Mathematical modeling reveals modulation of both nuclear influx and efflux of Foxo1 by the IGF-I/PI3K/Akt pathway in skeletal muscle fibers

Abstract

Foxo family transcription factors contribute to muscle atrophy by promoting transcription of the ubiquitin ligases muscle-specific RING finger protein and muscle atrophy F-box/atrogin-1. Foxo transcriptional effectiveness is largely determined by its nuclear-cytoplasmic distribution, with unphosphorylated Foxo1 transported into nuclei and phosphorylated Foxo1 transported out of nuclei. We expressed the fluorescent fusion protein Foxo1-green fluorescent protein (GFP) in cultured adult mouse flexor digitorum brevis muscle fibers and tracked the time course of the nuclear-to-cytoplasmic Foxo1-GFP mean pixel fluorescence ratio (N/C) in living fibers by confocal imaging. We previously showed that IGF-I, which activates the Foxo kinase Akt/PKB, caused a rapid marked decline in N/C, whereas inhibition of Akt caused a modest increase in N/C. Here we develop a two-state mathematical model for Foxo1 nuclear-cytoplasmic redistribution, where Foxo phosphorylation/dephosphorylation is assumed to be fast compared with nuclear influx and efflux. Cytoplasmic Foxo1-GFP mean pixel fluorescence is constant due to the much larger cytoplasmic than nuclear volume. Analysis of N/C time courses reveals that IGF-I strongly increased unidirectional nuclear efflux, indicating similarly increased fractional phosphorylation of Foxo1 within nuclei, and decreased unidirectional nuclear influx, indicating increased cytoplasmic fractional phosphorylation of Foxo1. Inhibition of Akt increased Foxo1 unidirectional nuclear influx, consistent with block of Foxo1 cytoplasmic phosphorylation, but did not decrease Foxo1 unidirectional nuclear efflux, indicating that Akt may not be involved in Foxo1 nuclear efflux under control conditions. New media change experiments show that cultured fibers release IGF-I-like factors, which maintain low nuclear Foxo1 in the medium. This study demonstrates the power of quantitative modeling of observed nuclear fluxes.

transcription of a given gene is controlled by numerous regulatory proteins, including transcription factors, which are primary activators of transcription, as well as numerous other activators, coactivators, repressors, and corepressors (12, 15a). These factors combine to form macromolecular transcriptional regulatory complexes assembled on the promoter regions of the gene in question (16). To participate in such regulatory complexes, the regulatory molecules must be resident in the nucleus. Thus an important aspect of transcriptional regulation is control of the nuclear-cytoplasmic distribution of these regulatory molecules, and the mechanisms controlling nuclear-cytoplasmic movements of transcriptional regulatory molecules can thus be important determinants of transcription. Posttranslational modification of transcriptional regulatory molecules can modulate their nuclear-cytoplasmic distribution by causing selective exposure of nuclear import signals or nuclear export signals, depending on the posttranslational modification status of the molecule (10, 15). This, in turn, results in nuclear entry or exclusion of the regulatory molecule and a corresponding effect on the transcriptional process. However, the roles of transcriptional modifications in the cytoplasm and relative to those in the nucleus have not been thoroughly studied. Here we establish new modeling and analysis techniques for identifying intranuclear or cytoplasmic phosphorylation/dephosphorylation events that modulate the nuclear-cytoplasmic distribution of the transcription factor forkhead box class O (Foxo) 1 in skeletal muscle fibers.

Skeletal muscle atrophy and wasting occur during disuse and aging or as an accompaniment of cancer, diabetes, heart disease, septicemia, or other severe systemic disease states and cause debilitating limitations on mobility and breathing. The Foxo transcription factors, including Foxo1 studied here, serve as key activators of muscle protein breakdown during atrophy by promoting transcription of the atrophy-related ubiquitin ligases muscle-specific RING finger protein (MuRF1) and muscle atrophy F-box (MAFbx)/atrogin-1, leading to increased protein breakdown via the proteosomal pathway, and by promoting the lysosomal autophagy pathway for protein breakdown (22). Foxo proteins shuttle into and out of muscle fiber nuclei, primarily depending on phosphorylation status (Fig. 1). In response to IGF-I, the pathway including IGF receptor → phosphatidylinositol 3-kinase (PI3K) → Akt (PKB) leads to phosphorylation of Foxo1 or Foxo3A (hereafter referred to collectively as “Foxo”) by Akt on three conserved residues (30), which causes the nuclear localization signal in Foxo to be masked and, thus, prevents nuclear entry (7) (Fig. 1). Dephosphorylation exposes the nuclear localization signal, allowing nuclear entry of dephosphorylated Foxo via the nuclear import system (20). Within the nucleus, Foxo can bind to DNA sites via its highly conserved DNA-binding domain, causing transcriptional activation. Phosphorylation of nuclear Foxo by Akt and the resulting conformational change cause unbinding of Foxo from DNA (6, 27) and are required for binding of Foxo to the chaperone protein 14-3-3 followed by chromosome region maintenance 1 (CRM1) and Ras-related protein (Ran) binding (7) and exposure of the nuclear export sequence, resulting in transport of the Foxo-(14-3-3)-Ran-CRM1 complex out of the nucleus via the nuclear export system. In several models of skeletal muscle atrophy, nuclear Foxo promotes expression of the ubiquitin ligases MuRF1 and MAFbx/atrogin-1, resulting in an increase in muscle protein breakdown and, thus, contributing to muscle wasting (5, 23, 29). Although steady changes in Foxo nuclear-cytoplasmic distribution have been studied in muscle, there has been little analysis of the kinetics of Foxo nuclear-cytoplasmic redistribution, and little is known regarding the relative roles of nuclear vs. cytoplasmic signaling events in regulating these changes, which we address here.

Fig. 1.

Cartoon representation of the pathway for regulation of forkhead box class O (Foxo) nuclear-cytoplasmic distribution based on phosphorylation status. IGF-I activation of the IGF receptor (IGFR) at the plasma membrane (top) activates phosphatidylinositol 3-kinase (PI3K), causing plasma membrane lipid phosphorylation and resulting in phosphoinositide-dependent kinase 1 (PDK1) activation and Akt phosphorylation. Active (phosphorylated) Akt phosphorylates Foxo in cytoplasm, and protein phosphatase 2A (PP2A) dephosphorylates Foxo in the cytoplasm and possibly in the nucleus as well (nuclear Akt and PP2A action are not indicated). Only dephosphorylated Foxo can enter the nucleus via the nuclear import system, and only phosphorylated Foxo can exit the nucleus via the nuclear export system. Intranuclear Foxo promotes transcription of the “atrogenes” muscle-specific RING finger protein (MuRF1) and muscle atrophy F-box (MAFbx)/atrogin-1.

In previous studies, we made extensive use of adult skeletal muscle fibers maintained in culture and transduced to express fluorescent fusion protein constructs of the transcriptional regulators nuclear factor of activated T cells (NFAT) and histone deacetylase (HDAC) to study the nuclear-cytoplasmic distribution of these transcriptional regulators under a variety of experimental conditions (17, 18). We recently applied the same general experimental approach to study the time course of nuclear-cytoplasmic distribution of Foxo1-green fluorescent protein (GFP) (24). Here we develop a simplified mathematical model that accounts for several of our previous observations concerning Foxo1-GFP nuclear movements. We previously observed a gradual increase in nuclear Foxo1-GFP with time after change from culture to control experimental conditions (24), which is now modeled as the time-dependent approach to a new steady state. This dynamic now provides values for the apparent rate constants for unidirectional nuclear influx and efflux of Foxo1-GFP in the model. By changing the values of the apparent rate constants for nuclear unidirectional influx and efflux, we also use the model to account for our previous observations of changes in the time course of Foxo1-GFP nuclear concentration after experimental modulation of the activity of the Foxo1 kinase Akt in cultured adult muscle fibers (24). Our modeling indicates that intranuclear phosphorylation of Foxo1 is potentiated more than sevenfold by application of the Akt-activating growth factor IGF-I.

METHODS

Monitoring the time course of Foxo1-GFP net nuclear movements in living cultured adult muscle fibers by live-cell confocal fluorescence imaging.

Recently, we described in detail our procedures for monitoring the time course of nuclear and cytoplasmic concentrations of Foxo proteins in living adult muscle fibers isolated from flexor digitorum brevis muscles and maintained in culture (24). All animal procedures were carried out according to a protocol approved by the University of Maryland School of Medicine Animal Care and Use Committee. Briefly, muscle fibers isolated from young adult mice were maintained in culture and adenovirally transduced to express Foxo1-GFP. After 2–3 days in culture, culture dishes were transferred from the tissue culture incubator to the microscope stage [room temperature (∼23°C)], and the bathing medium was changed from the serum-free medium in which the fibers were maintained in culture to fresh serum-free experimental solution, as we described previously (24), unless otherwise specified. A computer-controlled stage and focus were used to select a number of nuclei. The chosen nuclei were imaged at 10- to 20-min intervals throughout the recording period, with or without addition of various pharmacological agents to the experimental solution. With this protocol, we acquire a time series of high-spatial-resolution fluorescence confocal images of fibers previously adenovirally transduced to express exogenous Foxo1-GFP fusion protein. By image analysis, we then calculate the mean nuclear pixel fluorescence (N), which is proportional to the mean nuclear concentration of Foxo1-GFP, in selected nuclei in several fibers as a function of time. We normalize N to the mean cytoplasmic pixel fluorescence (C) in the same fiber image to control for differences in Foxo1-GFP expression from fiber to fiber. All graphs of the Foxo-GFP fluorescence time course presented here are nuclear-to-cytoplasmic pixel fluorescence ratios (N/C). Note that because the nuclear volume is only a small fraction (∼5% or less) (24) of the cytoplasmic volume, in the records considered here, C remained essentially constant over the course of an experiment, even though N may have changed considerably. We previously showed that adenovirally expressed Foxo1-GFP is a good model for endogenous Foxo1 (24).

Our previously reported studies of Foxo1-GFP, as well as our new photobleaching studies described below, were carried out on fibers cultured in serum-free MEM and studied experimentally in L-15 medium (13, 24). In one group of new experiments reported here (see Fig. 6), we used MEM with HEPES (catalog no. 12360-038), in which fibers could be transferred directly from the culture incubator to the microscope stage and then studied without need of solution change. For comparability with the MEM used in the previously reported experiments, we added 2 mM l-glutamine to the MEM with HEPES.

Fig. 6.

Effect of solution change and restoration of conditioned culture medium on the time course of Foxo1-GFP nuclear-cytoplasmic distribution. A: images of a fiber in the medium in which it was cultured for 2 days (relatively dim nuclei), 1 h after medium was changed to fresh medium (nuclei relatively bright), and 1 h after return of the fiber to the original (“conditioned”) medium (nuclei very dim). B: time courses of N/C in the presence of various bathing media. Cultured fibers were removed from the culture incubator 10 min prior to the start of the records. At time 0 (downward arrow), fibers were changed to fresh culture medium and left in fresh medium throughout the experiment (△), left in the original medium in which they were cultured (i.e., no change in medium; ●), or transferred to fresh medium at 0 min (□) and then returned to their original (conditioned) culture medium at 58 min (■). In all cases, fibers in conditioned medium exhibit relatively low N/C, whereas fibers in fresh medium exhibit increasing N/C with time, consistent with some component of the conditioned medium, possibly IGF-I or related growth factor secreted by the cultured fibers, causing Foxo1-GFP to remain out of the nuclei. C: apparent rate constants for nuclear influx and efflux obtained from the individual time courses making up the average time courses in B. **P < 0.01. D: addition of IGF-I-neutralizing antibody to conditioned medium eliminates IGF-I-like effects of adding back conditioned medium. Note that, unlike experiments in Figs. 3–5, these experiments used culture medium that was buffered by both bicarbonate and HEPES, so the fibers could be left in their culture medium without change in pH of the bathing solution. Also note that data in B–D and in Fig. 7 are from new experiments, not included in Ref. 24.

Protocol for cytoplasmic photobleaching of Foxo1-GFP.

To test for linearity of the nuclear uptake system for Foxo1, we monitored net nuclear uptake of Foxo1-GFP before and after partial photobleaching of cytoplasmic Foxo1-GFP. For these studies, the fibers were oriented parallel to the long axis of the confocal image, and the peripheral nucleus under study was positioned in the center of the long axis. Net nuclear uptake of Foxo1-GFP was monitored before and after bleaching of a large cytoplasmic rectangle that extended almost from end to end of the fiber image and included most of the fiber width in the image plane, but not the peripheral nucleus, under study. Photobleaching was carried out over a 1-min period using full laser power. For reference, all routine imaging (not photobleaching) exposures utilized ≤10% of full laser power. The Foxo1-GFP nuclear uptake rates before and after photobleaching were measured in the presence of leptomycin B to block nuclear efflux, so net influx was equal to unidirectional influx.

Data acquisition.

The time course of Foxo1-GFP was tracked simultaneously in multiple fibers by using a computer-controlled stage with positions for two culture dishes, so that fibers can be monitored under two different experimental conditions during the same experiment (see Fig. 3, A and C, Fig. 4, A and C, and Fig. 5, A and C). The previously unpublished new data in Fig. 6A were acquired using a chamber with the well partitioned into four compartments. Most of the Foxo1-GFP N/C time course data analyzed here were previously published by Schachter et al. (see Fig. 4, A–D, and Fig. 7 in Ref. 24) for control conditions and for pharmacological manipulation of Akt activation but are presented again for further analysis using the mathematical model developed here. These data were previously described qualitatively (24) but were not analyzed in terms of apparent rate constants for unidirectional nuclear influx and efflux to establish relative effects of nuclear compared with cytoplasmic phosphorylation/dephosphorylation; such an analysis is carried out in the present study. Other data presented here are from new experiments that were not previously reported (see Figs. 6 and 7).

