Cytoskeletal signaling: is memory encoded in microtubule lattices by CaMKII phosphorylation?

Motor protein transport along MTs also depends on other types of MAPs, e.g. optimally spaced between ∼15 and 25 nm (equivalent to the length of two to three dimers) apart along the protofilament length [47], [48]. MAP tau (whose hyperphosphorylation and detachment from MTs correlates with Alzheimer's disease) appears to signal motor proteins precisely where and when to disengage from MTs and deliver their cargo [49] (see Figure 8 D). Consequently, patterns of CaMKII-induced tubulin phosphorylation in MT lattices could regulate motor protein transport along MTs directly, and/or by MAP attachment locations.

Cytoskeletal structures including MTs were first suggested to process information and regulate cellular function by Sir Charles Sherrington [16]. Computational interaction among discrete tubulin states [18], [50], and signals propagating along MT protofilaments and/or helical winding pathways have been proposed [15] (e.g. in dendritic and somatic integration in ‘integrate-and-fire’ brain neurons, regulating axonal firings and synaptic plasticity [51]).

Both MTs and MAPs can support solitary waves of ionic transport flow, e.g. via C-terminal tails on MT exteriors [52]. Signals may also propagate through MTs by intra-tubulin conduction pathways defined by electron resonance rings of aromatic amino acids [15]. Evidence suggests highly conductive electron and/or soliton conductance through MT lattices, including via helical pathways [53]. Signaling has also been proposed to occur in MT hollow cores [54].

MTs appear to have intrinsic electro-mechanical vibrations, e.g. coherent oscillations in the low megahertz range [55]. MT assembly is enhanced by 6 orders of magnitude when exposed to an applied 1 to 3 megahertz (radio frequency) alternating current, emits laser-like 3.1 to 3.8 megahertz radiation, and exhibits some form of electron condensation [53]. Signal propagation will be influenced by, and depend on MT lattice vibrations and related effects.

Dynamic interplay among CaMKII phosphorylation, communicative signaling/vibration, and MAP attachment sites are rich opportunities for regulation of synapses and intra-neuronal processes. Analogies to musical instruments have been suggested, e.g. MAP attachments acting as ‘frets’, as in a guitar [56], or the “guitar string hypothesis” [57] altering wavelengths and changing MT resonant frequency with consequences for functional processes.

In LTP, high frequency inputs (e.g. 50 to 100 Hz) are required for prolonged post-synaptic response. Kumar et al [58] showed that memory formation in dendrites depends on synchronized inputs, with an optimal frequency near 50 Hz. Density and patterns of CaMKII-induced tubulin phosphorylation in post-synaptic MT lattices would depend on frequency and synchrony of inputs.

Clusters of phosphorylated tubulin, and/or MAP attachment may serve as logic gates for propagating information. Figures 9 and 10 demonstrate two types of Boolean logic gates, an AND gate and an exclusive OR gate (XOR) in which MAPs convey inputs, with output along tubulin pathways. Figures 11 and 12 show AND and XOR gates in which MAPs convey output of inputs and processes in tubulins within the MT. The combination of XOR and AND logic gates forms a universal set for computation in which all other logic gates (NOT, OR etc.) can be conceived. Signals propagating through MT-MAP logic circuits may extend throughout cytoskeletal networks, regulating synaptic function, cognition and behavior.

‘Encoding’ is conversion of information from one form to another, each form requiring a geometry in which information is represented, e.g. as ‘bits’. Computer pioneer Alan Turing described a one-dimensional geometry, a string of information on a linear tape. Another computer pioneer John Von Neumann and others [59] developed two-dimensional cellular automata, lattice surfaces of interactive ‘cells’, or units. Simple rule-based interactions among neighbor cell states updating at discrete time steps can lead to complex patterns, computation and self-organization in cellular automata. Recently, three-dimensional assemblies of duroquinone molecules have been shown to function as ‘molecular’ automata (‘nanobrains’) [60].

Biology uses dynamical three-dimensional information via membrane dynamics, concentration gradients of various cytoplasmic components (e.g. reaction-diffusion) and cytoskeletal processes. How information is shared or encoded among these levels in a common framework is largely unknown.

