A Simple Kinetic Model with Explicit Predictions for Nuclear Transport

Abstract

Molecular exchange between the cell nucleus and cytoplasm is one of the most fundamental features of eukaryotic cell biology. The nuclear pores act as a conduit of this transport, both for cargo that crosses the pore autonomously as well as that whose translocation requires an intermediary receptor. The major class of such receptors is regulated by the small GTPase Ran, via whose interaction the nucleo-cytoplasmic transport system functions as a selective molecular pump. We propose a simple analytical model for transport that includes both translocation and receptor binding kinetics. The model is suitable for steady-state kinetics such as fluorescence recovery after photobleaching. Time constants appear as a combination of parameters whose effects on measured kinetics are not separable. Competitive cargo binding to receptors and large cytoplasmic volume buffer the transport properties of any particular cargo. Specific limits to the solutions provide a qualitative insight and interpretation of nuclear transport in the cellular context. Most significantly, we find that under realistic conditions receptor binding, rather than permeability of the nuclear pores, may be rate-limiting for nucleo-cytoplasmic exchange.

Introduction

Eukaryotic cells regulate molecular exchange between nucleus and cytoplasm through the nuclear pore complexes (NPC) by a dedicated biochemical machinery (1,2). The major nuclear transport factors are known, and extensive simulations have been performed (3–7). However, we lack an intuitive analytical model in which kinetic parameters of transport can be related directly to measured properties of nuclear accumulation. Early experimental studies of receptor-mediated nuclear import focused mainly on maximal rates using purified components (8). In live cells or cell extract assays, however, a steady state may be established. Fluorescence recovery after photobleaching (FRAP) then measures exchange kinetics (9–11).

We propose here a general kinetic model to account for nucleo-cytoplasmic exchange via importin receptors. The model incorporates both nuclear envelope permeability and receptor binding kinetics and affinities as measured in vitro (12–14) to predict cellular transport rates. In addition, it takes into account a passive (autonomous) flux relevant to smaller substrates. The equations are solvable analytically at steady state, with direct relevance to kinetics measured by FRAP. This approach clarifies differences among kinetic measurements made under distinct conditions such as FRAP, substrate titration, and redistribution by recovery from depletion conditions (15).

According to the accepted paradigm, nuclear import is dictated by the presence of nuclear-localization signal (NLS) peptides on the protein cargo. Delivery via the nuclear pore is mediated by receptors known as importins, whose interactions with NLS-bearing cargo are regulated by the small GTPase Ran. Ran auxiliary proteins produce an asymmetry with RanGTP in the nucleus and RanGDP in the cytoplasm. RanGTP releases cargo to the nucleoplasm while the receptor-RanGTP complex returns to the cytoplasm.

Model and Method

A given molecular cargo [C_i] may pass the NPC autonomously, following its concentration gradient, or in a receptor-mediated manner following instead the gradient of the receptor-cargo complex. The permeabilities relevant to the two modes (passive and active) are p_i and a_i, respectively. Note that these represent the entire nuclear envelope rather than single nuclear pores, so average permeabilities per pore should be scaled by the number of pores. (The permeabilities will of course depend on the particular cargo indexed i.) In addition, the nuclear transport system involves distinct kinetics for receptor binding to cargo as well as to RanGTP. We express these in terms of reversible chemical binding, described by simple on- and off-rates k and k′, respectively, indexed i for cargo or R for RanGTP. (The equilibrium dissociation affinity is K = k′/k.) The model is given in Fig. 1 and expressed in equation form as