Fig. 3.

Time course of nuclear Foxo1-green fluorescent protein (GFP) after transfer from culture medium to “control” experimental solution. A: average time course of nuclear-to-cytoplasmic Foxo1-GFP mean pixel fluorescence ratio (N/C) recorded under control conditions. Here and in all subsequent figures except Fig. 7, 30 min prior to the start of recording, isolated flexor digitorum brevis fibers were removed from the tissue culture incubator and transferred to the control experimental solution. Mean pixel fluorescence was then monitored in cytoplasm (C) and nucleus (N), and the mean value of N/C was plotted against time. A single exponential + constant (line) was fit to the N/C time course. Resulting fit parameters provided values for the apparent unidirectional rate constants kI′ = 0.110 min⁻¹ and kE′ = 0.013 min⁻¹ (N/C_ss = 8.45, where ss is steady state). Data from all nuclei under control conditions in Fig. 4 of Ref. 24 are now included in A, except for 2 nuclei, where the individual nucleus N/C time course could not be fit by a single-exponential time course. B: mean values for the respective apparent unidirectional rate constants kI′ = 0.119 ± 0.014 min⁻¹ and kE′ = 0.014 ± 0.003 min⁻¹ for Foxo1-GFP nuclear influx and efflux, obtained by fitting individual records (not shown) that were averaged to give data points in A. Mean value of N/C_ss from the same fits was 8.96 ± 0.69. Data are from 14 nuclei in 13 fibers. Error bars, SE.

Fig. 4.

Time course and apparent rate constants for Foxo1 nuclear-cytoplasmic unidirectional fluxes determined under control conditions and in the presence of pharmacological inhibitors of the IGF-I/PI3K/Akt pathway. A and C: time course of Foxo1-GFP N/C under control conditions (□) and in the presence of pharmacological inhibitors (△) of Akt (Akt inhibitor VIII; A) or PI3K (LY-294002, 25 μM; C) (27). A single exponential + constant was fit to the control data (□) over the entire recording interval and to the time course in the presence of pharmacological inhibitors (△) starting 20 min after they were added, giving good fits to control (black line) and drug addition (gray line) records. Parameter values for single exponential + constant fits (lines) to the mean N/C time courses in the control condition and in the presence of pharmacological inhibition gave kI′ = 0.031 and 0.112 min⁻¹ and kE′ = 0.0058 and 0.0101 min⁻¹ (N/C_ss = 5.44 and 11.1), respectively, in A, and kI′ = 0.089 and 0.241 min⁻¹ and kE′ = 0.011 and 0.021 min⁻¹ (N/C_ss = 8.28 and 11.3), respectively, in C. B and D: apparent rate constants for nuclear influx (kI′) and efflux (kE′) under control conditions and during application of inhibitors of Akt (Akt1,2-I; B) or the upstream kinase PI3K in the pathway to Akt activation (Ref. 27) (D). In B and D, kI′ and kE′ values are means of values obtained from fits to the individual records averaged to give the records in A and C, except control values for Akt1,2-I were obtained from a fit to the mean time course (and, thus, have no error bars and no corresponding P values), since the individual records were noisy. *P < 0.05, **P < 0.01. Each pair of values (control and drug) for the same apparent rate constant in B and D is displayed on a different vertical scale, but all control values were scaled so as to have the same height. Values of apparent rate constants kI′ and kE′ are thus displayed normalized to their value in the paired control using the normalized vertical scale shown at right in B and D. Data in A are from Fig. 4D in Ref. 24; data in C are from Fig. 4C in Ref. 24.

Fig. 5.

Time course and apparent rate constants for Foxo1-GFP N/C under control conditions and during application of IGF-I, the activator of the IGF-I/PI3K/Akt pathway. General layout is similar to Fig. 4. A and C: time course of N/C in control conditions and in the presence of IGF-I (100 ng/ml). Standard sampling interval of 10 or 20 min used in A and other figures was decreased to 2 min in C to capture the rapid time course of decay of N/C after addition of IGF-I. No control record was obtained at the higher sampling rate (C), since there was very little change of N/C at this time scale under control conditions. Parameter values for the single exponential + constant fits (lines) to the mean N/C time courses in control conditions and in the presence of IGF-I gave kI′ = 0.126 and 0.064 min⁻¹ and kE′ = 0.0132 and 0.137 min⁻¹ (N/C_ss = 9.57 and 0.47), respectively, in A and kI′ = 0.0241 min⁻¹ and kE′ = 0.109 min⁻¹ (N/C_ss = 0.22), respectively, in the presence of IGF-I in C (note that there was no matching control record for the IGF-I record in C). B and D: apparent rate constants for Foxo1 nuclear-cytoplasmic unidirectional fluxes determined under control conditions or in the presence of IGF-I. Since the rate of decay of N/C was fast compared with the standard 20-min sampling interval used in the preceding figures, there was insufficient information to carry out single-exponential fits to the time course data from individual nuclei, so kI′ and kE′ bars in B lack error bars and P values. Since there was no control record matched to the IGF-I record in C, control apparent rate constants in D were those from all control fibers (Fig. 3B). **P < 0.01. Data in A are from Fig. 4A in Ref. 24; data in C are from Fig. 7 in Ref. 24.

Fig. 7.

Linearity of Foxo1-GFP unidirectional nuclear influx. A: a fiber after exposure to leptomycin B for 1 h and imaged just prior to cytoplasmic photobleaching (left) and the same fiber imaged shortly after photobleaching of the large rectangular area, which does not include the peripheral nucleus (right). Areas of interest for tracking the nuclear (N) and cytoplasmic (C) fluorescence before and after bleaching are indicated. B: relative suppression of the rate of Foxo1-GFP nuclear influx plotted as a function of cytoplasmic fluorescence after relative to before photobleaching. Dashed line, exact proportionality; solid line, fit to the data, including 9 values at (1,1) to indicate prebleach relative values. Regression line (Y = Y₀ + aX; solid line) fit to the data, including an equal number of data points at (1.0, 1.0) to denote the values of relative influx rate and “relative C” before the photobleach gave Y₀ = 0.084 and a = 0.084 (r = 0.98), showing that, under the conditions of our studies, nuclear influx is close to proportional to the cytoplasmic level of Foxo1-GFP, thus supporting the assumption of linearity for the nuclear import of Foxo1-GFP.

Values are means ± SE. Student's t-test was used to determine the significance of difference. P < 0.05 was considered significant.

THEORY: KINETIC MODELS FOR FOXO NUCLEAR FLUXES

A four-state model for Foxo nuclear fluxes.