We address information processing in cytoskeletal MTs, polymers of the protein ‘tubulin’, in both A-lattice and B-lattice configurations. Under proper conditions tubulin self-assembles into MTs, cylindrical hexagonal lattices, which directly participate in cell organization. Sherrington in 1951 [16] first proposed the cytoskeleton might serve as a cellular ‘nervous system’, and others have suggested microtubule-based information processing, e.g. with individual tubulins representing bit-like information states [18], [19], [56]. For example MTs have been modeled as von Neumann-type cellular/molecular automata (‘microtubule automata’) in which tubulin subunits interact with neighbor tubulin states by rules based on dipole coupling strengths [50], [61]. In such proposed microtubule automata, tubulin states interact and update coherently at discrete time steps attributed to theorized [62] or experimentally-observed coherent MT resonances, e.g. in the low megahertz range [53], potentially resulting in millions of synchronized updates per second [63].

To simulate microtubule automata, two-dimensional hexagonal MT lattices are slightly skewed according to lattice geometry and wrapped into a three-dimensional cylinder. Simulations show oscillating wedge-shaped, triangular and hexagonal patterns of tubulin states, which evolve, compute and can learn [50]. Because of cylindrical MT lattice geometry, such patterns reverberate, interfere and change nonlinearly. Thus microtubule automata are in principle capable of information processing. How could they interact with membranes and beyond, with extra-cellular processes? How could microtubule automata receive inputs and express outputs?

In this paper we evaluated possible information inputs to microtubules in the context of brain neuronal memory encoding and long-term potentiation (LTP). A key intermediary in LTP involves the hexagonal holoenzyme calcium-calmodulin kinase II. When activated by synaptic calcium influx, the snowflake-shaped CaMKII extends sets of 6 foot-like kinase domains outward, each domain able to phosphorylate a substrate or not (thus convey 1 bit of information). As CaMKII activation represents synaptic information, subsequent phosphorylation by CaMKII of a particular substrate may encode memory, e.g. as ordered arrays of 6 bits (one ‘byte’). We used molecular modeling to examine feasibility of collective phosphorylation (and thus memory encoding) by CaMKII kinase domains of tubulins in a microtubule lattice.

We show, first, complementary electrostatics and mutual attraction between individual CaMKII kinase domains and tubulin surfaces. We also demonstrate two plausible sites for direct phosphorylation of tubulin by a CaMKII kinase domain, and calculate binding energies in the range of 6 to 36 kcal/mol per CaMKII-tubulin phosphorylation event. This indicates encoding which is robust against degradation, yet inexpensive, requiring on the order of 2% of overall brain metabolism for maximal encoding in all 1011 neurons.

We then compare size and hexagonal configuration of the six extended foot-like kinase domains of activated CaMKII with hexagonal lattices of tubulin proteins in MTs. We find that CaMKII size and geometry of 6 extended kinase domains precisely match hexagonal arrays of tubulin in both A-lattice and B-lattices.

We quantified the potential information capacity for CaMKII hexagonal encoding of MT lattice regions, specifically in lattice ‘patches’ of 7 (A lattice) or 9 (B lattice) tubulin protein dimers. The simplest case was taken as CaMKII phosphorylation of an A lattice patch of 7 tubulin dimers (central ‘address’ dimer unavailable for phosphorylation, 6 available dimers). Each kinase domain can phosphorylate one tubulin dimer, either its α- monomer or its β- monomer equivalently. For either dimer, phosphorylation = 1, no phosphorylation = 0. Sets of 6 CaMKII kinase domains interacting with 6 tubulin dimers can then provide 6 binary bits (64 possible states), comprising one byte.

On an A lattice we also consider each tubulin dimer being phosphorylated either on its α-tubulin monomer (0), α-tubulin monomer (1), or neither (2), resulting in 6 ternary states (‘trits’) or 729 possible encoding states (per ‘tryte’).

We also considered ternary states in a B lattice with 9 tubulin dimers in a patch. With the central ‘address’ dimer unavailable for phosphorylation, sets of 6 CaMKII kinase domains can choose and phosphorylate α- (0), β- (1), or neither (2) tubulin monomer in any 6 of 8 available dimers. This yields 5,281 possible information states per neighborhood patch of 9 tubulin dimers. There are approximately 1019 tubulins in the brain. Consequently, potential information capacity for CaMKII encoding of hexagonal MT lattices is enormous.

