Log-log profiles of the MSD under the self-crowding condition with a_DT = 0.001 for an extended duration of 4000ms.

<p>The log-log profiles are shown for the select few <i>a</i>_<i>CRO</i> = 0.0 (A), 0.25 (B) and 0.40 (C). It is evident that the profiles exhibit sharp decay at longer time, which was not clearly noticeable in the log-log profiles for the time duration of 2000<i>ms</i> shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005984#pcbi.1005984.g003" target="_blank">Fig 3D</a>.</p

The effect of varying DT-PRO binding energy and PRO density on the anomalousity of tracer diffusion, under different self-crowding conditions.

<p>(A) Weak DT-PRO binding. The profiles of variation in anomalous exponent, <i>α</i>, of the tracer diffusion under the different conditions of self-crowding are plotted together across the increasing <i>a</i>_<i>PRO</i>. <i>α</i> remains equal to one across the entire span of <i>a</i>_<i>PRO</i> for the no-self-crowding condition and the self-crowding conditions with <i>a</i>_<i>DT</i> ≤ 0.001. For <i>a</i>_<i>DT</i> = 0.005, <i>α</i> gradually rises from zero and approaches 1 with increase in <i>a</i>_<i>PRO</i>. However, the rise in <i>α</i> becomes significantly slower for the higher <i>a</i>_<i>DT</i> = 0.01, such that it remains less than 1 even at <i>a</i>_<i>PRO</i> = 0.8. In continuation, for very high <i>a</i>_<i>DT</i> = 0.1, effect of increasing <i>a</i>_<i>PRO</i> is visibly insignificant and <i>α</i> remains close to zero. (B) Intermediate DT-PRO binding. <i>α</i> remains 1 across the entire span of <i>a</i>_<i>PRO</i> for the no-self-crowding condition and the self-crowding conditions with <i>a</i>_<i>DT</i> ≤ 0.01. It is only for <i>a</i>_<i>DT</i> = 0.1 that, with increase in <i>a</i>_<i>PRO</i>, <i>α</i> sharply rises and approaches 1 by <i>a</i>_<i>PRO</i> = 0.6. (C) Strong DT-PRO binding. <i>α</i> remains 1 across the entire span of <i>a</i>_<i>PRO</i> for the no-self-crowding condition as well as all values of <i>a</i>_<i>DT</i>. (A-C) It is evident that increase in binding reduces anomalousity of tracer diffusion arising from self-crowding. However, the intensity of <i>a</i>_<i>PRO</i>-dependent amelioration of anomalousity further depends on the intensity of self-crowding.</p

Log-log profiles of the MSD of tracer diffusion for the varying DT-PRO binding energy and PRO density under different self-crowding conditions.

<p>Three levels of DT-PRO binding energy, 2, 6 and 10 <i>k</i>_<i>B</i><i>T</i>, are considered to represent the situations of weak, intermediate and strong DT-PRO binding, respectively. Four increasing PRO densities viz. <i>a</i>_<i>PRO</i> = 0.2, 0.4, 0.6 and 0.8 are sampled to account for a range of sparse to dense submembranous crowding of PROs at the PSD. Subsequently, these combinations are examined for the different conditions of self-crowding. Here, the log-log profiles under the different conditions of self-crowding are laid together in a single plot. (A-D) Weak -PRO binding. The profiles for no-self-crowding condition and <i>a</i>_<i>DT</i> ≤ 0.001 are consistently flat (representing perfectly normal diffusion) regardless of the <i>a</i>_<i>PRO</i> as well as fairly overlap with each other. For <i>a</i>_<i>DT</i> ≥ 0.005, the profiles indicate strong anomalous diffusion for the lower <i>a</i>_<i>PRO</i> = 0.2. Increase in <i>a</i>_<i>PRO</i> causes a gradual transition of the profiles from strongly anomalous to normal diffusion, though the intensity of the anomalousity relaxation further depends on the level of <i>a</i>_<i>DT</i>. For example, the profile for <i>a</i>_<i>DT</i> = 0.005 acquires perfectly normal behaviour by <i>a</i>_<i>PRO</i> = 0.8 and overlaps with that of the lower <i>a</i>_<i>DT</i>. However, the profile for <i>a</i>_<i>DT</i> = 0.01 carries slightly anomalous behaviour even at <i>a</i>_<i>PRO</i> = 0.8. The anomalousity in the profile for <i>a</i>_<i>DT</i> = 0.01 remains insignificantly affected by the increase in <i>a</i>_<i>PRO</i>. (E-H) Intermediate DT-PRO binding. The profiles for the no-self-crowding condition and <i>a</i>_<i>DT</i> ≤ 0.01 consistently exhibit normal diffusion regardless of <i>a</i>_<i>PRO</i> and fairly overlap with each other. Notably, <i>a</i>_<i>DT</i> = 0.1 is associated with strongly anomalous behavior for the lower <i>a</i>_<i>PRO</i> = 0.2. However, the anomalousity steeply decreases with the increase in <i>a</i>_<i>PRO</i> and the log-log profile approaches closer to that for the <i>a</i>_<i>DT</i> ≤ 0.01. (I-L) Strong DT-PRO binding. The profiles for the no-self-crowding condition and all values of <i>a</i>_<i>DT</i> consistently exhibit normal diffusion regardless of the <i>a</i>_<i>PRO</i> and remain sufficiently overlapping with each other. (A-L) Altogether, increase in binding ameliorates anomalousity of tracer diffusion arising from their higher self-crowding. Furthermore, increase in <i>a</i>_<i>PRO</i> and binding energy cause the log-log profiles to shift to lower levels. In general, this has implications to the decrease in tracer mobility in terms of the effective diffusion coefficient.</p