Our starting model for Foxo nuclear-cytoplasmic movements (Fig. 2A) considers four states of Foxo in muscle fibers: phosphorylated Foxo in the nucleus (P_n), unphosphorylated Foxo in the nucleus (U_n), phosphorylated Foxo in the cytoplasm (P_c), and unphosphorylated Foxo in the cytoplasm (U_c). We assume that Foxo molecules can be enzymatically unphosphorylated or phosphorylated, with respective rate constants k_Uc or k_Pc in the cytoplasm and k_Un or k_Pn in the nucleus. Nuclear influx and efflux of Foxo occur via two parallel and independent unidirectional transport processes that transport Foxo in opposite directions through the nuclear pores (Fig. 2) (1, 9, 20). Only unphosphorylated Foxo is carried into the nuclei by the nuclear import system, and only phosphorylated Foxo is carried out of the nuclei by the nuclear export system (3, 23, 30; see above). Nuclear influx is thus proportional to the cytoplasmic concentration of unphosphorylated Foxo (U_c), whereas nuclear efflux is proportional to the nuclear concentration of phosphorylated Foxo (P_n). Nuclear transport is driven by the RanGTP concentration gradient across the nuclear envelope, as well as by concentrations of the cotransported molecules: importins for import and 14-3-3 and CRM1 for export (20) (not illustrated in Fig. 1 or 2; assumed to contribute to determining the respective constitutive values of the corresponding nuclear influx and efflux rate constants k_I and k_E). The vector sum of the unidirectional nuclear influx and efflux via the two separate transport systems (Fig. 2A; influx − efflux) equals the net flux of Foxo into or out of the muscle fiber nuclei, which determines the time course of change of nuclear Foxo. Using Foxo-GFP fusion proteins, together with live muscle fiber confocal imaging, we can experimentally image and quantify the total nuclear and cytoplasmic concentrations (N and C) of Foxo-GFP as a function of time in living fibers (see below), but we cannot experimentally distinguish the separate contribution of P_n and U_n to N or the contribution of P_c and U_c to C.

Fig. 2.

Kinetic reaction schemes for Foxo nuclear-cytoplasmic movements. A: 4-state scheme, including Foxo phosphorylation/dephosphorylation in cytoplasm (C) and nucleus (N), with nuclear influx of only dephosphorylated cytoplasmic Foxo (U_c; unidirectional influx rate = k_IU_c) and nuclear efflux of only phosphorylated nuclear Foxo (P_n; unidirectional nuclear efflux rate = k_EP_n). P_c, phosphorylated cytoplasmic Foxo; U_n, nuclear unphosphorylated Foxo. B: reduced 2-state scheme, where c is total cytoplasmic Foxo (i.e., P_c + U_c; see A) and n is total nuclear Foxo (i.e., P_n + U_n; see A). Apparent rate constants for nuclear influx and efflux are given by kI′ = k_I[U_c/(U_c + P_c)] and kE′ = k_E[P_n/(P_n + U_n)]. See text for further details.

A two-state “reduced” model for Foxo nuclear-cytoplasmic fluxes.

If the rates of phosphorylation and dephosphorylation in the cytoplasm and nuclei are rapid compared with the nuclear influx and efflux rates, then phosphorylation/dephosphorylation will be essentially at equilibrium in the cytoplasm and nucleus. In this case, the four-state model in Fig. 2A reduces to an effective two-state model (Fig. 2B), where C is the total concentration of cytoplasmic unphosphorylated + phosphorylated Foxo (C = U_c + P_c) and N = U_n + P_n is the total nuclear concentration of Foxo. Note that when Foxo1-GFP is used, C and N are proportional to the cytoplasmic and nuclear mean pixel fluorescence, respectively, the parameters that we measure experimentally as a function of time using confocal imaging of live muscle fiber.

Mathematical description of nuclear fluxes of Foxo1-GFP in muscle fibers.

The measured net rate of change dN/dt of nuclear mean pixel fluorescence (N) due to nuclear fluxes of Foxo1-GFP at any instant in time is given by the difference between the rate of nuclear influx (I) and the rate of nuclear efflux (E)

$$\text{dN}/\text{d}t=\text{I}\hspace{0.17em}-\hspace{0.17em}\text{E}$$ (1)

Unphosphorylated cytoplasmic Foxo1 (U_c), but not phosphorylated cytoplasmic Foxo1 (P_c), is carried into the nucleus by the nuclear import system, whereas phosphorylated nuclear Foxo1 (P_n), but not unphosphorylated nuclear Foxo1 (U_n), is transported out of the nucleus by the nuclear export carrier. If it is assumed that influx is proportional to U_c and efflux is proportional to P_n, I = k_IU_c and E = k_EP_n, where k_I and k_E are the rate constants for unidirectional Foxo1 nuclear influx and efflux, respectively. Thus

$$\text{dN}/\text{d}t=k_{\text{I}}{\hspace{0.17em}\text{U}}_{\text{c}}-k_{\text{E}}{\hspace{0.17em}\text{P}}_{\text{n}}$$ (2)

Although, in principle, we could explicitly include the individual time courses of U_c and P_n in a mathematical model, in our experiments we can only directly measure the total cytoplasmic mean pixel fluorescence (C) due to U_c + P_c (which is proportional to the total cytoplasmic concentration of Foxo1-GFP) and the total nuclear mean pixel fluorescence (N) due to P_n + U_n (which is proportional to the total nuclear concentration of Foxo1-GFP), where P_c is the cytoplasmic phosphorylated Foxo1-GFP and U_n is the nuclear unphosphorylated Foxo1-GFP. We thus introduce the apparent rate constants kI′ and kE′, such that kI′ = k_IfU_c and kE′ = k_EfP_n, where the fraction (f) of cytoplasmic Foxo that is unphosphorylated is fU_c = U_c/(U_c + P_c) and the fraction of nuclear Foxo1 that is phosphorylated in the nucleus is fP_n = P_n/(P_n + U_n), so

$$\text{dN}/\text{d}t=k_I^{\prime }\text{C}\hspace{0.17em}-\hspace{0.17em}{k}_E^{\prime }\text{N}$$ (3)

C is the total measured cytoplasmic fluorescence (proportional to U_c + P_c), and N is the total nuclear fluorescence (proportional to P_n + U_n). Note that dN/dt, C, and N are directly measurable from the fluorescence image at any time during the course of an experiment.

