ReDiClus
ReDiClus - Receptor Diffusion & Cluster Model
Simulation Space
ReDiClus Model Space
ReDiClus is simulated on a 2D surface in 3D space
- The surface area represents a dendritic membrane with two synaptic spines
- Baseline dimensions are scaled to real-world values
- these values are based on empirical observations of distal dendrites
- base dimensions are set to 60x30 units
- Scale
- 1 unit ≃ 100 nm
- 10 units ≃ 1 µm
- 2D space: 2.3 µm x 4.6 um
- PSD: 0.3 x 0.3 µm
- peri-PSD: 0.3 x 0.3 µm
- PSD separation: 2.0 µm
- The Z axis is only 2 levels: 0 and 1
- 1 represents the membrane surface
- 0 represents intracellular space
Particle Types
- There are 2 types of particles in the simulation
- 'Red' particle dots represent AMPA receptors
- Red dots can randomly diffuse anywhere on the X-Y plane
- Red dots only diffuse on the surface Z = 0
- 'Blue' particle dots represent PSD-95 molecules
- Blue dots are contained in predefined PSD areas and cannot leave
- Blue dots can exist at the surface Z = 0 or intracellularly Z = -1
Particle Diffusion
Simulating Molecular Diffusion
ReDiClus Diffusion
Particle diffusion is generated from Einstein's equations on Brownian motion. This allows the model to generate real-world diffusion at rates that are empirically relevant. There are currently 5 different regions in the model that can each independently scale the diffusion rate: the extrasynaptic space (ES), post-synaptic density 1 (PSD-1), post-synaptic density 2 (PSD-2) and the perisynaptic PSD-1 region (pPSD-1) and PSD-2 region (pPSD-2). The PSD and pPSD diffusion rates (D_{psd}) can be automatically scaled in real-time by the number of PSD-95 SAP molecules currently expressed in a PSD-cluster region. For most simulations the starting SAP cluster size is 7x7 yielding 49 total SAP molecules. The amount of SAP dynamically fluctuates. It can hold a fairly steady number of about 50 SAPs, but it can also be made to grow and shrink to values ranging from 10 to 100 SAPs. The PSD diffusion rate can be scaled from these SAP values. The function for this scalar can be seen to the left. Given a range of 10 to 100 SAPs, the PSD diffusion rate values will range from 0.03 um²/s - 0.003 um²/s.
- Base ExtraSynaptic Diffusion rate D (D_{es})
- D_{es} ≃ 0.3 um²/s
- Base PSD Diffusion rates (D_{psd})
- D_{psd} ≃ 0.03 um²/s
- ↑ to ↓
- D_{psd} ≃ 0.003 um²/s
- D_{psd} SAP scalar function
- D_{psd} ≃ D_{es}/SAP
- D_{psd} ≃ 0.3/10 ≃ 0.03 um²/s
- ↑ to ↓
- D_{psd} ≃ 0.3/100 ≃ 0.003 um²/s
Diffusion Equations
Optional Subroutines
in1 ≃ 1; % in1: do PSD S-clusters
in2 ≃ 0; % in2: do homeostatic
in3 ≃ 0; % in3: do calcium
in4 ≃ 1; % in4: do FRAP
in5 ≃ 0; % in5: do 1dot plot
in6 ≃ 0; % in6: do manual step size
in7 ≃ 0; % in7: do track MSD
in8 ≃ 0; % in8: do track step sizes
in9 ≃ 0; % in9: do MainPlot
in10 ≃ 0; % in10: do GluR1
in11 ≃ 0; % in11: do 3D Plot
1 %=========================================================%
2 % STARTING PARAMETERS
3 %---------------------------------------------------------%
4 D = 3; % Diffusion Constant [2d*D*t]
5 d = 2; % N dimensions
6 dT = 1; % time delay between measurements
7 k = sqrt(d*D*dT); % stdev of step size distribution D
8 MSD = 2*d*D*dT; % mean squared displacement
9 muN = k*sqrt(2)/sqrt(pi); % mean of half normal distribution k=stdev
10 Ld = sqrt(2*d*D); % average diagonal XY step size
11 LdA = Ld/sqrt(2); % average linear X or Y step size
12 DSc = 10; % D Scalar: DSc[10, 100] equals D[0.1, 0.01]
13 LdS = 1/sqrt(DSc); % D Scalar Function, adjusts LdA LdSfun(i) = 1/sqrt(i)
14 Dn = D/DSc; % Local D value after being scaled
15
16 % SET POTENTIATION LEVELS