$$\begin{array}{c}\frac{d{\left[C_i\right]}^N}{dt}=-\frac{p_i}{v_N}\left({\left[C_i\right]}^N-{\left[C_i\right]}^C\right)+\left(k_i^{\text{'}}{\left[T{C}_i\right]}^N-k_i{\left[T\right]}_{SS}^N{\left[C_i\right]}^N\right),\\ \frac{d{\left[C_i\right]}^C}{dt}=\frac{p_i}{v_C}\left({\left[C_i\right]}^N-{\left[C_i\right]}^C\right)+\left(k_i^{\text{'}}{\left[T{C}_i\right]}^C-k_i{\left[T\right]}_{SS}^C{\left[C_i\right]}^C\right),\\ \frac{d{\left[T{C}_i\right]}^N}{dt}=-\frac{a_i}{v_N}\left({\left[T{C}_i\right]}^N-{\left[T{C}_i\right]}^C\right)+\left(-k_i^{\text{'}}{\left[T{C}_i\right]}^N+k_i{\left[T\right]}_{SS}^N{\left[C_i\right]}^N\right),\\ \frac{d{\left[T{C}_i\right]}^C}{dt}=\frac{a_i}{v_C}\left({\left[T{C}_i\right]}^N-{\left[T{C}_i\right]}^C\right)+\left(-k_i^{\text{'}}{\left[T{C}_i\right]}^C+k_i{\left[T\right]}_{SS}^C{\left[C_i\right]}^C\right),\end{array}$$ (1)

where tags N and C refer to nucleus and cytoplasm, v to compartment volumes, and T to the transport receptor. Total numbers of cargo and receptors are conserved, i.e., protein expression and degradation are neglected on the timescale of transport. We also presume that protein concentrations are uniform within each of the two compartments.

Figure 1.

Schematic overview of the nuclear import kinetics with bidirectional, receptor-mediated transport and autonomous passive diffusion across the nuclear pore. (Shaded) Nonfluorescent NLS cargoes such as endogenous cellular proteins.

At steady state, two more constraints apply:

1. The high-affinity binding partner RanGTP is in equilibrium with the available receptor in the nucleus; and

2. The total concentrations of receptors T_SS in each compartment are constant.

$$k_R^{\text{'}}{\left[TR\right]}_{SS}^N-k_R{\left[T\right]}_{SS}^N{\left[R\right]}_{SS}^N=0,$$ (2)

$${\left[T{C}_i\right]}^N+\sum _{i\ne j}{\left[T{C}_j\right]}^N+{\left[TR\right]}_{SS}^N+{\left[T\right]}_{SS}^N=T_{SS}^N,$$ (3)

$${\left[T{C}_i\right]}^C+\sum _{i\ne j}{\left[T{C}_j\right]}^C+{\left[T\right]}_{SS}^C=T_{SS}^C.$$ (4)

Here, [R] refers to RanGTP; k_R and k_R′ refer to the on- and off-rates for receptor-RanGTP binding (equilibrium affinity K_R=k_R′/k_R); and [C_j] refers to other cargoes that compete for receptors. To model FRAP, we consider [C_i] to be the traced fluorescent substrate and the set of [C_j] to represent the nonfluorescent background. Upon photobleaching, the darkened fraction of [C_i] must be reindexed (e.g., i′) among the dark species such that the total quantity is conserved. The measured fluorescence includes both free and receptor-bound cargo, i.e., [C_i]+[TC_i].

The coupled set of Eq. 1 can be rewritten in matrix form and diagonalized (see the Supporting Material). Under steady-state conditions, all coefficients in the matrix are time-independent, in particular the free receptor concentrations ${\left[T\right]}_{SS}^N$ and ${\left[T\right]}_{SS}^C$. This is directly relevant to FRAP where the fluorescent or bleached state has no bearing on the receptor interaction. The steady-state level of cargoes and receptors in nucleus and cytoplasm are found by setting the time derivatives in Eq. 1 to zero. With four equations, we may expect kinetic solutions in a superposition of exponentials with up to four distinct rates:

$${\left[C_i\right]}^N=A_1{e}^{-t/{\tau }_1}+A_2{e}^{-t/{\tau }_2}+A_3{e}^{-t/{\tau }_3}+A_4{e}^{-t/{\tau }_4}+B.$$

In fact, conservation reduces the number of independent equations to three. The remaining characteristic equation will be third-order and, in principle, analytically solvable. Eigenvalues of the matrix yield inverse time constants τ_k⁻¹ for the exchange kinetics. If one time constant is significantly longer than the others, it will dominate the measurable quantities. In considering the connection with experiment we also distinguish between closed systems where the cytoplasmic volume is finite (live cells), and open systems (e.g., permeabilized cell assays) in which the cytosol is essentially infinite.