If the cytoplasmic volume is large compared with the nuclear volume, then C will be essentially constant, and

$$\text{d(N}/\text{C})/\text{d}t=k_I^{\prime }-k_{\text{E}}^{\prime }\text{N/C}$$ (4)

If the phosphorylation and dephosphorylation rate constants, which depend on the Foxo kinase [presumably Akt (14)] and phosphatase [presumably protein phosphatase 2A (PP2A) (2)] activities, respectively, remain constant during a particular time interval and if phosphorylation/dephosphorylation in the nucleus and cytoplasm is fast compared with nuclear influx and efflux, then kI′ and kE′ will be constants during the given time interval and the differential equation (Eq. 4) can be solved to give

$$\text{N/C}\text{(}t\text{)}\hspace{0.17em}=\hspace{0.17em}({\text{N/C}}_{\text{start}}-k_I^{\prime }/k_E^{\prime }\text{) exp(}-k_{\text{E}}^{\prime }t\text{)}\hspace{0.17em}+k_I^{\prime }/k_{\text{E}}^{\prime }$$ (5)

where the steady-state (ss) condition [when d(N/C)dt = 0] is N/C_ss = kI′/kE′ and N/C_start is the value of N/C at the start of the analyzed interval.

Note that Eq. 5 applies only when the cytoplasmic volume is very large compared with the total nuclear volume, so C is constant. The appendix presents a general solution that applies to any ratio of nuclear to cytoplasmic volume, which can be applied to nonsyncitial cells, where cytoplasmic volume does not far exceed the nuclear volume, as is the case for the skeletal muscle fibers studied here.

The apparent rate constants kI′ and kE′ for nuclear influx and efflux provide information regarding cytoplasmic and nuclear phosphorylation/dephosphorylation of Foxo, respectively. The apparent rate constants kI′ for nuclear influx and kE′ for efflux in Fig. 2B and above are given by kI′ = fU_ck_I and kE′ = fP_nk_E. Since in the phosphorylation equilibrium kU_c/kP_c = U_c/P_c and kU_n/kP_n = U_n/P_n we can also write fU_c = kU_c/(kU_c + kP_c) and fP_n = kP_n/(kP_n + kU_n), the values of kI′ and kE′ provide information regarding the relative extents of phosphorylation/dephosphorylation of Foxo1 in cytoplasm and nuclei and, in turn, the phosphorylation/dephosphorylation rate constants and the corresponding enzymatic activities that determine these fractions.

RESULTS

Determining the apparent rate constants for Foxo1-GFP nuclear influx and efflux during the approach to steady state after change from culture medium to experimental recording conditions.

We previously showed that “control fibers” that were not exposed to any agent other than transfer from their culture medium to the “control experimental solution” exhibit a relatively slow, gradual rise in nuclear Foxo1-GFP (Fig. 3, replot of control data previously published in Fig. 4 of Ref. 24). We hypothesized that the slow rise in N/C after change of solutions under control conditions is due to removal of humoral factors, including IGF, that were previously secreted by the cultured fibers, accumulated in the bathing medium during culture, and then were removed with the solution change from culture medium to recording solution (24) (here substantiated in Fig. 6).

We first use our mathematical model to determine the apparent unidirectional rate constants for Foxo fluxes under the control conditions using the previously published data (45) shown again in Fig. 3A. On the basis of the solution (Eq. 5) of the differential equation (Eq. 4) for the rate of change of N/C, after a change to a new steady Foxo phosphorylation fraction in the cytoplasm and/or the nucleus, the system will approach a new steady state along a single-exponential time course in which the final (steady-state) level, (N/C)_ss, of N/C is given by kI′/kE′ and the rate constant for approach to the steady state is kE′. Fits of a single exponential + a constant to the approach to steady state gives values for these two parameters, which provides kI′ = kE′(N/C)_ss.

The theoretical time course generated by fitting a single exponential + a constant to the mean control data in Fig. 3A closely reproduces the observed Foxo1-GFP N/C mean time course under our control conditions. The fit to the average control time course, the apparent rate constants for nuclear influx and efflux under control conditions, were kI′ = 0.110 min⁻¹ and kE′ = 0.013 min⁻¹, with (N/C)_ss = 8.45. In addition, each individual N/C time course (not shown) that was included in the average in Fig. 3A was also separately fit by a single exponential + a constant. From the parameter values for fits to the individual control time courses, the rate constants were kI′ = 0.119 ± 0.014 (SE) min⁻¹ and kE′ = 0.014 ± 0.003 min⁻¹ (average values for 14 nuclei from 13 fibers; Fig. 3B), with (N/C)_ss = 8.96 ± 0.69. The parameter values obtained by fitting the mean N/C time course (Fig. 3A) agree within a few percentage points of the mean values of the same parameters obtained by fitting the individual N/C time course records (not shown), but the parameters obtained from fits to individual records also provide estimates of the SE and, thus, are used whenever available for graphical display of the apparent rate constants kI′ and kE′ (Fig. 3B). The considerably larger (∼8.5- to 9-fold) value for kI′ than kE′ under control conditions (Fig. 3B) is consistent with the value of the predicted final level (N/C)_ss of N/C of 8.45 (Fig. 3A) or 8.96 (Fig. 3B) obtained from the single-exponential fits of the average (Fig. 3A) or individual (Fig. 3B) N/C time courses.

We note again that the Foxo1-GFP N/C time course data in Fig. 3A, as well as the time course data presented in Fig. 4, A and C, and Fig. 5, A and C, have been published previously (24) but were not fit by any model. We introduce a mathematical model to interpret such data and fit the model to our previously published data sets to obtain new insights regarding effects of pharmacological manipulations on apparent rate constants for unidirectional nuclear influx and efflux of Foxo1-GFP, as shown in Fig. 3B, Fig. 4, B and D, and Fig. 5, B and D.

Determining the apparent rate constants for Foxo1-GFP nuclear influx and efflux during the approach to steady state after direct or indirect inhibition of Akt.

We next used our mathematical model to determine the apparent unidirectional rate constants for Foxo fluxes under various conditions of modulation of activity of the IGF-I/PI3K/Akt pathway, again using our previously published data (24). We first consider the effects of suppression of Akt activity. Figure 4, A and C, presents the time courses of Foxo1-GFP N/C in two sets of control fibers and two corresponding sets of fibers studied for 30 min under control conditions and then for 80 min during exposure to a direct (Fig. 4A) or an indirect (Fig. 4C) inhibitor of Akt. Blocking the IGF-I/PI3K/Akt pathway with an inhibitor of Akt [Akt1/2-I (24), 1 μM; Fig. 4A] or an inhibitor of PI3K (LY-294002, 25 μM; Fig. 4C) caused a net increase in Foxo1-GFP nuclear uptake compared with the matched controls (24), consistent with block of Akt activity and consequent decreased phosphorylation of Foxo1-GFP, which would be expected to result in increased unidirectional nuclear influx and/or decreased unidirectional nuclear efflux of Foxo1-GFP. Our model now allows us to quantitatively evaluate these anticipated changes in influx or efflux apparent rate constants.