17 PSD0 = 1; PSD3 = 1; % ESS D Base Diffusion Rate of ExtraSynaptic Space
18 PSD1DSc = 100; % PSD-1 D Scalar base
19 PSD2DSc = 100; % PSD-2 D Scalar base
20 PSD1D = D/PSD1DSc; % PSD-1 D value after being scaled
21 PSD2D = D/PSD2DSc; % PSD-2 D value after being scaled
22 PSD1 = LdSfun(PSD1DSc); % PSD-1 D Scalar Function, LdSfun(i) ≃ 1/sqrt(i)
23 PSD2 = LdSfun(PSD2DSc); % PSD-2 D Scalar Function, LdSfun(i) ≃ 1/sqrt(i)
Homeostatic Scaling
PSD-95 SAP Cluster Scaling
- From Tatavarty:
“ | Synaptic scaling is a cell-autonomous process in which neurons detect changes in their own firing through a set of calcium-dependent sensors, and then slowly increase or decrease the accumulation of synaptic AMPARs to compensate (Turrigiano et al., 1998; Ibata et al., 2008; Goold and Nicoll, 2010). | ” |
- From Sheng:
“ | PSD-95 family molecules outnumber AMPARs (up to 20-fold). Quantitative MS counted 60 copies of AMPAR subunits (GluR1, GluR2, GluR3) in the average PSD, which equates to 15 tetrameric AMPAR channels, of which >80 percent appears to be GluR1/GluR2 heteromers. Fifteen may be an underestimate because some postsynaptic AMPAR channels might be extracted by Triton during purification of PSDs. | ” |
Given those two observations, I wrote a matlab function that allowed for homeostatic scaling. Based on Sheng's review (and a few other sources), on average, there are about 20 AMPARs per PSD. In our current model we have 2 PSD areas, so there should be about 40 total receptors (combined) in those PSDs, on average.
To anthropomorphise, we want our modeled neuron to be homeostatically-content when there are 40 receptors in its synapses (content/satisfied meaning all diffusion parameters are running at some predefined baseline value, or whatever we specify). We can also specify a range of values at which our neuron is content; I arbitrarily set this range to 25-55 receptors. Just to be clear, that is 25-55 total combined receptors -- let's call this value PSDT. For example, PSD1 could have 35 receptors and PSD2 could have 10 receptors, making PSDT ≃ 45. If both PSD1 and PSD2 have 16 receptors each, PSDT ≃ 32. Our modeled neuron would be satisfied with either of those scenarios. However, if PSD1 had 35 and PSD2 had 30, making PSDT ≃ 65, our neuron will not be happy, and should take some action to decrease the total number of AMPARs being expressed at its PSDs.
- Homeostatic Scaling Function
IF PSDT > 55 THEN increase the PSD diffusion rates (make less sticky)
IF PSDT < 25 THEN decrease the PSD diffusion rates (make more sticky)
Remember the model functions are modular, so we could actually do something as simple as above, and directly alter the PSD diffusion rates. But we also want our modular functions to "play nice" with each other -- so if we are scaling PSD diffusion rates based on SAP expression ( function doSAP ≃ SAPfunc(on) ), instead of directly setting D_{psd} we'll instead want to increase or decrease SAP expression. Doing this will automatically update the diffusion rate of respective PSD areas. The easiest way to incorporate this function is by manipulating the SAP repulsion lattice constant (L) based on PSDT.
- When L≃2 the SAP cluster is relatively stable
- When L≃1 the cluster starts to grow
- When L≃3 the cluster starts to shrink.
So when PSDT > 55 then L ≃ 3 and when PSDT < 25 then L≃1 and when PSDT is between 25 and 55 then L≃2. And I should note that these are global changes to L (both PSD areas get the same L), because in a real-life scenario the neuron only cares that it's firing too much or too little, not necessarily which of its hundreds of PSDs are responsible. Does this make sense?
