1. Molecular mechanisms of long-term memory
Spine
Shaft of Dendrite
Axon
Presynaptic
Postsynaptic
Synapse
PSD

2. LTP: an increase in synaptic strength
Long-term potentiation (LTP)
Time (mins) 60
Postsynaptic current
LTP protocol induces postynaptic
influx of Ca2+
Bliss and Lomo J Physiol, 1973

3. LTP: an increase in synaptic strength
Long-term potentiation (LTP)
LTP protocol induces postynaptic
influx of Ca2+
with CaMKII
inhibitor or
knockout
Postsynaptic current
Time (mins) 60
Lledo et al PNAS 1995, Giese et al Science 1998

4. Calcium-calmodulin dependent kinase II (CaMKII)
One holoenzyme = 12 subunits
Kolodziej et al. J Biol Chem 2000

5. Model of bistability in the CaMKII-PP1 system:
autocatalytic activation and saturating inactivation.
a) Autophosphorylation of CaMKII (2 rings per holoenzyme): P0 P1
slow P1 P2
fast
Lisman and Zhabotinsky, Neuron 2001

6. b) Dephosphorylation of CaMKII by PP1 (saturating inactivation)
= phosphatase, PP1 k1 k2
k-1
Total rate of dephosphorylation can never exceed k2.[PP1]
Leads to cooperativity as rate per subunit goes down
Stability in spite of turnover

7. Bistability in total phosphorylation of CaMKII
[Ca2+]=0.1M (basal level)
Rate of dephosphoryation
Total reaction rate
Rate of phosphorylation
No. of active subunits
12N

8. Phosphorylation dominates at high calcium
[Ca2+] = 2M (for LTP)
Rate of dephosphoryation
Rate of phosphorylation
Total reaction rate
No. of active subunits
12N

9. The “Normal” State of Affairs
(one stable state, no bistability)

10. How to get bistability 1) Autocatalysis: k+ increases with [C]
2) Saturation: total rate down, (k-)[C], is limited

11. Reaction pathways
14 configurations of phosphorylated subunits per ring
P P1 P2 P3 P P5 P6

12. Phosphorylation to clockwise neighbors
P P1 P2 P3 P P5 P6

13. Phosphorylation to clockwise neighbors
P P1 P2 P3 P P5 P6

14. Random dephosphorylation by PP1
P P1 P2 P3 P P5 P6

15. Random dephosphorylation by PP1
P P1 P2 P3 P P5 P6

16. Random turnover included
P P1 P2 P3 P P5 P6

17. Stability of DOWN state
= PP1 enzyme

18. Stability of DOWN state
= PP1 enzyme

19. Stability of DOWN state
= PP1 enzyme

20. Stability of UP state
= PP1 enzyme

21. Stability of UP state
= PP1 enzyme

22. Stability of UP state
= PP1 enzyme

23. Stability of UP state
= PP1 enzyme

24. Stability of UP state
= PP1 enzyme

25. Protein turnover
= PP1 enzyme

26. Stability of UP state with turnover
= PP1 enzyme

27. Stability of UP state
= PP1 enzyme

28. Stability of UP state
= PP1 enzyme

29. Stability of UP state
= PP1 enzyme

30. Stability of UP state
= PP1 enzyme

31. Stability of UP state
= PP1 enzyme

32. Stability of UP state
= PP1 enzyme

33. Stability of UP state
= PP1 enzyme

34. Stability of UP state
= PP1 enzyme

35. Stability of UP state
= PP1 enzyme

36. Stability of UP state
= PP1 enzyme

37. Small numbers of CaMKII holoenzymes in PSD
Petersen et al. J Neurosci 2003

38. Simulation methods
Stochastic implementation of reactions, of rates Ri(t)
using small numbers of molecules via Gillespie's algorithm:
1) Variable time-steps, ∆t: P(∆t) = ∑Ri exp(-∆t ∑Ri)
2) Probability of specific reaction: P(Ri) = Ri/∑Ri
3) Update numbers of molecules according to reaction chosen
4) Update reaction rates using new concentrations
5) Repeat step 1)

39. System of 20 holoenzymes undergoes stable LTP1
Pulse of high
Ca2+ here
Fraction of subunits phosphorylated 10 20
Time (yrs)

40. Slow transient dynamics revealed
Fraction of subunits phosphorylated
Time (mins)

41. Spontaneous transitions in system with 16 holoenzymes
Fraction of subunits phosphorylated
Time (yrs)

42. Spontaneous transitions in system with 4 holoenzymes
Fraction of subunits phosphorylated
Time (days)

43. Average lifetime between transitions increases
exponentially with system size

44. Large-N limit, like hopping over a potential barrier
Reaction rates
Effective potential
No. of active subunits
12N

45. 1) Chemical reactions in biology:
x-axis = “reaction coordinate”
= amount of protein phosphorylation
2) Networks of neurons that “fire” action potentials:
x-axis = average firing rate of a group of neurons

46. Why is this important? Transition between states = loss of memory
Transition times determine memory decay times.

47. Something like physics
Barrier height
depends on area
between “rate on”
and “rate off”
curves, which
scales with
system size.

48. Physics analogy: barriers with noise ...
Inherent noise because reactions take place
one molecule at a time.
Rate of transition over barrier decreases exponentially
with barrier height ...
(like thermal physics, with a potential barrier, U and
thermal noise energy proportional to kT ) ?

49. General result for memory systems
Time between transitions increases exponentially
with scale of the system.
Scale = number of molecules in a biochemical system
= number of neurons in a network
Rolling dice analogy:
number of rolls needed, each with with probability, p
to get N rolls in row, probability is pN
time to wait increases as (1/p)N = exp[N.ln(1/p)]

50. Change of concentration ratios affects balance
between UP and DOWN states.
System of 8 CaMKII holoenzymes:
7 PP1 enzymes
9 PP1 enzymes
Phosphorylation fraction
Time (yrs)
Time (yrs)

51. Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes

52. Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes

53. Analysis: Separate time-scale for ring switching
Preceding a switch down
In stable UP state
Turnover
Turnover
No. of active subunits, single ring
Total no. of active subunits
Time (hrs)
Time (hrs)

55. Analysis: Separate time-scale for ring switching
Goal
Rapid speed-up by converting system to 1D and solving analytically.
Method
Essentially a mean-field theory.
Justification
Changes to and from P0 (unphosphorylated state) are slow.

56. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

57. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

58. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

59. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

60. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

61. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

62. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

63. Analysis: Project system to 1D
1) Number of rings “on” with any activation, n.
2) Assume average number, P, of subunits phosphorylated for all rings “on”.
3) Calculate reaction rates for one ring, assuming contibution of others is
(n-1)P.
4) Calculate average time in configurations with these reaction rates.
5) Hence calculate new value of P.
6) Repeat Step 2 until convergence.
7) Calculate rate to switch “on”, r+n, and “off”, r-n.
8) Continue with new value of n.

64. Analysis: Solve 1D model exactly
r+n-1
r+n
r+n+1 N0
n-1 n
n+1
n+2 N1
r-n+2
r-n
r-n+1
Time to hop from N0 to N1
Use: r+n Tn = 1 + r-n+1Tn+1 for N0 ≤ n < N1
r+n Tn = r-n+1Tn+1 for n < N0
Tn = for n ≥ N1
Average total time for transition, Ttot = ∑Tn

65. Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes

66. Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes