Lobe Specific Ca2+-Calmodulin Nano-Domain in Neuronal Spines: A Single Molecule Level Analysis
[['Calmodulin ( CaM ) is a ubiquitous Ca2+ buffer and signaling molecule in cells .', 'In the excitatory synapse of hippocampal CA1 pyramidal neurons , the activation of CaM dependent enzymes results in the induction of synaptic plasticity ( e . g . , long-term potentiation ( LTP ) and long-term depression ( LTD ) ) [1] .', 'The induction of NMDA receptor dependent LTP and LTD require increased Ca2+ and subsequent activation of CaM-dependent downstream enzymes: CaM-dependent protein kinase II ( CaMKII ) and calcineurin .', 'Injection of CA1 pyramidal cells with peptides that block CaMKII activity inhibited the induction [2] , [3] , but not maintenance [4] of LTP , while injection of the activated form of the enzyme also produced LTP-like plasticity [5] , [6] .', 'LTD is also critically dependent on Ca2+ and it appears that the CaM-dependent phosphatase , protein phosphatase 2B ( calcineurin ) is involved in LTD induction [7] .', 'The simplest correlative explanation for these results is that LTD is induced by intermediate levels of Ca2+ that activate CaM and subsequently calcineurin but not CaMKII .', 'Conversely , higher levels of Ca2+ initiate CaM-dependent CaMKII activation and autophosphorylation , leading to LTP induction .', 'However , it is still unknown how Ca2+ and CaM regulate two opposing processes as distinct as LTP or LTD in such a precise and controlled manner .', 'Besides being a major signaling molecule , CaM also functions as a primary Ca2+ buffer in CA1 pyramidal neurons [8] .', 'In fact , most CA1 pyramidal neurons contain CaM but not other EF-hand Ca2+ binding proteins ( e . g . , parvalbumin and calretinin ) ( reviewed in [9] ) .', 'An exception is calbindin-D28K , which is expressed in a subpopulation of CA1 pyramidal neurons but only in rat ( [10] , [11] ) .', 'CaM binds four Ca2+ ions , two in its N-terminal lobe and two in its C-terminal lobe [12] .', 'The binding sites in the N-terminal lobe are lower affinity [13] but exhibit faster kinetics as opposed to the higher affinity , slower kinetics of the C-terminal lobe sites [14] , [15] .', 'Surprisingly little is known as to how such a protein with multiple Ca2+ binding sites influences the diffusion of Ca2+ in the cell .', 'Most pre-existing theories of Ca2+ binding and diffusion assume a fast binding of Ca2+ and single Ca2+ binding site for the buffer ( see reviews by [16] ) .', 'In addition , recent experimental data suggest that each lobe of CaM has different affinity toward its downstream target ( CaMKII and calcineurin ) [17] , [18] , [19] .', 'As each lobe differentially responds to Ca2+ signals and downstream targets , it is possible that these lobe specific properties play distinct biological roles in synaptic spines ( see Discussion for more details ) .', 'This motivated us to dissect the spatial-temporal dynamics of lobe specific Ca2+-CaM interaction in detail at the single molecule level .', 'Many elegant experimental measurements have been made of dendritic spine Ca2+ [20] , [21] , [22] , [23] , [24] .', 'These measurements largely rely on a spatially averaged Ca2+ signal generated from fluorescence imaging of dyes whose quantum efficiency changes upon Ca2+ binding .', 'As such , they contain no direct information relative to the issue of possible micro- or nano-domains of intracellular Ca2+ .', 'The problem is exacerbated by the high diffusion coefficients of free and dye bound Ca2+ which additionally smears the spatial signal in time frames relevant for Ca2+-imaging experiments .', 'These and other caveats related to dye-based Ca2+-imaging experiments were recently reviewed [25] .', 'In addition , we do not have an effective fluorescence reporter to detect and monitor Ca2+ binding to each lobe of CaM at the single molecule level .', 'As such , mathematical models and computer simulations are presently the only tractable means of investigating this critical aspect of synaptic physiology .', 'Furthermore , in a medium size dendritic spine ( i . e . , sphere-shaped spine head of 500 nm diameter ) , the concentration of 1 µM of any chemical species corresponds to ∼40 molecules .', 'The basal ( resting ) level of spine Ca2+ is 50∼100 nM which corresponds to 2∼4 molecules of Ca2+ ions .', 'Under such a circumstance , the behavior of single molecules within synaptic spines is not well described by the concentration-based mathematical approach such as reaction diffusion equation .', 'Here we report the single molecule