# Actin

Actin filaments are quasihelical structures, which can be defined as a double-stranded helix or a single stranded spiral. There are 26-28 actin protomers per 360 degree rotation, which spans 72-74 nm. Thus there are ~ 0.37 actin/nm filament or 2.7 nm filament/actin protomer. Alternatively the actin filament can be considered to have a right-handed helical structure with two strands slowly twisting around each other. Each actin monomer is rotated 166 degrees with respect to its nearest neighbors across the strand (Holmes 1990). Within the strand, subdomains 2 and 4 contact subdomains 1 and 3 in the next monomer in the strand, and each monomer reaches across to the other strand through a hydrophobic plug that links the two strands together.
Luo Robinson 2011 [1]
.

Actin Parameter Value Reference
Gactin monomer diameter 5 nm Luo Robinson 2011
Luo Robinson 2011 [2]
Factin protomer filament diameter 8 nm Luo Robinson 2011
Luo Robinson 2011 [3]
protomers per filament helical pseudo-repeat 13 protomers Luo Robinson 2011
Luo Robinson 2011 [4]
helical pseudo-repeat length 37 nm Luo Robinson 2011
Luo Robinson 2011 [5]
Factin protomer structure left-handed helical Luo Robinson 2011
Luo Robinson 2011 [6]
nm of filament per protomer 37 nm / 13 p = 2.85 nm/p Luo Robinson 2011
Luo Robinson 2011 [7]
protomers per nm of filament 13 p / 37 nm = 0.35 p/nm Luo Robinson 2011
Luo Robinson 2011 [8]
protomers per 360 degree turn 28 protomers Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [9]
filament length per 360 degree turn 72 nm per 360 degrees Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [10]
nm of filament per protomer 72 nm / 28 p = 2.57 nm/p Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [11]
protomers per nm of filament 28 p / 72 nm = 0.39 p/nm Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [12]
PROTOTYPICAL: nm of filament per protomer 0.369 p/nm (Factin protomers per nm of filament) Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [13]
PROTOTYPICAL: protomers per nm of filament 2.71 nm/p (nm of filament per Factin protomer) Schutt et al. 1993, Nature 365:810
Also in: Lodish, Principles of Molecular Biology [14]

## Actin Concentration

If the prototypical dendritic spine (defined to have a volume of 0.1 µm³) had an actin concentration of 125 µM the spine would have a total actin monomer count of:

125 µM * 6e23/1e6 * 1e-16 = 7500 (p)articles

that is

${\displaystyle {\frac {125\mu mol}{1L}}\times 10^{-16}L\times {\frac {6.022e17p}{1\mu mol}}=7500particles}$

## F-actin Filament Lengths

Given that each actin particle contributes 2.667 nm to the filament (and each nm of filament is 0.375 actin protomers), if all the actin in the spine was in polymerized form it would equate to a total of:

125 µM = 7500 p

7500 p * 1 nm/.375 p = 20,000 nm of filament

## Actin Polymerization Rates

Prototypical Actin Polymerization Rates
- k +   µM-1s-1 k −   s-1 Cc   µM
barbed (+) end 10 1.6 0.16
pointed (-) end 1 0.6 0.6
Overall Keq free actin: 0.2 µM
• Note that only the on-rate (Ka) is concentration dependent.
• The critical concentration (Cc) is calculated by dividing the off-rate by the on-rate.

Actin rate dynamics is well characterized, both experimentally and via modeling. Simple Ka/Kd on-rate and off-rate values for prototypical actin filament polymerization are given above. From this we can see that actin will polymerize extremely quickly when the free actin concentration is above the critical concentration for polymerization. For example, if 125 µM actin was suddenly made available in a prototypical dendritic spine, at the barbed (+) end the initial polymerization rate would be:

125 µM * 10 p/µM*s = 1250 p/s

Within 30 seconds, the 125 µM free actin would have already dropped to under 1 µM, with 19850 (of 20000) nm filament added to the network, while losing a nominal 30 monomers back into the free-actin pool.

vol. conversions

• 1 µm³ = 1 femtoliter (fL)
• 1 fL = 10-15 L
• 1 µM = 10-6 mol/L

The actin filament network will achieve steady-state when free Ga actin levels are at the Cc. Given that the majority of (-)ends are capped (by capping protein or Arp2/3), we can see that the critical concentration is ~0.16 µM (the Cc for (+)end polymerization). Thus, at steady-state, in a prototypical spine volume of 0.1 µm³, the expected number of free Ga monomers will be:

0.16E-6 mol/L * 6E23 p/mol * 0.1E-15 L = 10 particles

## Force Generated by Actin Polymerization

During polymer assembly, the filament can generate a force, measured at 1 pN
Luo Robinson 2011 [15]
, which is close to the theoretical estimate (for a given concentration of G-actin) according to:

where δ is the length increment of one monomer addition, k+(k-) is the on(off) rate of polymerization at the polus end, and c_A is actin concentration.

