Molecular mechanisms of long-term memory

Spine

Shaft of Dendrite

Axon

Presynaptic PostsynapticSynapse

PSD

LTP: an increase in synaptic strength

Long-term potentiation (LTP)

Time (mins)0 60

Pos

tsyn

apti

c cu

rren

tLTP protocol induces postynaptic

influx of Ca2+

Bliss and Lomo J Physiol, 1973

LTP: an increase in synaptic strength

Long-term potentiation (LTP)

Time (mins)0 60

Pos

tsyn

apti

c cu

rren

tLTP protocol induces postynaptic

influx of Ca2+

Lledo et al PNAS 1995, Giese et al Science 1998

with CaMKII inhibitor or knockout

Calcium-calmodulin dependent kinase II (CaMKII)

One holoenzyme = 12 subunits

Kolodziej et al. J Biol Chem 2000

Model of bistability in the CaMKII-PP1 system: autocatalytic activation and saturating inactivation.

P0 P1

P1 P2

slow

fast

a) Autophosphorylation of CaMKII (2 rings per holoenzyme):

Lisman and Zhabotinsky, Neuron 2001

E = phosphatase, PP1

b) Dephosphorylation of CaMKII by PP1 (saturating inactivation)

k2

k1

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

Bistability in total phosphorylation of CaMKII

0 12NNo. of active subunits

Tot

al r

eact

ion

rate

0

Rate of phosphorylation

Rate of dephosphoryation

[Ca2+]=0.1M (basal level)

Phosphorylation dominates at high calcium

0 12NNo. of active subunits

Tot

al r

eact

ion

rate

0

Rate of phosphorylation

Rate of dephosphoryation

[Ca2+] = 2M (for LTP)

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

How to get bistability

1) Autocatalysis: k+ increases with [C]2) Saturation: total rate down, (k-)[C], is limited

Reaction pathways

14 configurations of phosphorylated subunits per ring

P0 P1 P2 P3 P4 P5 P6

Phosphorylation to clockwise neighbors

P0 P1 P2 P3 P4 P5 P6

Phosphorylation to clockwise neighbors

P0 P1 P2 P3 P4 P5 P6

Random dephosphorylation by PP1

P0 P1 P2 P3 P4 P5 P6

Random dephosphorylation by PP1

P0 P1 P2 P3 P4 P5 P6

Random turnover included

P0 P1 P2 P3 P4 P5 P6

Stability of DOWN state

= PP1 enzyme

Stability of DOWN state

= PP1 enzyme

Stability of DOWN state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Protein turnover

= PP1 enzyme

Stability of UP state with turnover

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Stability of UP state

= PP1 enzyme

Small numbers of CaMKII holoenzymes in PSD

Petersen et al. J Neurosci 2003

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 ∑R

i)

2) Probability of specific reaction: P(Ri) = R

i/∑R

i

3) Update numbers of molecules according to reaction chosen

4) Update reaction rates using new concentrations

5) Repeat step 1)

Simulation methods

Time (yrs)0

010 20

1

Fra

ctio

n of

sub

unit

s ph

osph

oryl

ated

Pulse of high Ca2+ here

System of 20 holoenzymes undergoes stable LTP

Time (mins)

Fra

ctio

n of

sub

unit

s ph

osph

oryl

ated

Slow transient dynamics revealed

Spontaneous transitions in system with 16 holoenzymes

Time (yrs)

Fra

ctio

n of

sub

unit

s ph

osph

oryl

ated

Spontaneous transitions in system with 4 holoenzymes

Time (days)

Fra

ctio

n of

sub

unit

s ph

osph

oryl

ated

Average lifetime between transitions increases exponentially with system size

Large-N limit, like hopping over a potential barrier

0 12NNo. of active subunits

Rea

ctio

n ra

tes

Eff

ecti

ve p

oten

tial

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

Why is this important?

Transition between states = loss of memoryTransition times determine memory decay times.

Something like physics

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

Physics analogy: barriers with noise ...

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 )

Inherent noise because reactions take place one molecule at a time.

?

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)]

Change of concentration ratios affects balance between UP and DOWN states.

System of 8 CaMKII holoenzymes:

Time (yrs) Time (yrs)

Pho

spho

ryla

tion

fra

ctio

n

7 PP1 enzymes 9 PP1 enzymes

Number of PP1 enzymes

Ave

rage

life

tim

e of

sta

te

10 yrs

1 yr

1 mth

1 day

Optimal system lifetime is a trade-off between lifetimes of UP and DOWN states

UP state lifetime

DOWN state lifetime

Number of PP1 enzymes

Ave

rage

life

tim

e of

sta

te

10 yrs

1 yr

1 mth

1 day

Optimal system lifetime is a trade-off between lifetimes of UP and DOWN states

UP state lifetime

DOWN state lifetime

Analysis: Separate time-scale for ring switching

Turnover

Preceding a switch down In stable UP state

Time (hrs) Time (hrs)

Tot

al n

o. o

f ac

tive

sub

unit

s

No.

of

acti

ve s

ubun

its,

sin

gle

ring

Turnover

Analysis: Separate time-scale for ring switching

GoalRapid speed-up by converting system to 1D and solving analytically.

MethodEssentially a mean-field theory.

Justification Changes to and from P0 (unphosphorylated state) are slow.

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.

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.

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.

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.

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.

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.

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.

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.

Analysis: Solve 1D model exactly

Time to hop from N0 to N

1

Use: r+n T

n = 1 + r-

n+1T

n+1 for N

0 ≤ n < N

1

r+n T

n = r-

n+1T

n+1 for n < N

0

Tn = 0 for n ≥ N

1

Average total time for transition, Ttot

= ∑Tn

N0

N1n n+1n-1 n+2

r-n+1

r+n

r-n+2r-

n

r+n-1 r+

n+1

Number of PP1 enzymes

Ave

rage

life

tim

e of

sta

te

10 yrs

1 yr

1 mth

1 day

Optimal system lifetime is a trade-off between lifetimes of UP and DOWN states

UP state lifetime

DOWN state lifetime

Number of PP1 enzymes

Ave

rage

life

tim

e of

sta

te

10 yrs

1 yr

1 mth

1 day

Optimal system lifetime is a trade-off between lifetimes of UP and DOWN states

UP state lifetime

DOWN state lifetime