level analysis of Ca2+-CaM interaction within a dendritic spine using a novel particle-based event-driven Monte Carlo algorithm , which we call Cellular Dynamics Simulator ( CDS , [26] ) .', 'Unlike other commonly used Monte Carlo simulation ( e . g . , MCell , [27] ) , it explicitly takes account of volume exclusion and collision between diffusing molecules in order to accurately simulate chemical reactions in the cellular interior .', 'Using this simulator and first passage time theory , we dissect the mechanisms that influence the dynamics of Ca2+-CaM interaction at the single molecule level .', 'We use a model of CaM built upon detailed kinetic data and ask if the lobe specific spatial-temporal micro-domain of Ca2+-CaM activation can exist and if so how it is biophysically regulated in a small sub-cellular compartment like dendritic spines .', 'We employ a statistical spatial point pattern analysis [28] to understand the spatial profile of Ca2+-CaM interactions .', 'The combination of spatial point pattern analysis and particle based Monte Carlo simulation is a unique computational strategy used in this study .', 'Our analysis shows a higher sensitivity of the N-terminal lobe to the location and influx rate of Ca2+ from typical receptor/channel sources .', 'Each lobe of CaM functions as distinct Ca2+ sensors and responds differentially to Ca2+ influx both in space and in time .', 'Coupled with the experimental knowledge that different enzymes bind preferentially to either the N- or C-lobes of Ca2+ saturated CaM , we propose a possible explanation for how two opposing Ca2+/CaM-dependent enzymes can be differentially activated .'], ['Fig . 1A illustrates the Ca2+ binding and unbinding pathway for each lobe of CaM .', 'As shown , Ca2+ binding to the N-terminal lobe and the first Ca2+ binding event to the C-terminal lobe are diffusion limited while the second Ca2+ binding to the C-terminal lobe is the rate-limiting step in achieving the fully Ca2+-saturated state .', 'If this Ca2+ binding step at the C-terminal lobe is much slower than the diffusion of Ca2+ , the majority of Ca2+ ions that entered the spine head will have moved away from the channel without saturating local CaM molecules .', 'The spatial profile of the C-terminal lobe or full Ca2+ saturation of CaM may then be less sensitive to the location of Ca2+ channels .', 'On the other hand , if the N-terminal lobe Ca2+ saturation is fast as compared to the Ca2+ diffusion , its Ca2+ saturation may be more closely localized to the Ca2+ channels .', 'Thus , three biophysical factors become important in understanding the spatial domain of Ca2+-CaM interactions .', 'The first is how fast each lobe of CaM becomes Ca2+ saturated with a given concentration of Ca2+ .', 'The second is how fast Ca2+ ions escape from the spine .', 'The third is how steep or flat the gradient of Ca2+ ion distribution will be in the spine head with a given Ca2+ injection rate through Ca2+ channels .', 'In this section , we analyze the first biophysical factor , which we call the ( mean ) first passage time: the ( average ) length of transition time required for each lobe of CaM molecule to reach the Ca2+ saturated state from a basal ( apo- ) state .', 'In fact , a mathematical formula is already available to calculate this mean first passage time ( Equations 5 , 29 in [29] ) .', 'In their single molecule biophysical analysis , Shaevitz et al . [29] used an algebraic recursive method to derive the Laplace transform of the first passage time distribution .', 'Fig . 1B and Eq .', '1∼2 explain their formalism applied to Ca2+-CaM interactions .', 'Here we define State “0” as a Ca2+ free ( apo ) form , State “1” as one Ca2+ ion bound form , and State “2” as a two Ca2+ ion bound form of a given lobe .', 'The symbols kXij in Fig . 1B denotes the rate constant between State i and State j ( i , j\u200a=\u200a0 , 1 , 2 ) of lobe X ( =\u200aN or C ) .', 'Thus , each lobe has three states and the whole CaM molecule has nine states ( Fig . 1C ) .', 'The resultant Laplace transform of the distribution of first passage time is: ( 1 ) where [Ca] is the given concentration of Ca2+ ( Note , in order to apply Eq . 29 in [29] , we needed to multiply the association rate constant by the concentration of Ca2+ ) .', 'Here we assume the system is well-stirred and the concentration of Ca2+ is constant ( time-invariant ) .', 'Then , the mean first passage time ( <t> ) can easily be found through differentiation ( see Eq . 