INTRODUCTORY BIOPHYSICS A. Y. 2014-15
10. ENZYMES, TRANSITION STATE THEORY AND SOME MORE THERMODYNAMICS
Edoardo Milotti Dipartimento di Fisica, Università di Trieste We have already met oxidative stress, and we know that it is harmful to cells.
Example: oxidative stress is part of the normal life of cells, for instance an electron- carrying riboflavin (vitamin B2) can encounter an oxygen molecule and convert it to - the dioxide(1−) anion O2
- +
These dioxide anions are highly reactive and can damage other molecules, DNA in particular.
How do cells defend themselves?
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 There are several important defense mechanisms against oxidative stress.
One of them involves superoxide dismutase (SOD) which converts the - dioxide(1−) anion O2 into oxygen (O2) or hydrogen peroxide (H2O2)
Another one is catalase, which converts the still dangerous hydrogen peroxide into water and oxygen (O2).
Still another is peroxiredoxin which also converts hydrogen peroxide into water and oxygen (O2).
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 The structure of catalase in human red blood cells: a protein with four polypeptide chains, each more than 500 aminoacids long. EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Superoxide dismutase and catalase are enzymes, i.e., biological catalysts.
Enzymes greatly accelerate specific chemical reactions.
Enzymes are mostly proteins, however some of them are based on RNA (ribozymes).
Our understanding of enzyme action is largely based on Transition State Theory (TST), and we start with a short refresher of thermodynamic concepts
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 A chemical thermodynamics refresher
1. Enthalpy
Recall that a change in internal energy is the sum of the heat absorbed and of the work done by the system
ΔU = ΔQ − ΔW which is the first principle of thermodynamics, and that work can be further subdivided into work due to volume expansion (useless) and all the other work (non-PV work):
ΔW = PΔV + ΔW ′
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Then
ΔU = ΔQ − ΔW = ΔQ − PΔV − ΔW ′ and we can formally restore the form of the first principle by defining the state function enthalpy
H = U + PV
if no non-PV work is done on so that the system, then the enthalpy change corresponds to the ΔH = ΔU + Δ(PV ) = ΔQ − ΔW ′ heat absorbed by the system
at constant pressure Δ(PV) = PΔV, as in most chemical reactions in the laboratory
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 2. Helmholtz free energy
According to the second principle of thermodynamics
ΔQ ≤ T ΔS and therefore
ΔU = ΔQ − ΔW ≤ T ΔS − ΔW = Δ(TS) − SΔT − ΔW
Therefore, when we define F = U − T S we find
ΔF = Δ(U − TS) ≤ −ΔW − SΔT
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Therefore, in isothermal processes (where the system exchanges heat with a heat bath)
F U TS W S T W Δ = Δ( − ) ≤ −Δ − Δ = −Δ
isothermal process
and therefore the work done by the system is less or equal than the decrease of free energy
ΔW ≤ −ΔF
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Therefore, in processes where no work is done or absorbed by the system
0 = ΔW ≤ −ΔF i.e.
ΔF ≤ 0 and this is the condition for a spontaneous process with no work involved.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 3. Gibbs free energy
The Gibbs free energy is like the Helmholtz free energy, for processes where the pressure is held constant:
G = H − TS = U + PV − TS = F + PV
and we find again that the condition for a spontaneous process, with no work involved, is
ΔG ≤ 0
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Let’s summarize it again, just for clarity ...
ΔU = ΔQ − ΔW
= ΔQ −[PΔV + ΔW ′] = ΔQ − ⎣⎡Δ(PV ) −VΔP + ΔW ′⎦⎤
≤ T ΔS − ⎣⎡Δ(PV ) −VΔP + ΔW ′⎦⎤
= Δ TS − SΔT − Δ PV −VΔP + ΔW ⎣⎡ ( ) ⎦⎤ ⎣⎡ ( ) ′⎦⎤ then Δ(U + PV − TS) ≤ −SΔT +VΔP − ΔW ′ and therefore for transformation at constant temperature and pressure and no non-PV work H = U + PV F = U − TS ΔG ≤ 0 G = F + PV = H − TS
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 ΔG = ΔH − T ΔS ≤ 0 ΔH ≤ T ΔS
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 4. Concentrations, Gibbs free energy, and chemical kinetics
Entropy of mixing in binary solutions
n1 molecules of solvent n2 molecules of solute
N = n1 + n2
Then the number of configurations is
N! Ω = n1!n2 !
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 N! Ω = n1!n2 !
lnΩ ≈ (N ln N − N ) − (n1 lnn1 − n1 + n2 lnn2 − n2 )
= N ln N − n1 lnn1 − n2 lnn2
= (n1 + n2 )ln(n1 + n2 ) − n1 lnn1 − n2 lnn2
n1 n2 = −n1 ln − n2 ln n1 + n2 n1 + n2
= −N (X1 ln X1 + X2 ln X2 ) X1,2 are the volume fractions
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Therefore the entropy change due to mixing is
S k ln ln1 k N X ln X X ln X Δ m = B ( Ω − ) = − B ( 1 1 + 2 2 ) and, assuming that there is no change in contact energy when the molecules of solvent and solute mix, the corresponding Gibbs free energy change is
ΔG = −T ΔSm = kB NT (X1 ln X1 + X2 ln X2 )
= nRT (X1 ln X1 + X2 ln X2 )
= X1ΔG1 + X2ΔG2
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 We see that we can associate a free energy to each substance A in solution
GA = nA RT ln XA (nA = nXA ) and in particular, if we consider the free energy change with respect to standard conditions concentrations volume fractions (mole/l)
X A G − G(0) = n RT ln A = n RT ln [ ] A A A (0) A XA [A] 0 and if we let [A]0=1M (0) 1 mole/l GA − GA = nA RT ln[A]
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 5. Chemical kinetics
Elementary reaction
A → P it can occur via a sequence of elementary reactions, with intermediates, e.g.,
A → I1 → I2 → P
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Rate equations
The rate at which a reaction proceeds is proportional to the probability of bringing all the reactants in the same place at the same time, i.e., it is proportional to their concentrations, therefore the rate of the general elementary reaction
aA bB zZ P + +…+ → is a b z rate = k A B Z n = a + b + + z [ ] [ ] …[ ] …
rate constant order of the reaction (notice that the rate constant has units adapted to the order of the reaction)
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Example: first order reaction
⎧ d[A] ⎪ = −k[A] ⎪ dt A = A exp −kt ⎪ A + P = A [ ] [ ]0 ( ) A → P ⇒ ⎨ [ ] [ ] [ ]0 ⇒ P = A ⎡1− exp(−kt)⎤ ⎪ [A] = [A] [ ] [ ]0 ⎣ ⎦ ⎪ t=0 0 ⎪ P = 0 ⎩ [ ]t=0 The concentration of A decreases and it is exactly half the initial concentration when
ln2 [A] = [A] exp(−kt1/2 ) = [A] 2 ⇒ t1/2 = 0 0 k
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Example: second order reaction
⎧ d[A] 2 ⎪ = −k[A] ⎪ dt 1 1 kt ⎪ 1 = + [A]+ [P] = [A] [A] [A]0 2A → P ⇒ ⎨ 2 0 ⇒ ⎪ [P] = 2 [A]0 −[A] ⎪ [A]t=0 = [A]0 ( ) ⎪ P = 0 ⎩⎪ [ ]t=0 The concentration of A decreases and it is exactly half the initial concentration when
2 1 1 = + kt1/2 ⇒ t1/2 = [A]0 [A]0 k[A]0
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 6. Equilibrium constants
We apply these concepts to the reversible chemical reaction
aA + bB cC + dD
and we note that at equilibrium
a b c d k f [A] [B] = kb [C] [D]
i.e. the forward rate is equal to the backward rate, or also
c d [C] [D] k f a b = = Keq [A] [B] kb
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Then, the free energy change for the i-th species with respect to the standard state, per mole, is
0 G − G( ) = RT ln i i i [ ] and therefore, in a reaction, the Gibbs free energy change splits into parts that take into account the chemical bonds and the concentration changes
ΔG = ΔG0 + cΔGC + dΔGD − aΔGA − bΔGB
= ΔG0 + cRT ln[C]+ dRT ln[D]− aRT ln[A]− bRT ln[B]
C c D d = ΔG + RT ln [ ] [ ] = ΔG + RT ln K 0 [A]a [B]b 0 eq
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 At equilibrium the free energy change vanishes
C c D d ΔG = ΔG + RT ln [ ] [ ] = ΔG + RT ln K = 0 0 [A]a [B]b 0 eq
⎛ ΔG ⎞ K = exp − 0 eq ⎝⎜ RT ⎠⎟
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 ⎛ ΔG0 ⎞ K = exp − eq ⎝⎜ RT ⎠⎟ (R ≈ 8.314 J K −1mol−1, RT ≈ 2.5 kJ mol−1 @ 300 K)
this is close to the binding energy of hydrogen bonds in water ≈ 5 kcal/mole ≈ 21 kJ/mole
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Transition-state theory (TST)
The idea is that a better description of chemical reactions can be obtained introducing a temporary, unstable state, the transition state, as in the following example of hydrogen exchange:
H − H + H → H − H − H → H + H − H A B C A B C A B C transition state
The decomposition of the transition state is assumed to be the rate- determining step of the reaction
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 The reaction coordinate
In chemistry, a reaction coordinate is an abstract one-dimensional time-like coordinate which represents progress along a “reaction pathway”. It is usually a geometric parameter that changes during the conversion of one or more molecular entities, and can sometimes represent a real coordinate (such as bond length, bond angle...)
