfer potential than ATP.
• Enzyme, pyruvate kinase, provides the coupling mechanism and increases rate.
• NOTE: The actual DG' (whose components are the DG^o' for the reaction and the ln{actual mass action ratio} term) under cellular conditions determines the direction a reaction will go in vivo.

The bond attaching a group with a high group transfer potential to the rest of the compound (the bond whose hydrolysis has a large negative DG^o' is sometimes shown in shorthand as a "squiggle" ~ to emphasize where the group transfer will be occurring.

What does it really mean to say that a compound has a large negative free energy of hydrolysis?  Does breaking the bond actually liberate free energy?

ATP (ATP)

What makes the free energy of hydrolysis of the phosphoanhydride bond of ATP such a large negative value, i.e., WHY does that phosphoryl group have such a high group transfer potential? 

Factors involved in the large negative free energy change for hydrolysis of ATP
(all contribute to the fact that the reaction PRODUCTS have lower free energy than the reactants):

 

Relief of charge repulsion ATP's 3 phosphate groups' negative charges are close to each other. Conversion to ADP and (separate) Pi relieves some of the unfavorable electrostatic interaction.  (An equal amount is relieved when ADP is converted to AMP; hydrolysis of phosphoanhydride bond of ADP also has large negative DGo'.)

Greater resonance stabilization of products of hydrolysis reaction Inorganic phosphate ion can has more resonance forms than the terminal phosphoryl group of ATP, which gives it a higher entropy.  Therefore freeing Pi (by hydrolysis) is an entropically favorable process.

2 other factors involved in the negative free energy change in ATP hydrolysis:

• release of a proton product (H⁺) into the solvent, which has a very low H⁺ concentration (pH 7)
• greater degree of solvation of products (ADP and P_i) than of reactants (ATP), i.e., more noncovalent interactions formed with the solvent H₂O molecules

Why is ATP so important as the energy currency of the cell?

The Importance of METASTABILITY:
metastable compounds:

• thermodynamically unstable (equilibrium under the given conditions lies in the direction of breakdown), but
• kinetically stable (in absence of a catalyst, breakdown is very slow)
• e.g., ATP -- thermodynamically unstable but kinetically stable
  • useful characteristic -- energy stored in phosphoanhydride bond is not squandered by random hydrolysis; its energy is released only under controlled (catalyzed) conditions, when it can be usefully employed to do work

LIGAND BINDING:
The essence of protein function/action is BINDING (recognition of and interaction with other molecules). Proteins bind (just a few examples):

• small molecules & ions (including protons!)
• other proteins
• nucleic acids (DNA, RNA)
• membranes
• polysaccharides

All the binding is the result of specific, usually NONCOVALENT interactions between molecular surfaces, complementary in SHAPE (lots of van der Waals interactions) and in CHEMISTRY (hydrogen bonds, salt linkages, hydrophobic interactions).

LIGAND: a molecule or ion (usually small) that's bound by another molecule (usually large, e.g. a protein)

Binding equilibrium:

• concept fundamental in biochemistry -- same general concept as proton binding/dissociation
• Analysis of binding equilibria requires a few simple algebraic equations:
• chemical equation for the dissociation of the ligand (L) from the protein (P)
   
• expression for the equilibrium dissociation constant K_d for the reaction
  (The concentrations of free protein [P], free ligand [L], and the P•L complex [PL] in this expression are the equilibrium concentrations.)
  What are the units of K_d for this protein•ligand dissociation reaction?

Suppose a protein can bind either of 2 different ligands.  K_d is 0.1 µM for one ligand (A), and 1.0 µM for the other ligand (B).  Which ligand shows higher affinity binding (tighter binding, a stronger tendency to associate), ligand A or ligand B?

• A useful parameter for plotting data is q (greek letter theta), where q is the fraction of the total binding sites on the protein ([P]_total) that are actually OCCUPIED by ligand under the given conditions:
  
• What are the units of q?
• The value of q varies from 0 (all sites empty, at [L] = 0) to a maximum of 1.0 (all sites occupied, "saturation" conditions, [L] >> K_d); q = 1.0 isn't experimentally quite achievable, since the approach to site saturation is asymptotic.

Combining expressions for Kd and q gives . This is the equation of a rectangular hyperbola.  The concentration of the ligand, [L], for which q = 0.5 is equal to the Kd. It is difficult to accurately estimate q = 0.5 from inspection of the graph, because the saturation curve approaches q = 1.0 asymptotically.

• A linear transformation of the binding equation can be used to determine K_d by a graphical extrapolation to infinite [L], or a curve-fitting program on a computer or calculator can fit the data to the equation for a hyperbola and give a value for K_d that best fits the data.
• Once K_d has been determined, then the free energy of dissociation can be determined from
  DG^o = -RTlnK_d. The free energy of association or the binding free energy is - (DG^o of dissociation), because K_a = 1/K_d.
  NOTE: K_d is the concentration of ligand needed to HALF-SATURATE the binding sites.

Example

For the binding of a certain ligand to a protein at 25^o C and pH=7.0,  K_d = 1x10^-7 M. What is the free energy change for association of the ligand with the protein under standard conditions?

• For the dissociation reaction,  DG^o' = -RTlnK_d
• DG^o' = - (8.315x10^-3kJ/(mol•K))(298K)(-16.1) = + 39.9 kJ/mol
  Dissociation is not favored under standard conditions.
• For the association reaction,  DG^o' = -RTlnK_a = -RTln(1/K_d)
• DG^o' = - (8.314)(298)(16.1) = - 39.9 kJ/mol
  Association is favored under standard conditions.

Thermodynamics and the HIV virus

• During replication, the AIDS virus produces a large polypeptide chain, which is cleaved into smaller proteins required for the assembly of the virus.
• The cleavage of the the precursor protein is carried out by a specific enzyme called the HIV protease. 
• In the absence of the HIV protease, the virus can not mature and can not infect cells. 
• Therefore, the HIV protease has been the target of intense study to try to discover specific inhibitors that can prevent virus replication.
• A variety of different compounds have been developed, as shown below.

These compounds all have high affinity for the HIV protease, K_d = 5-15 x 10^-8 M (HIV).

The virus fights back!  

• The presence of the inhibitor places strong selective pressure on the virus to evolve an inhibitor-resistant form of the protease.  Apparently, the virus is more concerned with having a protein with a low affinity for the inhibitor than with a protein with high enzymatic activity.
• Binding of inhibitors to the wild type HIV protease and some resistant mutants:
  (Mutations are indicated with amino acid abbreviations; for example, I82V means that the wild-type HIV protease has an Isoleucine residue at the 84th amino acid position in the protein, and that mutant has substituted a Valine (V) residue at that position.)

• How much of a change in K_d is necessary to significantly alter the amount of inhibitor bound?
  

(Values of K_d on the plot are in units of nM.)