Effect of densities of CROs and DTs on the anomalousity and effective diffusion coefficient of tracer diffusion.

<p>Anomalousity of tracer diffusion is characterized through the anomalous exponent, <i>α</i>, of the diffusion, which is computed from the log-log profiles of the temporal evolution of MSD. <i>α</i> = 1 is associated with perfectly normal diffusion. Lower is the <i>α</i>, higher is the anomalousity of tracer diffusion. (A) The variation in <i>α</i> across increasing CRO density under different self-crowding conditions is shown. It is apparent that for lower DT densities, the profile of <i>α</i> exhibit a switch-like behaviour where the anomalousity steeply rises beyond a certain high level of <i>a</i>_<i>CRO</i> (i.e. 0.4 here) and leads to the strong sub-diffusion or confinement of the diffusing tracers. However, very high levels of <i>a</i>_<i>DT</i> consistently exhibit strong anomalous sub-diffusion regardless of the <i>a</i>_<i>CRO</i>. (B) The variation in <i>α</i> across increasing <i>a</i>_<i>DT</i> under different conditions of <i>a</i>_<i>CRO</i> is shown. For lower <i>a</i>_<i>CRO</i>, the profiles demonstrate a switch-like behaviour across increasing <i>a</i>_<i>DT</i> and there occurs a sudden transition to strongly anomalous diffusion beyond <i>a</i>_<i>DT</i> = 0.001. Increase in <i>a</i>_<i>CRO</i> appears to accentuate the sharpness of the transition. However, for very high <i>a</i>_<i>CRO</i>, the tracer diffusion is strongly anomalous regardless of the <i>a</i>_<i>DT</i>. (C) The effects of varying <i>a</i>_<i>CRO</i> and <i>a</i>_<i>DT</i> on the anomalousity of tracer diffusion are collectively summarized in the gray-scaled heat-map and is derived from the previous observations made in (A) and (B). It clearly shows that both high CRO density and/or high DT density may lead to strongly anomalous sub-diffusion and trapping of tracers. (D) The variation in effective diffusion coefficient, <i>D</i>_<i>eff</i>, of tracer diffusion across increasing CRO density under different self-crowding conditions is shown. For the lower values of <i>a</i>_<i>DT</i> and the no-self-crowding condition, increase in <i>a</i>_<i>CRO</i> leads to a consistent decrease in the <i>D</i>_<i>eff</i>. Moreover, their profiles of <i>D</i>_<i>eff</i> are highly overlapping and fairly identical. With increase in <i>a</i>_<i>DT</i>, the profile shifts to lower levels, such that very high <i>a</i>_<i>DT</i> is associated with almost negligible <i>D</i>_<i>eff</i> and, in concordance with <i>α</i>, indicates severely hampered and confined tracer mobility, regardless of <i>a</i>_<i>CRO</i>. Notably, at <i>a</i>_<i>CRO</i> = 0, <i>D</i>_<i>eff</i> of tracer diffusion for no-self-crowding conditions and <i>a</i>_<i>DT</i> ≤ 0.0001 is equivalent to the effective free diffusion coefficient of AMPARs (0.2<i>nm</i>².<i>μs</i>⁻¹) in the extrasynaptic membrane of excitatory synapses.</p

The effect of varying DT-PRO binding energy and PRO density on the effective diffusion coefficient of tracer diffusion, under different self-crowding conditions.