In the formalism of transition-state theory the reaction coordinate is that coordinate which leads smoothly from the configuration of the reactants through that of the transition state to the configuration of the products.
The reaction coordinate is typically chosen to follow the path along the gradient of potential energy from reactants to products.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Now consider the reaction
† K † k′ A + B X ⎯⎯→ P + Q
for which we can write k has units [time]-1[concentration]-1
the first step is fast d P † and nearly at [ ] = k′ ⎡X ⎤ = k A B ⎣ ⎦ [ ][ ] equilibrium dt k’ is a rate, units are [time]-1 and the equilibrium constant of the first step is
† ⎡X ⎤ k ⎛ ΔG† ⎞ K † = ⎣ ⎦ = ∝ exp − [A][B] k′ ⎝⎜ RT ⎠⎟
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 activation barrier
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Since
⎡X † ⎤ † † ⎣ ⎦ k ⎛ ΔG ⎞ K = = ∝ exp⎜ − ⎟ [A][B] k′ ⎝ RT ⎠ we find ⎛ ΔG† ⎞ k ∝ k′exp − ⎝⎜ RT ⎠⎟
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 k’ is a frequency and therefore hk’ is an energy; the only energy scale in the breakdown of the transition state is the thermal energy kBT , and therefore
k T k T k′ ∝ B ⇒ k′ = κ B h h
Finally
k T ⎛ ΔG† ⎞ k = κ B exp − h ⎝⎜ RT ⎠⎟
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Enzymes
Enzymes are biological catalysts which share several important properties
• they lead to much higher reaction rates for catalyzed processes (acceleration factors in the range 106 – 1012) • they make reactions happen under mild conditions • they are very specific • the catalysis can be controlled and tuned
In the context of TST, enzymes work by lowering the activation barrier.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Since
† kBT ⎛ ΔG ⎞ k = κ exp⎜ − ⎟ h ⎝ RT ⎠ if an enzyme lowers the energy barrier by ΔΔ G † , we see that the new value of the rate constant is
† † † kBT ⎛ ΔG − ΔΔG ⎞ ⎛ ΔΔG ⎞ kEnz = κ exp − = k exp h ⎝⎜ RT ⎠⎟ ⎝⎜ RT ⎠⎟
This is a very sensitive function of the barrier height and enzymes achieve large acceleration factors with small energy changes.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Michaelis-Menten kinetics
k ⎯⎯+ → E + S ES ⎯k⎯cat → E + P ←⎯⎯ k−
d[P] v = = kcat [ES] dt d[S] = −k+ [E][S]+ k− [ES]+ vS dt d[E] = −k+ [E][S]+ k− [ES]+ kcat [ES] dt d[ES] d[E] = k+ [E][S]− k− [ES]− kcat [ES] = − dt dt
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 We must add the equation for enzyme mass conservation
[E]0 = [E]+[ES]
When the system is in quasi-stationary conditions, we obtain
k ES k ES k E S k E ES S − [ ]+ cat [ ] = + [ ][ ] = + ([ ]0 −[ ])[ ] then, using ⎡ E S ⎤ = v k we find ⎣ ⎦ cat
v [E]0 [S] kcat [E]0 [S] Vmax [S] = [ES] = v = = k k− + kcat k− + kcat K + S cat +[S] +[S] m [ ] k+ k+
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Michaelis-Menten Vmax [S] v = equation Km +[S] where we let
k− + kcat Vmax = kcat [E]0 Km = k+
maximal velocity Michaelis constant
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 0.8
0.6 max
V 0.4 ê V
0.2
0.0 0 1 2 3 4 S K
@ Dê
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Notice that at small concentrations
Vmax [S] kcat [E]0 [S] kcat v = = → [E] [S] K + S K + S K 0 m [ ] m [ ] m
and therefore the ratio kcat/Km corresponds to the normal rate of a binary reaction and it can be used to evaluate the acceleration afforded by enzyme action.