## Articles

Testa Urban Hell • 2012 • Cell • PDF

Figure 3

RESOLFT Nanoscopy Reveals Actin Distribution within Dendritic Spines

Honkura, Matsuzaki, Noguchi, Ellis-Davies, Kasai • 2008 • Cell • FullText

SI video1

Movie S7. Two-photon imaging of the confinement of PAGFP-actin fluorescence after repetitive photoactivation and uncaging of MNI-glutamate (60 pulses of 0.6 ms duration at 1 Hz) at a point (square) distal to the apex of the spine shown in Figure 5D. The 2D images were acquired every 10 s for 7 min. Photoactivation was induced at the moment when the white square turns red. Scale bar, 1 μm.

SI video1

Movie S8. Two-photon imaging of PAGFP-actin fluorescence after repetitive photoactivation and uncaging of MNI-glutamate (60 pulses of 0.6 ms duration at 1 Hz) at a point (square) distal to the apex of the spine shown in Figure 5G in the presence of Lat A (0.1 μM). The 2D images were acquired every 15 s for 10 min. Photoactivation was induced at the moment when the white square turns red. Scale bar, 1 μm.

Fischer, Kaech, Knutti, Matus • 1998 • Cell • FullText

Hover mouse over icon to expand supplementary movies
• SI video1

Visualization of actin dynamics in migrating fibroblasts. A rat embryo fibroblast transiently transfected with EGFP was recorded crawling from top right to bottom left of the frame. In addition to the actin dynamics associated with membrane ruffles, note the reorientation of stress fibers as the cell turns slightly toward the left from its initial direction. Time marker shows hours, minutes, and seconds.
• SI video2

Growth cone on a 48-hr-old hippocampal neuron transfected with GFP-actin. In addition to the concentration of actin in the “palm” of the growth cone, local concentrations are associated with spots and lateral filopodia from the shaft of the neurite. Time marker shows hours, minutes, and seconds.
• SI video3

A large field from a GFP-actin expressing hippocampal neuron is shown at low magnification (40X objective lens). As well as showing the widespread nature of spine motility, this sequence also indicates the extent to which actin is concentrated in spines compared to the shaft domain of dendrites. Time marker shows hours, minutes, and seconds.
• SI video4

Dynamics in motile spines recorded at higher magnification (100X objective lens) from a GFP-actin transfected cell in another culture. Over these short recording times the actin-driven changes are limited to spine shape and involve the growth and shrinkage of miniature protrusions. Time marker shows hours, minutes, and seconds.
• SI video5

Even over the brief duration of this recording (90 s) in a GFP-actin transfected neuron, continuous changes in the configuration of spine actin are visible. Time marker shows minutes and seconds.
• SI video6

The cell shown in Figure 2a was recorded during an experiment to examine the susceptibility of spine motility to the actin polymerization inhibitor cytochalasin D. At the point indicated, medium containing the drug flowed into the observation chamber. Time marker shows hours, minutes, and seconds.

We find that induction of long-term potentiation (LTP) of synaptic transmission in acute hippocampal slices of adult mice evokes a reliable, transient expansion in spines that are synaptically activated, as determined with calcium imaging. Similar to LTP, transient spine expansion requires N-methyl- D-aspartate (NMDA) receptor-mediated Ca2 influx and actin polymerization. Moreover, like the early phase of LTP induced by the stimulation protocol, spine expansion does not require Ca2 influx through L-type voltage-gated Ca2 channels nor does it require protein synthesis. Thus, transient spine expansion is a characteristic feature of the initial phases of plasticity at mature synapses and so may contribute to synapse remodeling important for LTP.