<p>(A) Weak DT-PRO binding. For the no-self-crowding condition and self-crowding conditions with <i>a</i>_<i>DT</i> ≤ 0.001, the effective diffusion coefficient, <i>D</i>_<i>eff</i>, of tracer diffusion appears to consistently decrease with increase in <i>a</i>_<i>PRO</i> and their profiles also fairly overlap with each other. For <i>a</i>_<i>DT</i> = 0.005, <i>D</i>_<i>eff</i> initially rises with increase in <i>a</i>_<i>PRO</i> and closely approaches the <i>D</i>_<i>eff</i> for lower <i>a</i>_<i>DT</i> at <i>a</i>_<i>PRO</i> = 0.6. Subsequently, following the <i>D</i>_<i>eff</i> profile for lower <i>a</i>_<i>DT</i>, it decreases with further increase in <i>a</i>_<i>PRO</i>. For <i>a</i>_<i>DT</i> = 0.01, <i>D</i>_<i>eff</i> gradually rises with increase in <i>a</i>_<i>PRO</i> and closely approaches the profile of variation in <i>D</i>_<i>eff</i> for lower <i>a</i>_<i>DT</i> by <i>a</i>_<i>PRO</i> = 0.8. The rise in <i>D</i>_<i>eff</i> observed here with rising <i>a</i>_<i>PRO</i> is owing to the associated reduction in anomalousity and confinement of tracer diffusion. However, for <i>a</i>_<i>DT</i> = 0.1, <i>D</i>_<i>eff</i> remains significantly low across the entire span of <i>a</i>_<i>PRO</i>. (B) Intermediate DT-PRO binding. <i>D</i>_<i>eff</i> appears to consistently decrease with increase in <i>a</i>_<i>PRO</i> for the no-self-crowding condition and self-crowding conditions with <i>a</i>_<i>DT</i> ≤ 0.01. However, for <i>a</i>_<i>DT</i> = 0.1, <i>D</i>_<i>eff</i> sharply rises with increase in <i>a</i>_<i>PRO</i> and closely approaches the profile of variation in <i>D</i>_<i>eff</i> for lower <i>a</i>_<i>DT</i> by <i>a</i>_<i>PRO</i> = 0.4. Further increase in <i>a</i>_<i>PRO</i> leads to gradual decrease in <i>D</i>_<i>eff</i>. (C) Strong DT-PRO binding. <i>D</i>_<i>eff</i> consistently decreases with increase in <i>a</i>_<i>PRO</i> for the no-self-crowding condition as well as all values of <i>a</i>_<i>DT</i>. Further, the profiles of the variations in <i>D</i>_<i>eff</i> are sufficiently overlapping. (A-C) It is evident that the scale of the magnitude of <i>D</i>_<i>eff</i> significantly decreases with increase in the DT-PRO binding energy. Accordingly, increase in binding, either through increase in binding energy or PRO density or both, in general leads to reduction in tracer mobility. Only for the conditions of self-crowding where increase in binding leads to reduction in anomalousity of tracer diffusion, one may observe a relative increase in <i>D</i>_<i>eff</i>.</p

Log-log profiles of the ensemble-averaged MSD under different self-crowding conditions and CRO densities.

<p>The profile of variation in natural logarithm of the ratio of ensemble-averaged MSD to time,log (〈<i>r</i>²〉/<i>t</i>), with respect to the natural logarithm of time, log (<i>t</i>), is referred here as the log-log profile for simplicity. (A-G) Each plot demonstrates the log-log profiles of MSD across the increasing CRO density for a given condition of self-crowding. Perfectly normal diffusion is characterized by a flat or horizontal log-log profile across the entire duration of MSD measurement. However, profiles exhibiting sharp linear decrease at longer time characterize strongly anomalous sub-diffusion or confined diffusion and indicate the original dependence of MSD on time raised to a fractional power (power-law relation). There are profiles too which show transition from anomalous to normal diffusion. Evidently, tracer diffusion is observed to be confined and strongly anomalous at higher densities of CROs and DTs.</p

Temporal profiles of ensemble-averaged MSD under different self-crowding conditions and CRO densities.