k Since cat [ E] can also be interpreted as the rate of an K 0 m elementary reaction, the same ratio kcat/Km represents the acceleration factor per enzyme molecule of the spontaneous conversion of the substrate.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 (2001) 938-945 34 and M. J. Snider, “The Depth of Chemical Time and the Power Time “The Depth of Chemical and M. J. Snider, Wolfenden
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from R. Acc. Chem. Res. of Enzymes as Catalysts”, (2001) 938-945 34 and M. J. Snider, “The Depth of Chemical Time and the Time “The Depth of Chemical and M. J. Snider, Wolfenden from R. from R. Acc. Chem. Res. of EnzymesPower as Catalysts”,
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from R. Wolfenden and M. J. Snider, “The Depth of Chemical Time and the Power of
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Enzymes as Catalysts”, Acc. Chem. Res. 34 (2001) 938-945 Lock-and-key specificity
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from M. Garcia-Viloca et al, “How Enzymes Work: Analysis by Modern Rate Theory and ComputerEDOARDO MILOTTI Simulations, - INTRODUCTORY BIOPHYSICSScience - A.Y. 303 2014-15 (2004) 186 Thermodynamics and viral self-assembly
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Important topics in the biophysics of viruses
• viral shape • thermodynamics of viral self-assembly • dynamics of viral self-assembly • internal capsid pressure • packaging DNA (or RNA) inside capsid • genetic drift and the concept of quasispecies
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 The taxonomy of viruses
• DNA viruses • dsDNA • ssDNA
• RNA viruses • dsRNA • ssRNA(+) • ssRNA(-)
• Retroviruses • ssRNA (RT) • dsDNA(RT)
image from F. A. Murphy, “The nature of viruses as etiologic agents of veterinary and zoonotic diseases”in rd VeterinaryEDOARDO Virology, MILOTTI 3 - INTRODUCTORYed. Academic BIOPHYSICS Press - (1999).A.Y. 2014-15 3 Nobel prize winners among the authors ... guess who!!
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 On the basis of crystallographic observations, Crick and Watson (1956), and later Caspar and Klug (1962) set basic constraints on possible virus shapes.
CW:
• viral genome too short to code for a single large protein for the capsid • there must be many small tiles that make up the capsid • unit cell of virus crystals is cubic (Caspar) • observed virus shape is often spherical • observed symmetry constraints (5-3-2 fold symmetry) point to an icosahedral shape
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Aaron Klug
Born: 11 August 1926, Zelvas, Lithuania
Nobel Prize in Chemistry in 1982 "for his development of crystallographic electron microscopy and his structural elucidation of biologically important nucleic acid-protein complexes”
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from D. L. D. Caspar and A. Klug, “Physical Principles in the Construction of Regular Viruses”, EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Cold Spring Harbor Symposia on Quantitative Biology Vol. XXVII (1962) EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 The Tobacco Mosaic Virus (TMV)
Tobacco mosaic virus (TMV). (A) Systemic infections of Nicotiana tabacum cv. Turk plants showing TMV-associated mosaic. (B) Necrotic local lesions on N. tabacum Glurk leaf, demonstrating Holmes’ N- gene resistance following inoculation with TMV. Photo: K.-B. G. Scholthof. (from http://www.apsnet.org/ publications/apsnetfeatures/pages/tmv.aspx)
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Wendell Meredith Stanley
Born: 16 August 1904, Ridgeville, IN, USA
Died: 15 June 1971, Salamanca, Spain
Nobel Prize in Chemistry in 1946 with James B. Sumner and John H. Northrop, "for their preparation of enzymes and virus proteins in a pure form”
He received a Nobel Prize in Chemistry in 1946 for his work on the tobacco mosaic virus, begun in the 1930s and which he crystallized in 1935. The demonstration of the molecular properties of the virus gave impetus to a new research approach in virology: the study of viruses as large molecules. This was a departure from the predominant view of viruses as infectious agents causing disease.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from W. M. Stanley’s Nobel Lecture (1946) M. Stanley’s from W. EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15
Plant Pathology: Problems and Progress, 1908-1958
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 from Franklin, R. E., Caspar, D. L. D. and Klug, A. (1959) "The Structure of Viruses as (1959)A. "The Structure of Viruses and L. D. D. Klug, from Franklin, Caspar, E., R. In Determined by X-ray Diffraction." ed.), S., University(Holton, C. of Wisconsin Press, Madison, Wisconsin, pp. 447-461. EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Bacteriophage phi X 174
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Bacteriophage phi X 174 , RCSB “Molecule, RCSB of the month”, February 2000 Goodsell D. D.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Bacteriophage phi X 174
capsid protein
spike protein , RCSB “Molecule, RCSB of the month”, February 2000 Goodsell D. D.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 , RCSB “Molecule, RCSB of the month”, February 2000 Goodsell D. D.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Phage phi X 174 was the first DNA-based genome for which the full DNA nucleotide sequence was determined (in 1977, by F. Sanger and collaborators).