None of the above considerations depends on solution of the coupled equations, so the insights up to this point are completely general. Specific solutions range from the trivial (a = 0) to the unwieldy (p ≠ 0, a ≠ 0, and v_C finite). They are developed for progressively increasing complexity in the Supporting Material.

Results and Discussion

Passive-only transport

The approach is first developed with a standard exercise to treat passive transport. Cargo lacking NLS may cross the NPC autonomously at a rate dominated by size, but also affected by charge and hydrophobicity (16–18). At steady state, its concentration will reach a uniform distribution: [C_i]^N = [C_i]^C (see Eq. S7 in the Supporting Material). Kinetics are given by a single time constant (p/v_N + p/v_C)⁻¹, independent of initial concentrations, steady-state level, or experimental perturbation, e.g., substrate titration or photobleaching. Obviously kinetics are faster in the closed system than in the open one where v_C → ∞.

Receptor-mediated transport

In the case of purely receptor-mediated transport, steady state occurs with a distribution of receptor-cargo binding equivalent to that at equilibrium (see Eqs. S10 and S12 in the Supporting Material) in each compartment: nuclear and cytoplasmic receptor-cargo complex concentrations are equal, as postulated in our earlier experimental works (10,19). Receptor-cargo binding then follows a simple Langmuir form involving cargo of interest [C_i] as well as the affinity-weighted sum of all other cargoes [C_j]:

$${\left[T{C}_i\right]}_{SS}^N={\left[T{C}_i\right]}_{SS}^C=\frac{T_{SS}^C\frac{{\left[C_i\right]}^C}{K_i}}{1+\sum _j\frac{{\left[C_j\right]}^C}{K_j}},$$

$${\left[C_i\right]}_{SS}^N=\frac{T_{SS}^C{\left[C_i\right]}^C\left(1+\frac{{\left[R\right]}_{SS}^N}{K_R}\right)}{T_{SS}^N\left(1+\sum _j\frac{{\left[C_j\right]}^C}{K_j}\right)-T^C\sum _j\frac{{\left[C_j\right]}^C}{K_j}}\approx \frac{T_{SS}^C{\left[C_i\right]}^C\left(1+\frac{{\left[R\right]}_{SS}^N}{K_R}\right)}{\left(T_{SS}^N-T_{SS}^C\right)\left(1+\sum _j\frac{{\left[C_j\right]}^C}{K_j}\right)}.$$ (5)

Note that steady-state nuclear concentration depends neither on compartment volumes nor on pore permeabilities.

Predicted curves are plotted in Fig. 2, A and B, with numerical parameters specified in the caption (5,12–14). Free RanGTP, ${\left[R\right]}_{SS}^N/K_R,$ was adjusted to provide a steady-state nuclear to cytoplasmic concentration ratio (N/C) of 10, in agreement with previously published data for GFP-nucleoplasmin (10). This also yields a time constant consistent with the earlier observations. Accumulation is enhanced by higher free nuclear RanGTP (see Fig. S1 in the Supporting Material), whose level depends on cytoplasm to nucleus influx via the RanGDP-specific transport receptor NTF2 and the kinetics of the GTP exchange factor RanGEF (3,4). For this model, we require only its steady-state value. Nuclear accumulation further requires that the free receptor concentration in the nucleus exceeds that in the cytoplasm, ${\left[T\right]}_{SS}^N>{\left[T\right]}_{SS}^C$ and saturates sharply when the cargo concentration exceeds the total receptor concentration in the cytoplasm $T_{SS}^C$. The dotted black curve shows the case of only a single cargo [C_i] (see also Fig. S2). This is, however, unrealistic both biologically and conceptually because fluorescence recovery cannot be observed without some material having been darkened. Colored curves in Fig. 2 A shows the accumulation in the presence of unobserved competitors to the receptors, i.e., cargoes [C_j]. Because the effect of the hidden substrates is qualitative and dominant, we call them “dark matter”.