Hover mouse over icon to expand supplementary movies
• video1

Typical transient expansion.
Movie of a spine exhibiting typical transient expansion corresponding to Fig. 1B
(12 frames acquired every 0.5 min, viewed at 4 frames/s).
The appearance of the red circle represents stimulation with a 1-s, 100-Hz tetanus.
• video2

Movie 2. Asymmetric transient expansion.
Movie of a spine exhibiting asymmetric transient expansion
corresponding to Fig. 1D (12 frames acquired every 0.5 min, viewed at 4 frames/s).
The appearance of the red circle represents stimulation with a 1-s, 100-Hz tetanus.
• video3

Movie 3. Transient spine expansion on large- and small-diameter dendritic branches.
Movie of three dendritic branches with shafts of varying diameter from a single
CA1 pyramidal neuron exhibiting many spines undergoing transient expansion
(12 frames acquired every 0.5 min, viewed at 4 frames/s).
The appearance of the red circle represents stimulation with a 1-s, 100-Hz tetanus.

Fujiwara, Vavylonis, Pollard • 2007 • PNAS • FullText

Hover mouse over icon to expand supplementary movies
• SI movie1

SI Movie 1. Time lapse (230´) movie of the elongation of actin filaments by 3 mM Mg-ADP-actin
(30% Alexa-Green label) in polymerization buffer with 0.2 mM ADP,
viewed in a flow chamber coated with 50 nM NEM-myosin and 1% BSA.
• SI movie2

Time lapse (225) movie of the elongation and depolymerization of ADP-actin filaments (30% Alexa label).
The movie begins with filaments elongating in 5 mM Mg-ADP-actin in polymerization buffer.
At the point indicated by WASHOUT, free actin monomers were washed out of the
chamber with polymerization buffer with 0.2 mM ADP to allow depolymerization.

Koskinen and Hotulainen • 2014 • Frontiers • FullText

Measurements of actin turnover in dendritic spines: Fitting the data from individual measurements resulted in a mean stable component size of 18% as well as mean time constants of 51 sec for the dynamic component and 840 sec for the stable component.

Bindschadler, Osborn, Dewey, McGrath • 2004 • BiophysicalJ • PMC

Actin polymerization proceeds until only a small concentration (~0.1 µM) of unpolymerized actin (Gactin) remains. This ‘‘critical concentration’’ is also the minimum concentration required to form filaments (F-actin).

Both regulated and unregulated actin binding proteins modify the actin cycle in cells (Fig. 1). Barbed-end binding proteins block the assembly of G-actin at filament-barbed ends. The most abundant barbed-end binding proteins, capping protein (CP) and gelsolin (Isenberg et al., 1980; Yin et al., 1981), are inactivated by PIP2 and other polyphosphoinositides (Heiss and Cooper, 1991; Janmey and Stossel, 1987). Gelsolin, which also severs actin filaments (Yin and Stossel, 1979), requires micromolar calcium for its activity. CP, gelsolin, and Arp2/3 complex (Mullins et al., 1998), can nucleate new actin filaments. The processes of severing and nucleation help determine the number and length of actin filaments. Arp2/3 complex can also cap pointed ends (Mullins et al., 1998). Arp2/ 3 complex activities are greatly enhanced by the GTPase binding protein N-WASp (Machesky et al., 1999; Yarar et al., 1999). Inhibited by phosphorylation (Morgan et al., 1993), the ADF/cofilin family proteins bind preferentially to ADP containing subunits (Carlier et al., 1997). Cofilin destabilizes filaments by severing them (Maciver et al., 1991), by accelerating the rate of ADP subunit disassembly (Carlier et al., 1997), and by enhancing the rate of Pi release (Blanchoin and Pollard, 1999). Unregulated proteins of the b4-thymosin family bind actin monomers to maintain unpolymerized actin at hundreds of times the critical concentration (Safer et al., 1990). Unlike b4-thymosin, the monomer binding protein profilin has catalytic functions. Profilin accelerates the exchange of ADP for ATP on actin monomer 140-fold (Selden et al., 1999). Also unlike actin complexed with b4-thymosin, profilin-bound G-actin assembles at barbed ends but not pointed ends (Pollard and Cooper, 1984), releasing unbound profilin (Pantaloni and Carlier, 1993).