5 in [29] ) : ( 2 ) Note that the dissociation rate ( ) of the second Ca2+ is not included in the formula .', 'The latter rate determines the lifetime of fully Ca2+ saturated state of each lobe but it does not influence the first passage time .', 'Therefore , three kinetic rates ( , , ) and Ca2+ concentration determine the lobe specific first passage time .', 'Note that both lobes have similar association rates for the first Ca2+ ions ( ) ( Fig . 1A ) .', 'The difference in the second Ca2+ binding rates ( ) is large as compared to the dissociation of the first Ca2+ ion ( ) ( Fig . 1A ) .', 'Thus , in Eq .', '2 , the second Ca2+ binding rates ( , ) determine the difference of the first passage time between the N- and C-lobes .', 'Fig . 2A is a numerical display of this formula showing that the first passage time sharply increases as we decrease the Ca2+ concentration ( the unit of time , y-axis , is in seconds ) .', 'As predicted , the mean first passage time for the C-terminal lobe ( magenta ) is much longer than the N-lobe ( blue ) .', 'For comparison , we show the first passage time for full Ca2+ saturation of CaM; the mean first passage time to reach the state N2C2 in Fig . 1C .', 'As one can see from the diagram in Fig . 1C , this first passage time depends on all Ca2+ association and dissociation pathways for both lobes and is influenced by the lifetime of the Ca2+ saturated states of each lobe .', 'The corresponding mathematical formula will be much more complicated than Eq .', '1 and 2 and therefore , we calculated this quantity numerically using an extended version of the Gillespie type stochastic algorithm ( see [8] , [30] for more details ) .', 'The results presented in Fig . 2A suggest that the N-terminal lobe may respond to a short Ca2+ transient but the C-terminal lobe may not if the transient is shorter than the first passage time of C-lobe Ca2+ saturation .', 'For example , NMDA receptor type Ca2+ transients ( ∼1 µM peak with duration of ∼80–200 ms ) may not result in significant CaM saturation in the spine .', 'In fact , at a ∼1 µM Ca2+ concentration , the mean first passage time for the C-terminal lobe ( or full Ca2+ saturation of CaM ) is much longer than the duration of the Ca2+ transient ( Fig . 2A upper right inset ) .', 'Such straightforward interpretation of the first passage time analysis , however , could be misleading .', 'Note that we have only discussed the mean but not the entire distribution ( or standard deviation ) of the first passage time .', 'In addition , we ignored the fact that the number of Ca2+ ions may be limited in the dendritic spines and that their concentration is not constant as postulated in Eq .', '1∼2: the N-terminal lobe and the C-terminal lobes on the same or different CaM molecules will compete for the limited number of Ca2+ ions .', 'As for the stochastic fluctuation , we can derive the standard deviation of the first passage time using the same analytic method described above: ( 3 ) The resultant standard deviation is very close to the mean first passage time for all Ca2+ concentrations ( i . e . , the coefficient of variation is >0 . 9 for all [Ca2+]<10 µM ) .', 'The second term in the right-hand side of Eq .', '3 ( ) is small because the first Ca2+ binding rate ( ) for both lobes are high and therefore the ratio of the right-hand sides of Eq .', '3 and Eq .', '2 approaches 1 .', 'Fig . 2B and C show the histograms of the first passage time distribution for the N-terminal lobe ( blue ) and the C-terminal lobe ( magenta ) Ca2+ saturation , respectively , taken from a single stochastic simulation ( the same bin size , 5ms , for both lobes and the total number of CaM molecules is 400 ) .', 'Fig . 2B clearly shows that the Ca2+ saturation of the C-terminal lobe is possible even if the mean first passage time is shorter than that of Ca2+ transient .', 'However , the inset of Fig . 2B and 2C , i . e . , the histogram up to 80 ms , predict that the N-terminal lobe Ca2+ saturation predominates and precedes that of the C-terminal lobe during the short Ca2+ transient .', 'Knowing that two lobes of CaM compete for the limited amount of available Ca2+ ions in the dendritic spines , we predict that the N-terminal dominance for the short Ca2+ transient is more prominent in neurons .', 'This type of analysis , however , is further complicated when taking into account the non-homogeneous spatial distribution of molecules .', 'When Ca2+ ions enter the spine head through a Ca2+ channel , a steep spatial gradient of Ca2+may be formed around the channel mouth ( depending on the Ca2+ injection rate ) .', 'At a single molecule