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**Molecular mechanisms of long-term memory**

Spine Shaft of Dendrite Axon Presynaptic Postsynaptic Synapse PSD

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**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

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**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

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**Calcium-calmodulin dependent kinase II (CaMKII)**

One holoenzyme = 12 subunits Kolodziej et al. J Biol Chem 2000

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**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

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**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

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**Bistability in total phosphorylation of CaMKII**

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

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**Phosphorylation dominates at high calcium**

[Ca2+] = 2M (for LTP) Rate of dephosphoryation Rate of phosphorylation Total reaction rate No. of active subunits 12N

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**The “Normal” State of Affairs**

(one stable state, no bistability)

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**How to get bistability 1) Autocatalysis: k+ increases with [C]**

2) Saturation: total rate down, (k-)[C], is limited

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

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**Phosphorylation to clockwise neighbors**

P P1 P2 P3 P P5 P6

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**Phosphorylation to clockwise neighbors**

P P1 P2 P3 P P5 P6

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**Random dephosphorylation by PP1**

P P1 P2 P3 P P5 P6

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**Random dephosphorylation by PP1**

P P1 P2 P3 P P5 P6

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**Random turnover included**

P P1 P2 P3 P P5 P6

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**Stability of DOWN state**

= PP1 enzyme

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**Stability of DOWN state**

= PP1 enzyme

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**Stability of DOWN state**

= PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Protein turnover = PP1 enzyme

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**Stability of UP state with turnover**

= PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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Stability of UP state = PP1 enzyme

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**Small numbers of CaMKII holoenzymes in PSD**

Petersen et al. J Neurosci 2003

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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)

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**System of 20 holoenzymes undergoes stable LTP**

1 Pulse of high Ca2+ here Fraction of subunits phosphorylated 10 20 Time (yrs)

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**Slow transient dynamics revealed**

Fraction of subunits phosphorylated Time (mins)

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**Spontaneous transitions in system with 16 holoenzymes**

Fraction of subunits phosphorylated Time (yrs)

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**Spontaneous transitions in system with 4 holoenzymes**

Fraction of subunits phosphorylated Time (days)

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**Average lifetime between transitions increases**

exponentially with system size

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**Large-N limit, like hopping over a potential barrier**

Reaction rates Effective potential No. of active subunits 12N

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**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

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**Why is this important? Transition between states = loss of memory**

Transition times determine memory decay times.

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**Something like physics**

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

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**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 ) ?

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**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)]

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**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)

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**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

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**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

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**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)

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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

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**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

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**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

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