Actin
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 {{#info: Halavatyi 2009 [1]}}
- Flexuraly rigidity persistence strength/length of an actin filament is ~ 17.7 µm {{#info: Fredrick 1993 }}
- Hippocampal LTP is accompanied by enhanced F-actin content within the dendritic spine that is essential for late LTP maintenance in vivo. {{#info: Fukazawa [2]}}
- Theta stimulation polymerizes actin in dendritic spines of hippocampus {{#info: Lin Jneuro }}
- overexpressed PSD95 is coupled with a modest (but significant) increase in spine size {{#info: Ehrlich Malinow [3]}}
Articles
Honkura, Matsuzaki, Noguchi, Ellis-Davies, Kasai • 2008 • Cell • FullText
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Fischer, Kaech, Knutti, Matus • 1998 • Cell • FullText
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Lang • 2004 • PNAS • FullText
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.
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Fujiwara, Vavylonis, Pollard • 2007 • PNAS • FullText
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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).
Background Info
Actin Researchers
Carlier: empirical/experimental Pollard: empirical/experimental Pantaloni: empirical/experimental Kasai: empirical/experimental
Bindschadler: steady-state models Halavatyi: stochastic MCMC models (ActinSimChem) Yarmola: stochastic MCMC models
Actin Models
- Bindschadler
- steady-state models
The (attached) review by 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)
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- Yarmola
- stochastic MCMC models
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All three of the modelers... Bindschadler: steady-state models Halavatyi: stochastic MCMC models (ActinSimChem) Yarmola: stochastic MCMC models
Use the data from these guys... Carlier: empirical/experimental Pollard: empirical/experimental Pantaloni: empirical/experimental
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:
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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've attached the manuscript associated with this software, and the manual. Here's a screenshot:
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I validated our model (S3D) against ActSimChem 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 Starting Actin Monomers: 18066
| measure/model: | 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 our "SIMPLEST SHOVAL CLUSTER MODEL EVER" idea, 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:
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 moles/L * 6e23 particles/M * 1e-15 L = 6 particles
So... based on that, 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; which is up next...
Actin Diffusion and Dynamics
Excerpts from my qualifying exam manuscript
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).
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).
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.
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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