level , it is the transient local ( microscopic ) “concentration” of Ca2+ ( i . e . , the number of Ca2+ collision events ) felt by a CaM molecule that determines the probability of Ca2+ saturation of a given lobe of each CaM molecule .', 'A CaM molecule can experience much higher ( local ) Ca2+ “concentration” than indicated by the bulk Ca2+ transient depending on its location with respect to the Ca2+ source .', 'The present work aims to describe a detailed analysis of this spatial stochastic phenomenon .', 'However , before going into the detailed simulations , it is necessary to dissect each of the biophysical factors that we discussed at the beginning of this section .', 'The last two of these factors determine the space- and time- dependent Ca2+ profile in the spines .', 'Without such a systematic dissection , the interpretation of simulation results when trying to determine the spatial/temporal profile of CaM activation would not be possible .', 'We next explored how fast Ca2+ ions escape from the spine .', 'The second factor that will determine the spatial profile of CaM activation is the escape rate of Ca2+ from the spine .', 'Ca2+ ions that enter the spine through ion channels will eventually diffuse into the dendrites or be extruded by the Ca2+ pumps [23] .', 'Here we focus on the impact of spine geometry and Ca2+ pumps on the escape rate of Ca2+ from the spines .', 'We carry out this analysis in a stepwise manner .', 'We first analyze the escape of Ca2+ via pure diffusion without Ca2+ pumps ( or buffers ) and establish the impact of spine morphology on the Ca2+ escape rate ( Fig . 3A and B ) .', 'Then we add Ca2+ pumps to examine their impact ( Fig . 3C ) .', 'This way we can isolate and understand the contribution of each of these factors in the regulation of the Ca2+ escape rate .', 'In neurons , Ca2+ buffers such as CaM also influence this escape rate but in a highly complicated manner .', 'We will study the effect of Ca2+ binding proteins ( CaM ) in the later sections when we combine all known biophysical factors in the detailed simulations .', 'Fig . 3A shows the time courses of Ca2+ decay for three different spine neck geometries .', 'Here , we randomly placed a fixed number of Ca2+ ions ( =\u200a400 that corresponds to ∼10 µM ) in the head of a spherical spine and let them diffuse out of the spine to the dendrite .', 'The diffusion coefficient ( ) of Ca2+ was set to 200∼225 µm2/s ( nm2/µs ) [31] .', 'Each curve in Fig . 3A represents the average of 100 simulation runs .', 'Clearly , the longer and the narrower the neck , the slower the Ca2+ decay process .', 'This is a so-called narrow escape problem and has been extensively investigated [32] , [33] .', 'As predicted by these theoretical studies , the simulated Ca2+ decay transient is well approximated by a single exponential decay term .', 'These decay time constants fit well ( the relative error <5% ) with one of the pre-existing mathematical formula ( the left-hand side of Eq . 4 below ) : ( 4 ) where , , and are the volume of the spine head , the length and the radius of spine neck , and the volume of neck , respectively [32] .', 'Fig . 3B summaries our simulation results for different spine geometries .', 'We plot the narrow escape time ( ) against the ratio of spine head and neck volume ( ) ( x-axis ) .', 'As shown all data points are aligned on straight lines , indicating that the narrow escape time is a linear function of the volume ratio ( ) ( see the right-hand side of Eq . 4 ) .', 'Note that Eq .', '4 was previously tested against experimental data of molecular diffusion ( using photo-bleaching recovery of fluorescein-dextran and enhanced green fluorescent protein ) across spine-dendrite junctions in CA1 neurons [21] , [24] .', 'In other words , Eq .', '4 is consistent with escape of diffusing molecules from real spines on CA1 neurons .', 'Additional simulations confirm that Eq .', '4 fits well with real spines when morphologies from 3D EM reconstructions are used ( http://synapses . clm . utexas . edu/ ) ( data not shown ) .', 'Another biophysical factor that regulates the Ca2+ decay from spines is Ca2+ pumps [20] , [23] .', 'The main Ca2+ extrusion mechanisms in CA1 spines are Na+/Ca2+ exchangers ( NCX , NCKX ) and plasma membrane Ca2+ ATPase ( PMCA ) [20] , [34] .', 'We have modeled both of them using the kinetic scheme used in [35] ( see Methods for more details ) .', 'Fig . 