Assuming input information may be encoded and processed in hexagonal microtubule lattices, how could such information be expressed as output to regulate cytoplasmic and membrane activities? We list six possible mechanisms including a potential connection to reaction-diffusion systems in cytoplasm.

Concentration gradients of various molecules in cytoplasm regulate cellular activities. In 1953 Alan Turing [64] showed that stable patterns of molecular concentrations could emerge through reaction, diffusion and inhibition. Reaction-diffusion systems include periodic waves and hexagonal patterns, which oscillate, undergo phase transitions and are capable of information processing [41]. Turing patterns have been suggested in biological systems across scale, e.g. from intra-cellular processes [42] to neuronal network organization. Cortical neurons representation maps are organized hexagonally [65], and ‘grid cells’ in entorhinal cortex record spatial location as apparent hexagonal maps, with different spatial scales at different layers of entorhinal cortex [66], [67].

‘Scale-free’ implies self-similar information patterns repeating at different spatial and temporal scales (following 1/frequency power laws). Similar to ‘fractals’ and holograms, scale-free structures and processes arise commonly in nature and technology, and are inherently robust and resistant to disruption. In the brain, evidence suggests neuronal network structure, temporal dynamics, and representation of mental states are all ‘scale-free’, with self-similar patterns repeating at various temporal and spatial scales and locations [68]–[71]. Interference patterns of periodic, coherent reaction-diffusion waves in cytoplasm and larger spatial scales could account for scale-free information patterns regulating biological systems including the brain. Microtubules can generate three-dimensional reaction-diffusion patterns [70], and we suggest such patterns operate at multiple time scales to regulate biological systems.

We demonstrate a feasible and robust mechanism for encoding synaptic information into structural and energetic changes of microtubule (MT) lattices by calcium-activated CaMKII phosphorylation. We suggest such encoded information engages in ongoing MT information processes supporting cognition and behavior, possibly by generating scale-free interference patterns via reaction-diffusion or other mechanisms. As MTs and CaMKII are widely distributed in eukaryotic cells, the hexagonal bytes and trytes suggested here may reflect a real-time biomolecular information code akin to the genetic code.

Sequences of human αCaMKII kinase, autoregulatory, linker, and association domains, as defined by Dosemici et al. [72] and Tombes et al. [73] were used to build homology models. Crystal structures 1HKX [74] and 2VZ6, found in the Protein Data Bank (PDB) [75], were used as templates to build basic homology models of the association, kinase and autoregulatory domains, respectively, using MODELLER 9V6 [76]. The CaMKII holoenzyme structure was built with PYMOL 0.99rc6 [77] using the geometry described in Rosenberg et al. [78] with the linker region constructed as a linear chain of residues joining the autoregulatory and association domains.

PDB tubulin protein structure 1JFF [79] was repaired by adding missing residues from 1TUB [80]. The repaired 1JFF dimer was solvated, neutralized and energy-minimized using NAMD [81]. The minimized and repaired 1JFF structure was used as a template to build basic homolgy models of TUBA1A and TUBB3 using MODELLER 9V6 [82]. Using this dimer, MT A and B lattice structures were built with PYMOL 0.99rc6 [77] using MT geometry described in Li et al. [82] and Sept et al. [32]. PYMOL 0.99rc6 [77] was used to model and illustrate the positional geometry changes of CaMKII-tubulin/MT lattice interactions.

To analyze electrostatic matching, hydrogens were added, and protonation states set at pH 7 with PROPKA [83], via PDB2PQR [84], [85] for both CaMKII and MT lattice structures. The Poisson-Boltzmann equation was solved for the structures in given arrangement with the Adaptive Poisson-Boltzmann Solver (APBS) [86] at a grid spacing of less than 1 Å. Isosurfaces were generated at +0.5 kT/e and −0.5 kT/e, and field lines were drawn with gradient magnitude 3. Electrostatic and field line images were generated in PYMOL 0.99rc6 [77] and VMD 1.8.7 [87], respectively.