<p>(A-G) Each plot demonstrates the temporal profiles of ensemble-averaged MSD, 〈<i>r</i>²〉, across the increasing CRO density, <i>a</i>_<i>CRO</i>, for a given condition of self-crowding. The no-self-crowding condition refers to the conventional approach where self-crowding of diffusing tracers is not considered in the simulation and, hence, is regarded here as the standard benchmark for comparison. Otherwise, the different self-crowding conditions are recognized by the density of DTs, <i>a</i>_<i>DT</i>, on the lattice with which the diffusion simulation is performed. The tracer diffusion appears normal for lower densities of CROs and DTs where 〈<i>r</i>²〉 appears to increase linearly with time. However, it becomes strongly subdiffusive-anomalous and confined for the higher CRO and DT densities where 〈<i>r</i>²〉 reaches a saturation or plateau and no significant increase in MSD occurs further with temporal progression.</p

The simplified molecular composition of PSD and lattice model of AMPA receptor diffusion.

<p>(A) An schematic demonstration of the CROs and PROs under the broad classification adopted here for the various crowding elements present at the PSD of a typical excitatory synapse on a dendritic spine. CROs are generally the transmembrane proteins which interact with a diffusing receptor only to elastically repel it away on collision. PROs are typically the submembranous scaffold proteins which offer partial reflection as well as binding to diffusing receptors. The AMPAR lateral diffusion is modeled here using a transmembranous diffusing tracer (DT). (B) A part of the entire lattice illustrating a discrete space for the tracer diffusion. DT, CRO and PRO are localized on the lattice as point elements, with different respective area fractions. A tracer can diffuse randomly to either of the four directions. Δ<i>l</i> is the size of the lattice edges between any two lattice points. The entire lattice is an abstraction meant to be used here for representing receptor diffusion over either the entire PSD or a region within the PSD. (C) An illustration of the false self-blocking of two DTs, labelled 1 and 2, lying at the neighbouring lattice sites. The different computational sequences of performing the hopping of these DTs may lead to different new configurations of the tracers localizations on the lattice. One of these configurations is false, viz. Case I, with respect to the ideal concept of simultaneous diffusion of the tracers over a time-step of simulation. Its description is provided in the text.</p

Prefronto-cortical dopamine D1 receptor sensitivity can critically influence working memory maintenance during delayed response tasks

<div><p>The dopamine (DA) hypothesis of cognitive deficits suggests that too low or too high extracellular DA concentration in the prefrontal cortex (PFC) can severely impair the working memory (WM) maintenance during delay period. Thus, there exists only an optimal range of DA where the sustained-firing activity, the neural correlate of WM maintenance, in the cortex possesses optimal firing frequency as well as robustness against noisy distractions. Empirical evidences demonstrate changes even in the D1 receptor (D1R)-sensitivity to extracellular DA, collectively manifested through D1R density and DA-binding affinity, in the PFC under neuropsychiatric conditions such as ageing and schizophrenia. However, the impact of alterations in the cortical D1R-sensitivity on WM maintenance has yet remained poorly addressed. Using a quantitative neural mass model of the prefronto-mesoprefrontal system, the present study reveals that higher D1R-sensitivity may not only effectuate shrunk optimal DA range but also shift of the range to lower concentrations. Moreover, higher sensitivity may significantly reduce the WM-robustness even within the optimal DA range and exacerbates the decline at abnormal DA levels. These findings project important clinical implications, such as dosage precision and variability of DA-correcting drugs across patients, and failure in acquiring healthy WM maintenance even under drug-controlled normal cortical DA levels.</p></div

Effects of variation in D1R-sensitivity on the WM-robustness in terms of potential barrier (PB).

<p>Increase in <i>D</i>1<i>R</i>_<i>sens</i> causes a consistent decrease in the PB of any individual level of sustained activity either sampled from the pre-peak (A) or from the post-peak (B) set of the modulation profile of cortical sustained <i>a</i>_<i>PN</i> activity. The percentage activities are with respect to the peak (100%) sustained activity. (C) The percent decrease in the average PB of pre-peak and post-peak sets across increase in <i>D</i>1<i>R</i>_<i>sens</i> shows higher vulnerability of the pre-peak set to change in D1R-sensitivity.</p