3C shows a Ca2+ clearance process with standard spine morphology ( 500 nm spine head diameter , 500 nm spine neck length and 150 nm spine neck diameter ) with ( dashed black line ) and without pumps ( solid black line ) .', 'The fast decay time constant of Ca2+ in the presence of pump is ∼45% of the narrow escape time without pumps ( ∼5–6 ms ) .', 'In this analysis , we have included NCX/NCKX and PMCA at the concentration close to the highest level known in the literature to examine the maximal impact that Ca2+ pumps would have on Ca2+ clearance .', 'The Ca2+ transients with reduced number of pumps lie between the dashed and solid lines ( data not shown ) .', 'Overall , the analyses in Fig . 3 show that the narrow escape time of Ca2+ without buffers in a standard spine in the presence of pumps is ∼5 ms or shorter .', 'In the subsequent section , we will show that a major Ca2+ buffer in CA1 pyramidal neurons ( i . e . , CaM ) slows down the Ca2+ decay to ∼10∼20 ms ( the latter is close to that observed in the Ca2+ imaging analyses [20] .', 'It is this brief time window that each lobe of CaM becomes Ca2+ saturated or not during each Ca2+ spike .', 'The first passage time becomes a critical factor to understand the spatial profile of Ca2+-CaM interactions .', 'Having established the impact of spine geometry on the Ca2+ extrusion process , we now analyze the third biophysical factor that influences the spatial gradient of spine Ca2+: the Ca2+ injection rate of channels .', 'Since the kinetics of the voltage-gated Ca2+ channels and NMDA receptors are highly complicated , we used a “model stochastic Ca2+ channel” in this section .', 'A single stochastic Ca2+ channel was placed on the top of the head of a standard spine ( black circle in Fig . 4A; see Fig . 3C for the standard morphology of CA1 dendritic spine ) .', 'This channel injects Ca2+ at a given ( average ) rate and we examine the relation between the Ca2+ injection rate and the spatio-temporal profile of Ca2+ transients in the spine .', 'To realize the impact of Ca2+ injection rate in isolation on the spatiotemporal Ca2+ profile , there are no pumps or Ca2+ binding buffers in this model spine .', 'Once injected , Ca2+ ions travel via simple diffusion until they are absorbed from the compartment at the spine-dendrite boundary ( see the vertical arrow in Fig . 4A ) .', 'We varied the rate , but the total number of injected Ca2+ ions was set to 700 so that the peak Ca2+ concentration would be in a physiological range ( ∼6–16 µM , i . e . , ∼250–650 Ca2+ ions; see panel C of Fig . 4 ) .', 'Note these numbers are taken from the lowest estimated Ca2+ injection rate of NMDA receptors and the higher Ca2+ injection rates of voltage gated Ca2+ channels ( [36] , [37] , [38] ) .', 'Fig . 4 Panel A and B show the location of Ca2+ ions ( not to scale ) at designated time points after the start of Ca2+ injection .', 'The mean Ca2+ injection rates in Panel A and B are 1 . 4 and 0 . 07 Ca2+ ions per microsecond , respectively .', 'At the higher injection rate ( 1 . 4 ions/µs ) , there is a build-up of Ca2+ ions near the channel ( Panel A ) while such a build up is not evident in Panel B . Note that the time points chosen for Panel A and B are 20-fold different so that the total number of Ca2+ injected by the indicated time points in Panel A ( 10 , 20 , 100 , 200 µs ) and B ( 200 , 400 , 2000 , 4000 µs ) are identical .', 'Ca2+ ions can travel ∼140 nm from the channel via diffusion before the next Ca2+ ion exits the channel at injection rate of 0 . 07 ions/µs .', 'At a higher Ca2+ injection rate , Ca2+ ions will accumulate near the channel pore before they diffuse away ( red in Fig . 4C ) .', 'As anticipated , the lower Ca2+ injection rate ( black ) leads to a much lower peak Ca2+ number ( concentration ) than the higher Ca2+ injection rates .', 'Ca2+ ion can travel more than 1 µm away from the channel during 1 ms . During a 10 ms Ca2+ injection period , a significant fraction of Ca2+ ions has already left the spine .', 'Thus , we have lower Ca2+ peak than at the higher Ca2+ injection rate .', 'After the peak , the Ca2+ level decreases with a time constant of ∼7–8 ms for all Ca2+ injection rates .', 'This decay process is controlled by the diffusion and is consistent with the narrow escape rate we calculated in Fig . 3 .', 'Fig . 4 clearly shows the impact of Ca2+ injection rates on the spatial and temporal dynamics of Ca2+ transients in dendritic spines .', 'The relative lack of a Ca2+ gradient in Fig . 4B and the long first passage time of the C-terminal lobe of CaM in Fig . 