To interpret data from experiments such as those in Fig. 4, where an inhibitor of the IGF-I/PI3K/Akt pathway was added and the resulting time course of N/C was compared with a matched control time course, a single exponential + constant function of time was fit to the inhibitor records starting 20 min after drug addition to allow time for the full effect of the added agent to develop, whereas the corresponding control record was fit over the entire recording interval as in Fig. 3A, since there was no change in condition in the control case over the full recording interval. Over these specified time intervals, a single exponential + constant again provided an excellent fit to the control data (as in Fig. 3A) and to the data in the presence of added drug (Fig. 4, A and C).

Figure 4, B and D, presents mean values for the apparent rate constants kI′ and kE′ obtained from fitting the individual records that were averaged to give the corresponding drug and matched control time courses in Fig. 4, A and C. It is anticipated that application of inhibitors of Akt or PI3K, the upstream kinase for Akt activation, should decrease Akt activity, thereby decreasing Foxo1 phosphorylation and, thus, tending to increase nuclear accumulation of Foxo1-GFP (Fig. 4, A and C) (24). On the basis of comparison of mean parameter values from single exponential + constant fits to the individual N/C time courses averaged to give the time courses in Fig. 4, A and C, during drug application and the matched control runs, the direct Akt inhibitor caused a 3.7-fold increase in kI′ and the PI3K inhibitor (an indirect inhibitor of Akt) caused a 2.7-fold increase in kI′ (Fig. 4, B and D). An increase in kI′ is in the direction anticipated for decreased Foxo1-GFP phosphorylation (i.e., increased Foxo1-GFP dephosphorylation) in the cytoplasm, which should increase the rate of Foxo1-GFP unidirectional nuclear influx and, thus, increase the extent of Foxo1-GFP net nuclear accumulation, as observed. On the other hand, the apparent rate constant for nuclear efflux (kE′), which is expected to decrease when Akt activity is inhibited in the nucleus (thereby decreasing Foxo1 intranuclear phosphorylation and, thus, nuclear efflux), was hardly changed or, instead, even increased somewhat compared with matched controls upon addition of Akt inhibitor or PI3K inhibitor (Fig. 4, B and D). The lack of suppression of kE′ by the direct or indirect inhibition of Akt could indicate that, under control conditions, Akt is not the predominant enzyme that phosphorylates Foxo1 in the nucleus prior to Foxo1 nuclear efflux.

Relative phosphorylation of nuclear or cytoplasmic Foxo1-GFP under various experimental conditions.

As considered above for Akt inhibition and below for Akt activation by IGF-I, under various conditions of activity of the kinases and phosphatases that phosphorylate and dephosphorylate Foxo1, the values of fC_P and fN_P will vary. If it is assumed that the values of the actual rate constants k_I and k_E for Foxo nuclear influx and efflux are independent of Foxo phosphorylation status (i.e., that the properties of the nuclear import and export systems and the concentrations of all their cofactors are not altered by the applied drugs), the ratio k′_Ia/k′_Ib of kI′ values under two such conditions (a and b) equals the ratio fU_ca/fU_cb and the ratio k′_Ea/k′_Eb equals fP_na/fP_nb. These are the ratios of the fractional unphosphorylation of Foxo1-GFP in the cytoplasm and phosphorylated Foxo1-GFP in the nuclei, respectively, in condition a relative to condition b. These ratios thus reveal condition-dependent changes in Foxo1-GFP relative phosphorylation/dephosphorylation status in the cytoplasm and nuclei, respectively, which we evaluate here.

Apparent rate constants for Foxo1-GFP nuclear influx and efflux change dramatically during application of IGF.

Application of IGF-I caused a dramatic rapid, pronounced, and maintained decline of nuclear fluorescence to a small fraction of its control value (Fig. 5A; data replotted from Fig. 4A in Ref. 24). Since the decline of nuclear Foxo1-GFP was almost completed at the start of the fit at 20 min after application of IGF (Fig. 5A), another set of experiments in which IGF-I was added was carried out on different fibers at a higher sampling rate (2-min sampling interval; Fig. 5C) than the 10- or 20-min sampling intervals used in Fig. 5A and all previous figures. The high-time-resolution IGF-I experiment was not carried out with a paired control, so we compared the parameter values obtained for the high-time-resolution IGF-I response with the mean of the parameter values obtained from fits to each of the control records in Fig. 3 (24). The apparent rate constants for these experiments with IGF-I are presented in Fig. 5, B and D. IGF-I application would be anticipated to increase Akt activity in the cytoplasm, resulting in decreased fractional unphosphorylation of Foxo1 in the cytoplasm, which should decrease kI′, as observed, since unphosphorylated, but not phosphorylated, Foxo1 can be moved into the nucleus via the nuclear import system. Our data suggest that kI′ was decreased to ∼0.50 or 0.20 of the control values (Fig. 5, B and D, respectively).

Application of IGF-I also caused a pronounced increase in kE′ for each of the two types of experiments using IGF-I (Fig. 5, B and D). These results clearly and dramatically demonstrate a 10.4- or 7.9-fold increase in Foxo1 apparent nuclear efflux rate constant in the presence of IGF-I. This novel observation implies a dramatic increase in fractional phosphorylation of Foxo1-GFP within the nucleus, which indicates that, in the presence of IGF, Akt activity or, possibly, some other kinase activity activated in the presence of IGF-I has markedly increased within the muscle fiber nuclei.

Basis for the time-dependent increase in N/C under control conditions.

Foxo1-GFP N/C time courses recorded under our standard control conditions exhibited a gradual increase with time that appeared to approach a new steady-state level (Fig. 3A). Since our control N/C time courses were recorded starting 30 min after our culture medium (serum-free MEM) was changed to our experimental solution (L-15 medium), we hypothesized that the control records' approach to a new steady state at a higher level of N/C after the solution change represents the effects of a step decrease in the bath concentration of soluble factors that influence muscle fiber Foxo1-GFP nuclear-cytoplasmic distribution, probably due to washout of IGF-I or similar molecules in the culture solution as a result of accumulation of growth factors released from the muscle fibers during the previous time in culture (24).

In our previous studies we used a bicarbonate-buffered culture medium (MEM) that requires 5% CO₂ to maintain correct pH and, thus, had to be changed to an alternatively (i.e., nonbicarbonate) buffered solution (L-15 medium) for our experiments in room air (24). Therefore, to test our hypothesis regarding accumulation of soluble factors during fiber culture, we carried out a new series of experiments using a different serum-free culture medium that was buffered by bicarbonate, as well as by HEPES, so that pH was maintained in the 5% CO₂ tissue culture incubator and in room air environments and, thus, did not have to be changed at the start of the experiment. Using this medium, we found that fibers maintained in the solution in which they were previously cultured for 3 days did not exhibit a rise in nuclear Foxo1-GFP but, instead, showed a slight decline in N/C with time (Fig. 6B). In contrast, when the culture medium was replaced with fresh culture medium (upward arrow in Fig. 6B), N/C increased with time (Fig. 6B), as seen in our previous control studies, where the culture medium was changed prior to the start of the experiment (Fig. 3A). However, when the culture solution was changed to fresh medium at the start of recording, causing N/C to increase (Fig. 6B), and then changed back to the original medium (downward arrow in Fig. 6B) 58 min after the preceding change to fresh medium (Fig. 6B), N/C rapidly declined to the level seen without a change of the medium. These observations conclusively demonstrate that some factor that accumulated in the bathing medium during the culture period had a strong effect on keeping Foxo1-GFP out of the fiber nuclei and are highly reminiscent of our previously observed effects of IGF-I addition to the bathing medium (24; replotted here in Fig. 5, A and C).