## Primary Simulation Goals

Overall the goal is to create an accurate, spatially resolved, 3D simulation of actin polymerization and structural dynamics. As a proxy for whether the model is accurate, there are several key parameters that should match empirical measurements on actin steady-state behavior. Given X µM total actin, R µM total branching protein (Arp2/3), in a volume V (0.1 µm2), with F+ and F- free parameters effecting on and off rates, the model should attempt to fit the following:

• the number of Factin protomers in filaments
• the number of free Gactin monomers.
• average filament length
• the amount of Gactin-Factin turnover in a unit time
• the number of filaments evolved in a unit time

## Quick Facts

• ActinSimChem for modeling actin dynamics
Halavatyi 2009 [16]
• Flexuraly rigidity persistence strength/length of an actin filament is ~ 17.7 µm
Fredrick 1993
• Hippocampal LTP is accompanied by enhanced F-actin content within the dendritic spine that is essential for late LTP maintenance in vivo.
Fukazawa [17]
• Theta stimulation polymerizes actin in dendritic spines of hippocampus
Lin Jneuro
• overexpressed PSD95 is coupled with a modest (but significant) increase in spine size
Ehrlich Malinow [18]

## Background Info

### Actin Researchers

• Carlier: empirical/experimental
• Pollard: empirical/experimental
• Pantaloni: empirical/experimental
• Kasai: empirical/experimental
• Halavatyi: stochastic MCMC models (ActinSimChem)
• Yarmola: stochastic MCMC models

### Actin Models

Bindschadler provides a good overview of both the experimental and modeling work on on actin dynamics. Here's Fig. 1 from that review going over the basic factors that affect actin polymerization:

Halavatyi - stochastic MCMC models (ActinSimChem)

Yarmola - stochastic MCMC models

All three of the modelers use the data from the experimentalists listed above. Digging back through my simulation notes, it looks like I did too. For example, Carlier provides some data (right) on the effects of Arp2/3 branching protein on the polymerization rate of the actin filament network, which I used (left) to validate my parameters for Arp2/3 filament association rates:

To further validate my parameters, I compared our 3D stochastic model with the (very complex) MCMC model by Halavatyi. This was easy to do because they packaged their simulation into a runnable program called ActinSimChem which you can download and run on Windows. I validated our model (S3D) against ActinSimChem across both nonstructurally-resolved (nSRF) and structurally-resolved filament (SRF) models, and it shows very good concordance with both. Comparisons over a 50 min (real-world) time period revealed concordance in:

total actin in system: 18066   S3D     SRF     nSRF
Mean Factin protomers 18025 18003 18014
Mean Gactin monomers 41 63 52
filaments (branches) evolved 312 380 351
Mean filament length (actin/fil) 57 47 51

From this, I think it's safe to say that the parameters we'll be using to simulate actin dynamics are right up there with the most sophisticated models currently in the wild for simulating actin dynamics (which, as far as I can tell, are among the most sophisticated models for simulating any biological process). The numbers would probably be dead-on if I hadn't taken the liberty to simplify things a little bit (a lot actually), based on averaging over some empirical data on actin polymerization, to arrive at these basic (prototypical) polymerization/depolymerization rates:

So, the take-away, is that actin polymerizes at the the leading end (+end) at a rate of Ka = 10 actin/(uM*s), and depolymerizes at a rate of Kd = 1 actin/s. Central to the idea of a simple cluster model is that the Factin off-rate (Kd) is not concentration dependent -- something well accepted in the actin literature. Another take-home point is that using these prototypical values for the +end (and assuming the -end is capped or is a branch-point, which it usually is), the critical concentration (Cc) for filament polymerization is 0.1 uM; that is, the on-rate is the same as the off-rate:

   Ka = 10 actin/(uM*s)  *  0.1 uM  = 1 actin/s
Kd == 1 actin/s


This critical concentration of 0.1 uM in a prototypical spine volume of 0.1 um^3, equates to 6 free actin monomers (unless my conversions are off):

   0.1 µm^3 = 1e-15 L
1 mole = 6e23 particles (p)

   Ga = .1/1e6 mole/L * 6e23 particles/mole * 1e-15 L = 6 particles


Based on this information, we have a framework to think about how the unbinding of bound actin monomers can (stably) influence spine size and filament density at the PSD.