Figure 2.

Steady-state nuclear accumulation and exchange time constants. Colors indicate the concentration of nonfluorescent endogenous cargo (i.e., “dark matter”): ${\sum }_{j\ne i}{\left[C_j\right]}^C/K_j=0$μm³ (black dashed), 100 μm³ (red), 1000 (blue), and 10,000 μm³ (green) for panels A–C. (A) Total nuclear cargo concentration [C_i]^N+[TC_i]^N. (B) Nuclear to cytoplasmic ratio $N/C=({\left[C_i\right]}^N+{\left[T{C}_i\right]}^N)/({\left[C_i\right]}^C+{\left[T{C}_i\right]}^C)$. (C) Kinetic timescale in the closed system (v_C = 4000 μm³) and in the open system (dashed) as a function of cytoplasmic concentration [C_i]^C+[TC_i]^C. (D) The effect of cytoplasmic volume on kinetic rates, with cytoplasmic volume v_C = 800 μm³ (black), 4000 μm³ (red), 20,000 μm³ (blue), 100,000 μm³ (green), and ${\sum }_{j\ne i}{\left[C_j\right]}^C/K_j=1000$. Other parameters: k = 3 × 10⁷ M⁻¹ s⁻¹; k′ = 3 × 10⁻² s⁻¹; T^C = 4.5 μM; T^N = 5 μM; v_N = 800 μm³; a = 100 μm³/s; and ${\left[R\right]}_{SS}^N/K_R=5000$.

Kinetic timescales are shown in Fig. 2 C. Long characteristic times are represented as solid lines for a closed system and dashed lines for an open system, respectively; short times are represented as dot-dashed curves. Where one time constant is dominant, measurements will show single exponential behavior. Fluorescence recovery kinetics measure a complex function, including the on- and off-rates for binding to receptors and the permeability of receptor-cargo i complex (Eq. 6, open systems):

$${\tau }_i^{-1}=\frac{1}{2}\left\{\left(k_i{\left[T\right]}_{SS}^N+\frac{a_i}{v_N}+k_i^{\text{'}}\right)-{\left({\left(k{\left[T\right]}_{SS}^N+\frac{a_i}{v_N}+k^{\text{'}}\right)}^2-4\frac{a_i}{v_N}{k}_i{\left[T\right]}_{SS}^N\right)}^{\frac{1}{2}}\right\}.$$ (6)

The main effect of unobserved endogenous substrates is to smooth the sharp increase of both the steady-state accumulation and the time constant, as shown in Fig. 2. This buffering effect via ${\left[T\right]}_{SS}^N$ gives the transport system a more robust character compared to the specific cargo observed. As seen in Fig. 2 D, cytoplasmic volume also has a buffering effect on the time constant, but no influence on the steady-state ratio. The kinetics are again faster in the closed system than in the open one.

Nuclear accumulation requires an unbinding rate faster than that of rebinding,

$$k_i^{\text{'}}>k_i{\left[T\right]}_{SS}^N.$$

Commonly observed timescales of 10∼1000 s further imply that translocation is fast compared to receptor-cargo binding, i.e.,

$$\frac{a_i}{v_N}>>k_i^{\text{'}}>k_i{\left[T\right]}_{SS}^N.$$

Thus the rate monitored by FRAP reflects primarily receptor-cargo binding rather than NPC permeability (see Eq. S19 in the Supporting Material).

