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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) 0 60 Postsynaptic current LTP protocol induces postynaptic influx of Ca 2+ Bliss and Lomo J Physiol, 1973

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LTP: an increase in synaptic strength Long-term potentiation (LTP) Time (mins) 0 60 Postsynaptic current LTP protocol induces postynaptic influx of Ca 2+ Lledo et al PNAS 1995, Giese et al Science 1998 with CaMKII inhibitor or knockout

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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. P0 P1 P2 slow fast a) Autophosphorylation of CaMKII (2 rings per holoenzyme): Lisman and Zhabotinsky, Neuron 2001

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E = phosphatase, PP1 b) Dephosphorylation of CaMKII by PP1 (saturating inactivation) k2k2 k1k1 k -1 Total rate of dephosphorylation can never exceed k 2.[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 0 12N No. of active subunits Total reaction rate 0 Rate of phosphorylation Rate of dephosphoryation [Ca 2+ ]=0.1 M (basal level)

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Phosphorylation dominates at high calcium 0 12N No. of active subunits Total reaction rate 0 Rate of phosphorylation Rate of dephosphoryation [Ca 2+ ] = 2 M (for LTP)

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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 P0 P1P2 P3 P4 P5 P6

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Phosphorylation to clockwise neighbors P0 P1P2 P3 P4 P5 P6

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Phosphorylation to clockwise neighbors P0 P1P2 P3 P4 P5 P6

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Random dephosphorylation by PP1 P0 P1P2 P3 P4 P5 P6

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Random dephosphorylation by PP1 P0 P1P2 P3 P4 P5 P6

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Random turnover included P0 P1P2 P3 P4 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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Stochastic implementation of reactions, of rates R i (t) using small numbers of molecules via Gillespie's algorithm: 1) Variable time-steps, ∆t: P(∆t) = ∑R i exp(-∆t ∑R i ) 2) Probability of specific reaction: P(R i ) = 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

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Time (yrs) Fraction of subunits phosphorylated Pulse of high Ca 2+ here System of 20 holoenzymes undergoes stable LTP

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

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

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

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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 0 12N No. of active subunits Reaction rates Effective potential

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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... 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. ?

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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 p N 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: Time (yrs) Phosphorylation fraction 7 PP1 enzymes 9 PP1 enzymes

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Number of PP1 enzymes Average lifetime of state 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

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Number of PP1 enzymes Average lifetime of state 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

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

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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 Time to hop from N 0 to N 1 Use: r + n T n = 1 + r - n+1 T n+1 for N 0 ≤ n < N 1 r + n T n = r - n+1 T n+1 for n < N 0 T n = 0 for n ≥ N 1 Average total time for transition, T tot = ∑T n N0N0 N1N1 n n+1 n-1 n+2 r - n+1 r+nr+n r - n+2 r-nr-n r + n-1 r + n+1

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Number of PP1 enzymes Average lifetime of state 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

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Number of PP1 enzymes Average lifetime of state 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

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