## Actin Diffusion and Dynamics

morphology of dendritic spines via PALM/dSTORM imaging of photoactivatable actin. -Izeddin 2011
actin networks (cyan) of three dendritic spines that were imaged using electron microscopy. Actin is making contact with microtubules in the dendritic shaft (yellow) and the axon terminal of a presynaptic neuron (purple). White aggregates in the spine-head are SAP clusters. (n.b. these spines are a bit longer and have an actin network that is more sparse than would expected in vivo). Scale bar for all panels is 0.2 μm. -Korobova & Svitkina 2011

## Excerpts from my qualifying exam manuscript

Various factors influence actin association and dissociation rate constants: (a) Filament end: barbed (+)ends have faster on/off rates than pointed (-)ends. (b) ATP/ADP: the Ka/Kd rates depend on whether actin has ATP (T), ADP (D), ADP-Pi (Y). (c) Age: young filaments depolymerize rapidly, while older filaments depolymerize more slowly. (d) ABPs: the presence and concentration actin binding proteins can influence polymerization and branching in various ways: decrease polymerization rates [capP (CP) & thymosin-β4 (B)], promote filament breakage [ADF/cofilin (AC)], promote assembly [formin (F) & profilin (P)], or create branches off preexisting filaments [Arp2/3 (ARP)]. adopted from Bindschadler 2010

While overexpression led to a global receptor upregulation, the relative weights between spines were retained, such that large spines remained stronger than small spines. In fact, spine size may be one way to preserve synaptic weight information while the neuron performs global homeostatic scaling of SAPs and AMPARs. Indeed, spines were shown to undergo morphological changes prior to AMPAR accumulation, with the former preceded the latter by several minutes after chemically induced LTP 153. These structural (spine morphology) and ultrastructural (PSD / scaffold-protein remodeling) changes appear to depend on modifications to the actin cytoskeletal network that structurally supports the spine 154-157. LTP-induction stimuli are known to abruptly increase the amount of filamentous actin (Factin) in the spine; in-turn, the added filaments exert mechanical pressure on the surrounding membrane, promoting spine growth. Various studies have investigated questions concerning the relationship in synaptic strength and spine size, with the impetus of determining how synaptic strength remains directly tied to spine size given the stochastic noise inherent in a system governed by a relatively small number of molecules. Interestingly, Kopec et al. (2007) found that the two pathways (structural and receptor insertion) involved are not coupled until downstream of the LTP stimulus 144. Specifically they found that synaptic entry of GluR1 Ctails does not, itself, evoke spine enlargement; it is however required for the spine to stably increase in size during a time-window after an LTP-induction stimulus has been delivered. Again, this is emblematic of the interesting and complex role of GluR carboxyl termini in AMPAR trafficking and LTP regulation, and supporting the Malinow model previously described. Bosch et al. (2014) recently studied the precise chronology of several spine events after LTP induction 141. Consistent with that of Kopec and coworkers, Bosch found that cytoskeletal remodeling and changes in PSD proteins were independent to a degree, but flowed sequentially in several distinct phases: [1] the actin network underwent rapid remodeling, apparently with the help of several actin-associated proteins (AAPs) that were greatly upregulated in the spine; [2] actin filaments formed stable complexes with AAPs; [3] SAP and other PSD proteins fluctuated in their synaptic amounts in a late protein-synthesis phase.

Actin Dynamics Most, if not all, manipulations that prohibit the cytoskeletal remodeling of actin filaments block LTP 184. Actin is a multi-functional protein present in all eukaryotic cells, and widely known for its role as a cytoskeletal component. In vivo concentrations of actin can exceed 100 uM, and by weight it accounts for approximately 12% of all protein in dendritic spines 79. Actin can be present as a free or bound monomer (Gactin or Ga) or as a filamentous polymer (Factin or Fa), and is involved in many diverse cellular processes including cell motility, division, contraction, cytokinesis, vesicular transport, and cell signaling. There is a long literature on the in situ actions of actin, rich with quantitative details that highlight the complexity of actin dynamics, even in purified form. More recently it has become clear that actin’s importance extends beyond its central role as a structural protein. Actin has been shown to be involved in synaptic plasticity, playing key roles in both maintaining synaptic weights and reorganizing dendritic spines to support LTP/LTD, which will be presented in more detail below.