The relatively slow decline of N/C in the fibers maintained in the conditioned culture medium throughout the course of the experiment in Fig. 6 has a much slower rate constant than the decline after return to conditioned medium, indicating the presence of a small fraction of N/C that may be regulated by a slow process not accounted for in our model, which may account for the slight drop in N/C near the end of the time course of fibers changed to and maintained in fresh medium.

In other experiments, fibers were washed for 60 min with fresh medium and then returned to conditioned medium with or without the addition of neutralizing antibody for IGF, and the rate of change of N/C was determined by fit of a straight line to the N/C time course over the next 60 min. When the fibers were maintained in the wash solution, the rate of change of N/C was quite low, but it increased markedly upon return to conditioned medium (Fig. 6D). In contrast, when an IGF-I-neutralizing antibody was added to the conditioned medium, the effect of adding back the conditioned medium was completely eliminated, indicating that IGF-I, or some highly similar molecule that was also neutralized by the anti-IGF-I antibody, was responsible for the decline of N/C caused by the conditioned medium.

Near linearity of nuclear import of Foxo1-GFP.

Our modeling assumes linearity of the nuclear import and export systems for Foxo1. We therefore used the drug leptomycin B, which eliminates Foxo1-GFP nuclear export (24), together with a photobleaching protocol to test for the linearity of the nuclear import of Foxo1-GFP. In the presence of leptomycin B, the net rate of nuclear influx directly provides the rate of unidirectional nuclear influx, since efflux is eliminated. For these studies, fibers were oriented parallel to the long axis of the image. We first determined the rate of nuclear influx over a 0.5-h period (using 10-min sampling intervals) in a nucleus selected to be at the periphery of the fiber and centered in the image (Fig. 7A) in the presence of leptomycin B. We next bleached a large cytoplasmic region of the fiber in the image field (Fig. 7A, right) but avoided including the nucleus in the bleached area. We again determined the rate of nuclear influx, now over a 10-min interval starting 2 min after photobleaching, using 2-min sampling intervals to minimize return of Foxo1-GFP from fiber regions outside the bleached segment. We then calculated the rate of Foxo1-GFP nuclear influx after relative to before bleaching and plotted it as a function of the relative cytoplasmic fluorescence, also after relative to before bleaching, for the same nuclei (Fig. 7B). The decrease in influx rate after bleaching was close to proportional (Fig. 7B, dashed line) to the decrease in cytoplasmic fluorescence. The solid line is the linear regression fit to the data, including an equal number of data points at (1.0, 1.0) to denote the values of relative influx rate and relative C before photobleaching. These results support the assumption of linearity for the nuclear import of Foxo1-GFP.

Model sensitivity analysis.

Figure 8 shows the effects of +40% or −40% changes in the best-fit values of kI′, kE′, or kI′ and kE′ under control conditions and upon application of PI3K inhibitor and IGF. For the control and PI3K inhibitor records (Fig. 8 A, B, D, and E), a +40% or −40% change in each of the individual parameter values causes marked deviation from the observed best-fit time course. Consistent with the fact that the final steady level of N/C is given by the ratio kI′/kE′, changing both kI′ and kE′ by the same (40%) factor causes no change in steady level but does change the rate constant for the approach to the final steady state by the 40% change in kE′ (Fig. 8, C and F), resulting in smaller, but still quite noticeable, deviation from the observed time course.

Fig. 8.

Sensitivity analysis of model fits to experimental data under various experimental conditions. A–C: control conditions; no added pharmacological agents. Optimal fit to the data (solid line) from Fig. 4. D–F: in the presence of inhibitor of PI3K, an indirect activator of Akt. Optimal fit to the data (solid line) from Fig. 5C. G–I: in the presence of IGF-I, an indirect activator of Akt. Optimal fit to the data (solid line) from Fig. 4. A, D, and G: effect of ±40% change in kI′. B, E, and H: effect of ±40% change in kE′. C, F, and I: effect of +40% change in kI′ and kE′ and −40% change in kI′ and kE′.

For the IGF-I application (Fig. 8, G–I), the rate of approach to the final steady state is very much faster than for control or Akt inhibition, and the final level of N/C is close to zero. In this case, changing kI′ by 40% (Fig. 8G), which only changes the final level, causes little change in the observed time course. In contrast, changing kE′ by 40% (Fig. 8H) changes the rate constant and results in a detectable deviation from the observed time course. This difference would be more readily observed using an expanded time scale. When both kI′ and kE′ are changed by 40%, there is no change in the final level, but the change in the rate constant is maintained (Fig. 8I).

DISCUSSION

In the present study, we have made a number of advances in the analysis and understanding of Foxo1 nuclear-cytoplasmic movements and the simultaneous unidirectional nuclear influxes and effluxes that underlie the net nuclear movements of Foxo1 in adult skeletal muscle fibers maintained in culture and studied under various experimental conditions. We first developed a reduced two-state mathematical model for Foxo1 nuclear-cytoplasmic movements under the assumption that the rates of phosphorylation and dephosphorylation of Foxo1 in the nuclei and the cytoplasm are fast compared with the rates of nuclear influx and efflux. Our model also takes into account that the cytoplasmic volume is much larger than the nuclear volume and, as such, provides an essentially inexhaustible reservoir of constant concentration of Foxo1. We then used the reduced model to interpret the increasing nuclear-to-cytoplasmic ratio of Foxo1-GFP after transfer of muscle fiber cultures from the tissue culture incubator to our experimental apparatus. We further employed the reduced model to determine changes in apparent rate constants for nuclear influx and efflux of Foxo1-GFP under experimental conditions of suppressed or activated Akt/PKB and used these findings to attribute changes in nuclear-cytoplasmic distribution of Foxo1-GFP to intranuclear-cytoplasmic changes in Foxo1 phosphorylation ratios. Finally, although we recognize and applied the condition that, in muscle fibers, the cytoplasmic volume far exceeds the nuclear volume and, thus, serves as an effective constant reservoir for Foxo1 in muscle fibers, we present a general mathematical analysis of nuclear-cytoplasmic movements, including any possible ratio of nuclear to cytoplasmic volume.

Reduced model for nuclear-cytoplasmic Foxo1-GFP fluxes.