Mixed passive and receptor-mediated transport modes

Many NLS cargo can cross the NPC in both passive and receptor-mediated modes. As long as the NPCs remain unsaturated, coupling between such cargoes takes place via competition for free receptors. Therefore, we model the effect of dark matter simply as a reduced ${\left[T\right]}_{SS}^N$. Steady-state concentrations and N/C ratios are shown in Fig. 3. As p_i increases the ratio N/C reduces to 1, as expected. In the practically relevant case that

$$\frac{\left(a_i+p_i\right)}{v_N}>>k_i^{\text{'}}>k_i{\left[T\right]}^N,$$

the dominant time constant reflects the smaller of

$${\tau }_{i,1}^{-1}=\frac{a_i}{v_N}+k_i^{\text{'}}\text{or}{\tau }_{i,2}^{-1}={(\frac{p_i}{v_N}+k_i{\left[T\right]}_{SS}^N)}^{-1}$$

(see Eq. S23 in the Supporting Material). Thus the receptor binding kinetics also enters the crossover between active and passive transport behavior. The dominant timescale will therefore be given by ${\tau }_{i,2}^{-1}$ when nuclear accumulation is effective, i.e., when a_i > p_i and $k_i^{\text{'}}>k_i{\left[T\right]}_{SS}^N.$ This is consistent with purely receptor-mediated transport (p_i = 0) where the dominant time constant is

$${\left(k_i{\left[T\right]}_{SS}^N\right)}^{-1}.$$

However, when there is a significant passive permeability the timescale will be dominated by the term (p_i/v_N)⁻¹. Note that the timescale is faster the larger is p_i, yet concomitantly the steady-state ratio drops toward one.

Figure 3.

Steady-state nuclear accumulation and time constant for mixed passive and receptor-mediated transport. (A) Nuclear concentration versus passive permeability p. (B) Nuclear to cytoplasmic ratio. (C) Kinetic time in the open system. Colored curves indicate p = 0 μm³/s (cyan, active only), 0.1 μm³/s (black), 1 μm³/s (red), 10 μm³/s (blue), and 100 μm³/s (green, p = a). Other parameters are as in Fig. 1, except ${\sum }_{j\ne i}{\left[C_j\right]}^C/K_j=5000$.

Conclusions

As noted in the Introduction, the formal constraints of a mathematical model help to distinguish the various kinetics measured under different experimental conditions. The simple exponential solution shown here requires constant parameters in Eq. 1. For fluorescence recovery after photobleaching (FRAP) measurements, the steady-state conditions are maintained even as a given fluorescent population is converted from one cargo type (bright) to another (dark) because there is no difference in competition for receptor occupancy. Therefore the steady-state model is precisely relevant to FRAP measurement. The model’s kinetic predictions are approximately correct regarding addition (titration) of a new cargo, provided its concentration is low compared to that already present, i.e., when the dark matter already dominates the free receptor concentration:

$$\sum _{j\ne i}\frac{{\left[C_j\right]}^C}{K_j}\gg 1.$$

Then [T]^N and [T]^C will change negligibly as a result of the addition. At higher concentrations, coupling between distinct cargoes can be anticipated via an indirect effect on the free receptor concentrations (19). The model confirms and codifies a high sensitivity to cellular levels of transport factors, as reported recently in Kuusisto et al. (20). Other experimental frameworks such as metabolic recovery (15), may, however, measure very different kinetics.

Despite the familiarity of its assumptions, the model makes several provocative predictions. One is that in the case of purely receptor-mediated transport, the partitioning of receptors to cargoes and (in the nucleus) RanGTP is equivalent to the equilibrium Langmuir distribution, even though the system, as a whole, is out of equilibrium. This justifies our earlier Ansatz based on experimental observations (19), and implies a thermodynamic regulation of the receptor system for nuclear protein accumulation. Considering nuclear envelope permeability and cargo exchange with receptors, the latter appear to be rate-limiting for active transport that results in net nuclear accumulation. Apparently the number of nuclear pores is large enough that crossing them is not a cellular bottleneck. An intriguing conclusion is that the requirements on the nuclear pores per se are extremely weak to achieve a net nuclear accumulation: only the permeability to cargo-receptor complexes should be greater than that for cargo alone. This could explain the viability of cells with gross deletions or even inversions of key nucleoporin repeats (21,22).