Its DNA is 5386 nt long, and is wrapped in a circle.
The DNA encodes 11 genes, however since it is so short, the genes actually overlap.
In 2003, C. Venter and his group assembled the genome of phi X 174 in vitro, from synthetic nucleotides.
Its uncompressed genome has been shown to remain fully functional.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 A thermodynamic model of self-assembly
• adhesion energy
1 2 V (θ ) = V (0) + κ θ −θ * 2 ( ) • “mean field” term that accounts for the disk packing density of N disks
2 N ⎣⎡ρmax − ρ(N )⎦⎤
• N-disk Hamiltonian
z κ * 2 B 2 H (N ) = NV (0) + ∑(θi, j −θ ) + N ⎣⎡ρmax − ρ(N )⎦⎤ 2 2 i, j 2
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 z κ * 2 B 2 H (N ) = NV (0) + ∑(θi, j −θ ) + N ⎣⎡ρmax − ρ(N )⎦⎤ 2 2 i, j 2
Here z is the mean number of nearest neighbors, and B is proportional to the compression modulus.
If Φ ( N ) is the mole fraction of N-disk capsids, and
S = −kB ∑Φ(N )lnΦ(N ) N is the mixing entropy, then the free energy is
F = H (N ) − TS
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 It can be shown that the minimization of free energy leads to
Φ N = exp ⎡β µN − H N ⎤ ( ) ⎣ ( ( ) )⎦ and we can define the onset of self-assembly where half of the disks remain in solution while the others join .
It can also be shown that the number N in the dominant capsid structure is a function of T.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 First row: The T 1, T 3, T 4, and T 7 capsids produced by the Caspar-Klug construction with 12 pentamers and 10 (T 1) hexamers per capsid. T 1 is a dodecahedron and T 3 a truncated icosahedron. Second row: Optimal packing arrangements of N disks covering a sphere, known as the Tammes problem. Only N 12 has icosahedral symmetry; N 24 is an Archimedean solid known as the snub cube; N 32 could adopt T 3 icosahedral symmetry but in fact has D5 symmetry. Third row: The improved coverage for N 32, N 42, and N 72 when 12 of the disks adopt a diameter smaller than that of the other disks. (from R. Bruinsma et al., “Viral Self-Assembly as a Thermodynamic Process”, PRL 90 (2003) 248101) EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 Energy per capsid in the MC simulation (Zandi et al, PNAS 101 (2004) 15556) “Spherical” shapes obtained with the addition of smaller disks, with an additional conformational energy.
A modification of this conformational energy changes the shape from nearly spherical to cylindrical.
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15 References
• D. Voet and J. G. Voet, “Biochemistry, 4th ed.”, Wiley 2011 • D. Goodsell, RCSB “Molecule of the month: Catalase”, September 2004 • M. Garcia-Viloca et al, “How Enzymes Work: Analysis by Modern Rate Theory and Computer Simulations, Science 303 (2004) 186 • F. Crick and J. D. Watson, “Structure of Small Viruses”, Nature 177 (1956) 473 • D. L. D. Caspar and A. Klug: “Physical Principles in the Construction of Regular Viruses”, Cold Spring Harbor Symposia on Quantitative Biology, Vol. XXVII (1962) • D. Goodsell, RCSB “Molecule of the month”, February 2000 • W. H. Roos et al., “Physical virology”, Nature Physics 6 (2010) 733 • B.V. Venkataram Prasad and M. F. Schmid, “Principles of Virus Structural Organization”, in M.G. Rossmann and V.B. Rao (eds.), Viral Molecular Machines, Advances in Experimental Medicine and Biology 726 (2012) 17 • R. Bruinsma et al., “Viral Self-Assembly as a Thermodynamic Process”, PRL 90 (2003) 248101
EDOARDO MILOTTI - INTRODUCTORY BIOPHYSICS - A.Y. 2014-15