Cytoskeletal networks are formed via Gactin monomer assembly into filamentous Factin protomers. Converging evidence has revealed that a combination of factors influence the rate of Gactin monomer additions and removals from filament ends 185 , including filament age, the availability of ATP, the proteins bound to Gactin or Factin, the mechanical forces on the filament, and filament-end polarities (Fig. 16)i. Assembly at the two ends of an actin filament is regulated by a different set of kinetic properties (see Table 2). Conflux at the filament’s “barbed” (+)end is more active in relation to the “pointed” (−)end, which is most evident in the stark growth-rate differences observed at opposing tips. This gives rise to a phenomena known as treadmilling: when the availability of free-actin monomers is at the critical concentration for filament polymerization, monomers are added to the (+)end at the same rate they are removed from the (−)end. In mature spines this effect is abrogated by the presence of proteins that cap the free ends of actin filaments. In many cases the (−)end is attached to another actin filament by i branching proteins, giving rise to a highly branched cytoskeletal network of actin filaments in dendritic spines.

Actin concentrations in dendritic spines is very high 186, estimated to be around 12% of its total protein content at levels between 100 μM to 10 mM 187, 188; 100 μM actin in a prototypical spine volume (0.1 μm3) translates to a countable 6000 particles, while 0.1 μM (a level near the critical concentration for polymerization) computes to a mere 6 monomers (see Table 1). Monomers rapidly polymerize into filaments when free actin is above the critical concentration, because of this nearly all non-polymerized actin (Ga) in spines is sequestered by other proteins (e.g. thymosin-β4). Given the rapid transition of Ga to Fa, the scaffold network can be abruptly remodeled. If for example a signal released an additional 120 μM (7200 Ga), an additional 10,000 nm of filament would immediately be available (7200 Fa * 2.75 nm/Fa), ~850 nm of which would be added during the initial second of release, and nearly all 10,000 nm of filament would be added within 20 seconds (Fig. 17).

(A) Actin polymerizes rapidly. Over the course of just a few minutes thousands of individual Gactin monomers are incorporated into filaments. These figures are showing how polymerization would proceed if 120 μM were suddenly allowed to assemble in a prototypical spine volume of 0.1 μm3 (which equates to 7200 Ga monomers). Of the available 10,000 nm of filament (from 7200 monomers; 7200 Fa * 2.75 nm/Fa), ~850 nm would be added during the initial few seconds of release, and nearly all 10,000 nm of filament would be added within 20 s.
A simulation representing 20 min. of actin network growth and steady-state evolution was performed in a dendritic spine-shaped container. (A) Using empirical levels for spine actin and arp2/3 resulted in a cytoskeletal filament network that appeared optimal, as it uniformly filled the entire 0.1 μm2 spine volume. Other concentration ratios of actin and arp2/3 resulted in less ideal cytoskeletons, producing sparse filament distributions, or extreme branching in some spine regions while leaving others void. (B) Prototypical parameters resulted in full utilization of free Gactin monomers which were rapidly incorporated into the growing filament network. At the critical concentration for polymerization the filament network was stable but continued to undergo constitutive turnover of actin and arp2/3, as well as full branches. (C) Arp2/3 is incorporated into the filament network by associating with existing filaments and nucleating a new Factin branch. Thus, the rate at which arp2/3 monomers are converted to protomers depends upon, not just the monomeric concentration of arp2/3, but also the quantity of existing filaments. Since initially filamentous substrate is sparse, even though Arp2/3 concentrations are high, protomer conversion is slow. In short time however, this rate increases exponentially. Note that it's no coincidence that arp2/3 reaches a steady-state at the same time free-actin drops below its Cc for polymerization - plenty of arp2/3 monomers remain available to bind filaments, and they still do; but these nucleations are often transient, since the shortest branches have the highest probability of fully depolymerizing.

Various ABP-families facilitate this rapid polymerization, and aid in cytoskeletal remodeling by interacting with actin near the membrane. Actin Binding Proteins: WASP, Arp2/3, Cofilin The WASP-family of scaffold proteins connects the membrane to the cytoskeleton and is involved in facilitating the rapid polymerization of actin filaments. WASPs cap the barbed end of actin at the membrane and regulate polymerization; they also can detect signals and elicit cytoskeletal reorganization by activating the Arp2/3 complex 189. Arp2/3 activation resulting in a burst of filament polymerization due to its actions as a filament branch nucleator. Arp2/3 is an ABP complex located in many cellular compartments including dendritic spines (at ~10 μM, 600 copies) where it promotes filament nucleation. Arp2/3 also facilitates the rapidly assembly of branched actin networks by binding to the sides of preexisting filaments and nucleating new filament branches 190. In situ, Arp2/3 produced filament-branching at a rate of 9.7 Fbn/(ArpmM⋅s⋅Fμm) in the presence of WASPi and 2.5 Fbn/(ArpmM⋅s⋅Fμm) without WASP. That is 2.5 new filament branches per mM Arp2/3 per second per micron of existing filament. These rates translate to approximately 1 new filament branch forming each second in dendritic spines (Arp2/3 @ 9μM, 500μM Factin in a 0.1μm3 spine volume equates to 21.6 μm of filament).