The homeostatic function can be made more complex, or flexible, but it can really be this simple (and I know you are a proponent of eloquence), which is nice for when we are testing out predictions unrelated to homeostatic scaling. But from here, we can begin to consider what manipulations to this function would make it more true-to-life. For example, homeostatic scaling is a relatively slow process -- maybe we only want the model to check the PSDT value once every 10 minutes, maybe we want to make more subtle changes in L, maybe we want to change another parameter instead of L. The choice is ours. But ultimately, we want to update the model based on what we know happens in the real world.
Physical Properties
ReDiClus Physics
two independent processes
- In this model, there are two independently occurring processes.
- 1. Blue dots can be expressed at the surface or internalized within their PSD area
- The Blue dot internalization/externalization rate properties are set by the Shouval cluster model equations.
- 2. Red dots diffuse on the X-Y plane with brownian motion
- Each Red dot has an initial step size randomly drawn from a normal distribution with a mean = 1 and sd = .2
- The step size for Red dots is dynamically altered when it's located in a PSD area
- In a PSD, the step size is reduced by a by some factor based on the number of Blue dots currently expressed at the surface of that PSD
- The more Blue dots at the surface, the more the step size is reduced
- The current step size function is:
- f(R_{step}) = R * (10*(1 ⁄ B_{n}))
- where R_{step} is the baseline Red dot step size
- where B_{n} is number of Blue dots currently expressed at the PSD surface
- f(R_{step}) = R * (10*(1 ⁄ B_{n}))
MEAN SQUARED DISPLACEMENT
^{SEE ANIMATION}{{#info: REDICLUS ANIMATION }}
- Brownian Motion Mean Squared Displacement
- The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
- Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
- Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
- a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
- empirical observations show that reasonable values for MSD are:
- PSD 0.01 µm ²/s
- synaptic 0.05 µm ²/s
- extrasynaptic 0.1 µm ²/s
- given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
- spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s
ReDiClus in Matlab
Live 3D simulation space
2D surface plot with PSD areas marked with boxes
SAP cluster for PSD1. For most simulations the starting SAP cluster size for this PSD is 7x7 or 49 SAP molecules. The diffusion rate 'D' of the extrasynaptic area (D.es) is scaled in the PSD (D.psd) by the current total number of SAPs present (SAP.t), such that [D.psd = D.ec/SAP.t]. So if D.es = .3 µm²/s and there are 10 SAPs in PSD1, then D.psd1 = .03 µm²/s.
PSD1: post synaptic density area "1". This PSD area is 6x6 units (1 unit equals 1 nm). The AMPAR 'red' particle diffusion rate in the PSD areas are scaled as a function of the total SAP expressed at their surface (D.psd = D.ec/SAP). So if the extracellular diffusion rate is 0.3 µm²/s and there are 10 SAP molecules in PSD1, then PSD1 diffusion rate would be D.psd = .3/10 = 0.03 µm²/s which are empirically relevant values.
This is the diffusion rate meter (aka D-Ometer) for PSD1. It provides information on the moment-to-moment diffusion rate inside PSD1. Note the meter is log scaled from .001 µm²/s to 1.0 µm²/s.
This is the diffusion rate meter (aka D-Ometer) for PSD2. It provides information on the moment-to-moment diffusion rate inside PSD2. Note the meter is log scaled from .001 µm²/s to 1.0 µm²/s.
These bar graphs represent the number of AMPAR (red) particles in a defined region: EC (extracellular), PSD1, PSD2, and PSDT (the total combined number of receptors in both PSD areas). In most simulations, if PSDT > 50, PSD clusters tend to shrink, and if PSDT < 30 SAP clusters tend to grow. This is due to the repulsion lattice changes, globally, based on the total number of receptors in synapses.
This the the boundary for the periPSD area, aka the perisynaptic pad for PSD2 that represents a non-PSD portion of a dendritic spine. This area has its own diffusion rate, and separate D-rates for GluR1 and GluR2 particles
This is the boundary for the PSD area of PSD-2. This area has a lower diffusion rate (D) compared to the surrounding periPSD area box. It also has a lower diffusion rate for GluR2 than GluR1 during baseline conditions.
Malinow | Molecular Methods | Quantum Dots | Choquet | AMPAR |