2 suggest that a spatial gradient of the Ca2+-saturated C-terminal lobe may not form .', 'However , as mentioned at the beginning of this section , we need to include CaM and examine the combined effect of all of these biophysical factors on the spatial profile of Ca2+-CaM interactions .', 'The second half of Results provides this analysis .', 'In the previous sections , we studied the impact of three biophysical factors: the first passage time ( Fig . 2 ) , the narrow escape time ( Fig . 3 ) , and the impact of Ca2+ injection rate on the Ca2+ micro-domain ( Fig . 4 ) .', 'In this section , we wish to study the combined effects of these factors on the spatial-temporal pattern of Ca2+-CaM interaction .', 'As a first step , we placed a single “model Ca2+ channel” as in Fig . 4 but add CaM to assess the impact of Ca2+ injection rates on the Ca2+-CaM interaction .', 'Besides the “artificial” model channel , we included CaM and Ca2+ pumps .', 'We distributed 1600 molecules of CaM ( i . e . , 40 µM ) uniformly within the spine volume ( the estimated concentration of CaM in CA1 dendritic spines is 10∼100 µM , [8] ) .', 'Before injecting Ca2+ ions the entire system is equilibrated at basal Ca2+ conditions , i . e . , ∼40–46 Ca2+ bound CaM molecules with ∼2 free Ca2+ ions ( the latter correspond to 50 nM of basal free Ca2+ concentration ) .', 'At this basal condition , majority of CaM molecules are Ca2+ free or in a single Ca2+ bound form and none of their lobes are Ca2+ saturated .', 'The diffusion coefficient of CaM varies between 2∼20 µm2/s ( nm2/µs ) [30] , [39] .', 'In this section , we set it to 20 µm2/s ( nm2/µs ) ( but see our comments below ) .', 'The results in Fig . 5 show the dynamics of Ca2+/CaM with a channel of high Ca2+ injection rate ( 1 . 4 Ca2+ ions/µs and a total of 700 Ca2+ ions are injected as in Fig . 4 ) .', 'Fig . 5A shows the number of Ca2+ saturated N- and C-lobes ( blue and magenta , respectively ) and fully Ca2+-saturated CaM .', 'The number of free Ca2+ ions in the spine is shown in Fig . 5B .', 'Both Fig . 5A and 5B are taken from the same single simulation run .', 'The result of stochastic simulation varies from one simulation run to the other; however , the overall qualitative dynamics in Fig . 5A and 5B are similar among different simulation runs .', 'The N-terminal lobe of CaM binds Ca2+ much faster than the C-terminal lobe ( Fig . 2 ) .', 'As a consequence , the number of Ca2+ saturated N-terminal lobes increases rapidly as Ca2+ is injected ( blue line in Fig . 5A ) .', 'After the termination of Ca2+ injection ( at 500 µs ) , the N-terminal lobes quickly release Ca2+ and the C-terminal lobes slowly bind the available Ca2+ ( Fig . 5A ) .', 'Once bound , Ca2+ remains associated with the C-lobe for a relatively long time ( the decay time constant is ∼120 ms ) and the C-lobes therefore trap Ca2+ in the spine ( Fig . 5A ) .', 'The free Ca2+ level eventually returns to the basal level after a few hundred ms ( data not shown ) .', 'Another important point to note is that even at this high Ca2+ injection rate , the total number of fully Ca2+-saturated CaM molecule is less than ∼7 .', 'This number varies from simulation to simulation , but with a single Ca2+ channel , the number remains below 10 ( over 100 simulation runs ) , a remarkably low number .', 'Panels C , E , and G of Fig . 5 show the spatial dynamics of each lobe of CaM taken from 15 simulation runs .', 'During the early rising phase of their Ca2+ saturation , each lobe of CaM exhibits a nano-domain near the channel pore .', 'For example , in Fig . 5E , we record the location ( red circle ) of each CaM molecule when its N-lobe becomes first Ca2+ saturated .', 'We plot these accumulated locations of “first Ca2+ saturation event” up to the different designated time point in the figure ( note each lobe may undergo multiple cycles of Ca2+ saturation , but only the first one is recorded in Panel C , E , and G in Fig . 5 and in subsequent figures ) .', 'The formation of a Ca2+/CaM nano-domain is clear .', 'A similar but less obvious nano-domain is observed for the C-terminal lobe ( Panel C ) and for the fully Ca2+-saturated CaM .', 'To further confirm these observations , we performed spatial point pattern analysis ( see Methods and [28] , [40] , [41] ) .', 'In this statistical analysis , we counted the number of the Ca2+ saturation events ( e . g . , as shown in Fig . 5E for the N-terminal lobe ) and then randomly distributed the same number of points within the spine volume .', "We calculated a so-called ( Besag's ) L-function ( see Methods for details ) for this random point pattern .", 'We repeated this process 1000 times and calculated the mean and the maximum and minimum envelope of the L-function ( the black dotted lines in Fig . 