(A) Actin filaments are helical structures with 13-14 protomers every 36 nm. Arp2/3 can bind to existing filaments and nucleate branching; Arp2/3 produces new branches that radiate away from the existing filament at 70-degree angles. (B) In Euclidian space the new filament branch is translated 70-degrees from the existing filament (θ), at a rotation angle anywhere from 1- to 360-degrees about an Fa point vector (ψ). Seen here is an example of how the filament network was rendered by Matlab. Filaments were treated as vectors, upon which rotations and transformations were performed using a 3D-coordinate rotational matrix. (C) These geometric formulas were used in conjunction with rate parameters for actin/APB dynamics, to simulate the stochastic evolution of 3D actin filament networks in real-time.

Actin network assembly is autocatalytic, meaning that new filament branches act as substrate for Arp2/3 to catalyze the addition of even more filament branches; as a result, small increases to branching rate result in an exponential increase in Factin assembly. This branching volley is then offset by a concomitant decrease in Arp2/3 concentration and after <5min the branching rate would be cut in half (0.5 Fbn/s) and max out near the total Arp2/3 population of 540 units (Fig. 18). That said, a network of more than 500 filaments is indeed a healthy cytoskeleton from which to conduct spine-related business.Modeling Actin in 3D Actin filaments are helical structures with 13-14 Fa protomers for every 36 nm of filament length (Fig. 19a). The Arp2/3 complex can associate with existing actin filaments and nucleate branching; Arp2/3 produces new branches that radiate away from the existing filament at 70°-angles. In terms of Euclidian space, the new branch is translated 70-degrees away from the parent filament along the z-axis (theta rotation). Nucleation of these 70-degree branches can happen anywhere from a 1- to 360-degree revolution about an Fa point-vector (see Fig. 19b). In this model, filaments were treated as vectors with the location of their pointed (−)end representing the origin; the location of their barbed-end could be calculated using a direction vector with a magnitude proportional the number of Fa they contain. To appropriately position new branches, rotations and transformations were performed using a 3D rotational matrix 191. A combination of matrix-based rotational formulas and actin/APB rate parameters were codified to simulate the stochastic evolution of actin filament network dynamics in real-time (see Fig. 19c).

Modeling actin dynamics was the most complex element of this simulation due to the combination of stochastic processes and the analytical geometry required to render 3D geometrical structures. Fortunately actin has a vast pool of experimental studies, and several refined (time-hardened) quantitative models that provide excellent characterizations of actin polymerization kinetics. To simulate filament scaffolding in the unified model, I’ve developed what’s probably best described as a stochastic 3D model (S3D model) of actin dynamics based on parameters from previously established in steady-state (Bindschadler 2004192 Yarmola 2008193), monte carlo (Halavatyi 2008193) and stochastic (Mogilner 2006 194) models. Each of these formulations do well at modeling various facets of actin polymerization with a small set of parameters. The impetus for developing of these prior models was to provide a novel synthesis that could better explain actin polymerization behavior, typically utilizing some new bit of biochemistry information. My motivation for developing an actin model is not this, per se. Instead I attempted to harmonize with these established actin models wherever possible, to validate the actin dynamics component for use in the unified model. As far as I know, however, the ability to simulate the stochastic evolution of actin networks in 3D makes this model the first of its kind. As mentioned, I validated the model against these prior models using the open-access software ActSimChem (Halavatyi and coworkers, 2008193) which can simulate both the structurally-resolved (SRF) and nonstructurally-resolved (nSRF) filament models.

S3D is our model.

The spatially-resolved S3D model of actin dynamics that I developed displays excellent parity to both the SRF and nSRF models. The most important parameter-outputs for validating this S3D model are shown in Table 2, which is based on a 50- minute simulation using a prototypical spine volume and known molecular levels of actin and ABPs. Indeed the model had concordance across all primary components of actin polymerization and branching (shown in Table 3).

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