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# Precise gene expression patterns in complex neuronal morphologies from a simple local mechanism

Abstract

The spatial distribution of macromolecules and organelles in neurons is highly nonuniform. How cells achieve and maintain these expression patterns is unknown, but is believed to involve microtubular-based transport. Using mathematical analysis and numerical simulation, we show how reliable transport systems can be implemented in complex neuron morphologies. We derive a simple rule that relates local trafficking rates to the global steady-state distribution of cargo, and illustrate how this rule can be encoded by a second-messenger molecule, such as $$\text{Ca}^{2+}$$. Similar, but more flexible, transport strategies were developed for a model that included nonuniform activation or microtubular detachment of cargo. These models make several experimental predictions about the time scale of transport and cell-to-cell variability in spatial expression patterns. We illustrate these predictions in CA1 pyramidal cells, which rely on transport of activity-inducible mRNAs and proteins for long-lasting synaptic plasticity, and display linear expression gradients in HCN and potassium channels.

# Introduction

Neurons display diverse and elaborate morphologies that are inextricably linked to their physiological functions. Many studies have modeled how electrical signals propagate along realistic cellular morphologies (Almog 2013, Jarsky 2005, Hay 2011), but few have modeled the active transport of biomolecules and organelles within complex cell morphologies.

Ion channel expression patterns vary widely across cell types (Nusser 2009), altering how neurons process synaptic inputs and shaping their electrophysiological phenotype (Narayanan 2012). For example, the expression of HCN1 channels increases 60-fold from the proximal to distal portions of the apical dendrites in CA1 pyramidal cells (Lörincz 2002), which is accompanied by a linear increase in the hyperpolarization-activated cation current ($$I_{h}$$,(Magee 1998)). This pattern of $$I_{h}$$ expression appears to reduce the location-dependence of synaptic potentials at the soma in CA1 and neocortical neurons (Cash 1999, Magee 1999, Williams 2000). Microtubular transport contributes to the establishment, maintenance, and plasticity ion channel expression patterns, but it is unclear how transport mechanisms are organized to achieve this.

Microtubular-based transport is also critical for orchestrating somatic and peripheral reactions related to synaptic plasticity. Following synaptic activation, certain dendritic proteins are retrogradely transported to the nucleus to act as transcription factors, and blocking this transport impairs plasticity (Ch’ng 2012, Ch’ng 2011). Many long-lasting forms of synaptic plasticity also rely on the anterograde transport of mRNAs from the nucleus to the dendrites to support local protein synthesis (Kandel 2001, Puthanveettil 2008). Certain mRNAs are selectively transported to regions of heightened synaptic activity (Steward 1998, Steward 2001, Moga 2004) or to developing synaptic contacts (Lyles 2006), while others are uniformly distributed but undergo localized translation, resulting in nonuniform protein expression (Kim 2015).

A fundamental question is how low-level transport mechanisms, such as the biophysical parameters of molecular motors, enable and influence high-level physiological functions, such as activity-dependent plasticity and homeostasis. It is possible that the constraints of molecular transport conflict with theoretically-sound plasticity rules, but also enable previously unconsidered forms of plasticity. To address these possibilities, we developed transport models that produce desired protein expression patterns in complex dendritic trees. We outline the general principles illustrated by these models, their respective experimental predictions, and how their features might influence plasticity rules.

# Results

## Model description

Transport along microtubules is mediated by kinesin and dynein motors, which respectively mediate anterograde and retrograde transport (Hirokawa 2010, Gagnon 2011). Cargo is often simultaneously bound to both forms of molecular motors, resulting in stochastic back-and-forth movements with a net direction determined by the balance of opposing movements (Hancock 2014, Buxbaum 2014) (Fig. 1A). The biophysical mechanisms of single-cargo movements are active area of research (CITE), and the details of may vary across experimental preparations (CITE).

To remain agnostic to biophysical details, we model microtubule-based transport as a biased random walk along a one-dimensional cable (Bressloff 2006); for each time step, the cargo steps left, right, or remains the same. In the simplest model, the probabilities associated with each movement are fixed and independent across each time step (Fig. 1B, top panel). This model may accurately capture the movements of molecular motors that make short movements before changing directions. However, other motors exhibit sustained movements (CITE). This can be captured by incorporating hysteresis or history-dependence into the model; namely, the cargo is more likely to continue in the same direction as the previous time step (Fig. 1B, bottom panel).

While the position of individual cargoes can be highly stochastic, the net movement of a population along a neurite is more predictable (Fig. 1C). Figure 1D shows the distribution of 1000 molecules over time with (top panel) and without (bottom panel) history-dependence. The average behavior of both of these models can be fit by a mass-action model (Fig. 1E-F).

The rate of molecular transport is locally modulated by calcium, ADP, and other biochemical signals (Wang 2009, Mironov 2007), and globally modulated by transcriptional control of motor protein expression (Puthanveettil 2008).