5F ) for the set of 1000 randomly generated spatial patterns .', 'We then calculated the L-function for the original data point pattern of Ca2+ saturation and compared this ( the red line in Fig . 5F ) with that of complete spatial randomness ( the black lines in Fig . 5F ) .', 'The L-function of data ( red ) is outside of the maximum and minimum envelopes ( black dotted lines ) indicating that the given point pattern is not random .', 'In this case , L-function is larger than the maximum envelope and it is typical of spatial clustering .', 'We performed a similar analysis for the C-terminal lobe ( Fig . 5D ) and fully Ca2+ saturated CaM ( Fig . 5H ) and obtained the same conclusion ( non-randomness ) .', 'For all cases , we also performed ( two-sample ) Kolmogorov-Smirnov ( goodness-of-fit hypothesis ) test ( significance level\u200a=\u200a0 . 05 ) [28] to verify the conclusion of envelope test .', 'In summary , the high Ca2+ injection rate results in a transient Ca2+-CaM nano-domain ( for both lobes of CaM ) .', 'The N-terminal lobe responds to and senses the Ca2+ gradient much faster than the C-lobe ( blue Fig . 5A ) .', "The C-lobe's response is resistant to the Ca2+ gradient because of its longer first passage time ( i . e . , slow binding kinetics of Ca2+ ) .", 'Note we recorded and analyzed only the first Ca2+ saturation events for each lobe of each CaM molecules .', 'The relatively widespread C-terminal lobe Ca2+ saturation in Panel C , therefore , is not because the high affinity C-terminal lobe carries Ca2+ ions while diffusing away from the channel .', 'What if we reduce the Ca2+ injection rate ?', 'Fig . 4 indicates that the spatial gradient of Ca2+ is less prominent with a reduced Ca2+ injection rate .', 'One possible scenario is that , under such a condition , only N-terminal lobe with higher Ca2+ binding kinetics ( Fig . 2 ) can detect and sense the spatial gradient .', 'The Ca2+ saturation of C-terminal lobe and/or full Ca2+ saturation of CaM may show relatively homogeneous spatial patterns under this condition .', 'Fig . 6 shows results to test this prediction .', 'The simulation conditions are the same as in Fig . 5 except the Ca2+ injection rate is reduced to 0 . 07 Ca2+ per microsecond .', 'This is close to the lowest Ca2+ injection rate observed for a single NMDA receptor Ca2+ current [36] , [37] , [38] .', 'Panel A and B in Fig . 6 show the population dynamics of Ca2+ saturated N- and C-terminal lobe , fully Ca2+ saturated CaM ( A ) , and free Ca2+ ions ( B ) .', 'The difference in the rising phase of Ca2+ saturated N- and C- terminal lobes observed in Fig . 5A becomes less obvious at these lower rates of Ca2+ influx .', 'The Ca2+ saturated N- and C-terminal lobes increase at a similar rate but the N-terminal lobe exhibits a larger fluctuation due to its fast Ca2+ dissociation rate .', 'Again , the number of fully Ca2+ saturated molecules is small ( less than 5∼10 ) over the course of a 25 ms simulation experiment .', 'In addition , the location of Ca2+ saturation for each lobe becomes less localized around the channel ( Fig . 6C and 6E ) .', 'It still looks like the N-lobe exhibits a nano-domain but it is unclear by a simple inspection of the data as to whether a nano-domain exists for the C-terminal lobe .', 'Up to the time points 2 ms and 4 ms , the Ca2+ saturation of the C-terminal lobe takes place throughout the entire spine head .', 'The distribution of these points appears to be random .', 'To confirm whether this pattern is random or not , we carried out the same statistical analysis as that used in Fig . 5 ( panel D , F , H ) .', 'Clearly , the data point patterns in Panel D and F ( red line ) are closer to the maximum envelope ( black dotted line ) of complete spatial randomness but the N-terminal lobe data pattern shows a deviation from the complete spatial randomness .', 'This result was again confirmed by Kolmogorov-Smirnov test .', 'The spatial pattern of the C-terminal lobe and full Ca2+-CaM saturation lie within the maximum/minimum envelope and did not suggest significant deviations from the spatial randomness .', 'In conclusion , the N-terminal lobe exhibits a transient Ca2+-activated nano-domain at both lower and higher Ca2+ injection rates .', 'This indicates that the kinetic property of the N-terminal lobe ( Fig . 1 and 2 ) is the major determinant of the spatial pattern formation by the N-terminal lobe .', 'In fact , we repeated simulations used to produce Figs .', '5 and 6 with different spine morphologies ( with shorter and longer spine neck as shown in Fig . 3A ) and obtained similar results as to the N-terminal lobe specific nano-domain ( Fig . S1 and Fig . S2 ) .', 'We also set the diffusion coefficient of CaM to 2 µm2/s ( nm2/µs ) and repeated simulations in Fig . 5 and 6 ( Fig . S1 and Fig . S2 ) .', 'As long as CaM molecules are randomly distributed within the spine volume ( at time 0 ) , neither the diffusion coefficient nor the concentration of CaM ( even when reduced to 10 µM ) affected the high sensitivity of the N-terminal lobe to the Ca2+ influx .', 'It appears that the Ca2+ binding kinetics of CaM ( first passage time ) is the major determinant of the lobe specific spatial pattern formation during Ca2+ influx .', 'In addition , the spatial p
['Introduction', 'Results', 'Discussion', 'Methods']
['Calmodulin ( CaM ) is a ubiquitous Ca2+ buffer and second messenger that affects cellular function as diverse as cardiac excitability , synaptic plasticity , and gene transcription .', 'In CA1 pyramidal neurons , CaM regulates two opposing Ca2+-dependent processes that underlie memory formation: long-term potentiation ( LTP ) and long-term depression ( LTD ) .', 'Induction of LTP and LTD require activation of Ca2+-CaM-dependent enzymes: Ca2+/CaM-dependent kinase II ( CaMKII ) and calcineurin , respectively .', 'Yet , it remains unclear as to how Ca2+ and CaM produce these two opposing effects , LTP and LTD .', 'CaM binds 4 Ca2+ ions: two in its N-terminal lobe and two in its C-terminal lobe .', 'Experimental studies have shown that the N- and C-terminal lobes of CaM have different binding kinetics toward Ca2+ and its downstream targets .', 'This may suggest that each lobe of CaM differentially responds to Ca2+ signal patterns .', 'Here , we use a novel event-driven particle-based Monte Carlo simulation and statistical point pattern analysis to explore the spatial and temporal dynamics of lobe-specific Ca2+-CaM interaction at the single molecule level .', 'We show that the N-lobe of CaM , but not the C-lobe , exhibits a nano-scale domain of activation that is highly sensitive to the location of Ca2+ channels , and to the microscopic injection rate of Ca2+ ions .', 'We also demonstrate that Ca2+ saturation takes place via two different pathways depending on the Ca2+ injection rate , one dominated by the N-terminal lobe , and the other one by the C-terminal lobe .', 'Taken together , these results suggest that the two lobes of CaM function as distinct Ca2+ sensors that can differentially transduce Ca2+ influx to downstream targets .', 'We discuss a possible role of the N-terminal lobe-specific Ca2+-CaM nano-domain in CaMKII activation required for the induction of synaptic plasticity .']
['Calmodulin is a versatile Ca2+ signal mediator and a buffer in a wide variety of body organs including the heart and brain .', 'In the brain , calmodulin regulates intracellular molecular processes that change the strength of connectivity between neurons , thus contributing to various brain functions including memory formation .', 'The exact molecular mechanism as to how calmodulin regulates these processes is not yet known .', 'Interestingly , in other excitable tissues , including the heart , each of two lobes of calmodulin responds differentially toward Ca2+ influx and toward its target molecules ( e . g . , ion channels ) .', 'This way , calmodulin precisely controls the Ca2+ dynamics of the cell .', 'We wish to test if a similar mechanism may be operational in neurons so that two lobes of calmodulin interact differentially with Ca2+ ions to activate different downstream molecules that control the strength of connections between neurons .', 'We constructed a detailed simulation of calmodulin that allows us to keep track of its interactions with Ca2+ ions and target proteins at the single molecule level .', 'The simulation predicts that two lobes of calmodulin respond differentially to Ca2+ influx both in space and in time .', 'This work opens a door to future experimental testing of the lobe-specific control of neural function by calmodulin .']
['neuroscience/theoretical neuroscience', 'cell biology/cell signaling', 'biophysics/theory and simulation', 'computational biology/computational neuroscience', 'neuroscience/neuronal signaling mechanisms', 'biochemistry/cell signaling and trafficking structures', 'biochemistry/theory and simulation', 'computational biology/systems biology', 'chemical biology/chemical biology of the cell']