The first step in the analysis presented here was to develop a reduced two-state mathematical model for Foxo1 nuclear-cytoplasmic movements. Traditionally, the movement of Foxo1 between the cytoplasm and the nucleus is interpreted in terms of a kinetic scheme whereby unphosphorylated (but not phosphorylated) Foxo1 can be transported unidirectionally into the nuclei by the nuclear import system, whereas phosphorylated (but not unphosphorylated) Foxo1 can be transported unidirectionally out of the nuclei by the nuclear export system. Foxo1 phosphorylation/dephosphorylation occurs in cytoplasm and nuclei. This leads to a four-state, six-rate-constant model (Fig. 2A). However, under the assumption that the rates of phosphorylation and dephosphorylation of Foxo1 in nuclei and cytoplasm are fast compared with the rates of nuclear influx and efflux, the model is reduced to a system having two states and two apparent rate constants (Fig. 2B).

The reduced model predicts that, under steady conditions of phosphorylation/dephosphorylation of Foxo1 in cytoplasm and nuclei, the system will approach a steady-state nuclear-cytoplasmic distribution (N/C) of Foxo1 along a single-exponential time course. In a cellular system having a large excess cytoplasmic-to-nuclear volume, as in skeletal muscle fibers, the observed empirical rate constant for the approach to steady state is the apparent rate constant for nuclear efflux, and the steady-state level of N/C is given by the ratio kI′/kE′ of apparent rate constants for nuclear influx and efflux. The amplitude of the exponential component of the N/C time course is the difference between the starting and steady-state values of N/C. With this insight, the fits of a single exponential + constant function to the observed time course of N/C directly provides values for the two apparent rate constants kI′ and kE′ under the prevailing conditions in the fiber during the N/C recording.

Linearity of Foxo nuclear transport.

Our model formulation also assumes linearity of Foxo1 nuclear transport. By using leptomycin B to eliminate Crim1-dependent nuclear efflux, including efflux of Foxo1-GFP, together with measurements of Foxo1-GFP influx rates before and after cytoplasmic photobleaching, we were able to demonstrate that the relative decrease in influx rate after cytoplasmic photobleaching is very close to proportional to the relative decrease in cytoplasmic fluorescence (Fig. 7B). Thus there is no apparent saturation of the nuclear import system for Foxo1 in these studies. Future studies, including specific block of nuclear influx, would be needed to verify the linearity of the nuclear export system for Foxo1. Other further studies would be needed to test the linearity of the other reaction steps in the model. If certain steps are found in the future to exhibit saturation or to display other nonlinearities, then the model would have to be correspondingly modified.

Approach to new steady state after transfer of fibers from culture to control experimental conditions.

We previously observed that Foxo1-GFP exhibits a definite gradual increase in nuclear concentration with time after change from tissue culture (serum-free MEM culture medium at 37°C and 5% CO₂) to control experimental (serum-free L-15 medium at room temperature and ambient CO₂) conditions on our experimental apparatus (24). Using the reduced model, we now interpret this gradual rise in N/C as the approach to a new steady state, with the change from culture to control recording conditions apparently altering the Foxo1 phosphorylation status, such that the ratio of apparent rate constants for Foxo1-GFP unidirectional nuclear influx and efflux (kI′/kE′) is larger under the control conditions on the experimental setup than under the conditions prevailing prior to removal of the fiber from the incubator.

The two apparent rate constants for Foxo1 nuclear movement, one each for influx (kI′) and efflux (kE′), are equal to the product of an actual rate constant and the respective fractional dephosphorylation or phosphorylation of Foxo1 in the cytoplasm or the nucleus (k_IfC_U or k_EfN_P). The values of k_I and k_E depend on the intrinsic properties of the transport system for nuclear import and export and on the concentrations of the various cofactors for the respective transport systems. In principle, the properties and/or cofactor concentrations for the nuclear import or export transport systems might possibly change in the transition from the tissue culture incubator to the experimental apparatus and control solution. However, our laboratory previously made extensive use of adult skeletal muscle fibers that were maintained in culture and transduced to express fluorescent fusion protein constructs of the transcriptional regulators NFAT and HDAC to study the nuclear-cytoplasmic distribution of these transcriptional regulators under a variety of experimental conditions. In contrast to Foxo1-GFP, the NFAT- and HDAC-GFP fusion constructs exhibited a constant nuclear-to-cytoplasmic ratio (N/C) after the fiber culture was changed from culture medium to experimental solution and conditions [NFATc1-GFP (17) and HDAC4- and HDAC5-GFP (18)], indicating that changes in the transport systems and transport cofactor concentrations may not be occurring in muscle fiber cultures upon transfer from the incubator to the control solution and experimental apparatus, since these molecules likely utilize many of the same transport components as Foxo. Alternatively, the fractional phosphorylation of Foxo1-GFP in nuclei and/or cytoplasm is likely to have changed, possibly due to a change in the concentration of factors previously secreted by the cultured fibers, and then removed with the change from the incubator to the control recording solution. This scenario would be consistent with the secreted factors having little or no effect on nuclear-cytoplasmic distribution of NFATc1, HDAC4, or HDAC5 in adult muscle fibers under our culture and experimental conditions.

Our new experimental findings presented in Fig. 6 support our previous proposal that washout of growth factors accumulated during the culture period changes Foxo1 phosphorylation status, leading to a change in apparent rate constants for Foxo1 nuclear influx and efflux and the resulting change in Foxo1-GFP N/C. Using a culture medium that is buffered by bicarbonate and HEPES and, thus, does not need to be changed when fiber cultures are moved from the incubator to the experimental setup, we find that if the medium is not changed, N/C remains at a relatively constant, relatively low value. This indicates that if the solution is not changed, the fibers maintain the state they exhibited in culture prior to the start of recording. In contrast, if the bathing medium is changed to a fresh (i.e., not previously used for any culture) batch of the same culture medium, then Foxo1-GFP N/C increases toward a new steady-state level, indicating an altered fiber status. Additionally, if the fibers are reexposed to the original “conditioned” medium, N/C rapidly returns to the level seen prior to the initial solution change, very reminiscent of the rapid fall in N/C produced in other control fibers by addition of IGF-I.

As mentioned above, the time-dependent approach to a new steady-state N/C after change from the medium in which the fibers were maintained in culture to the control experimental recording solution in which the fibers were studied was not seen in our studies with the transcriptional regulators NFATc1 (13) and HDAC4 and HDAC5 (14). Similar to Foxo, the transcriptional effectiveness of these transcriptional regulators is also determined by their nuclear-cytoplasmic distribution, which is in turn determined by their phosphorylation status. However, in the case of NFATc1, HDAC4, and HDAC5, phosphorylation status is regulated by muscle fiber activity and the resulting Ca²⁺-dependent kinase or phosphatase activity, which apparently is not altered by change from culture to experimental solution, indicating a lack of effects of factors secreted by the fibers in culture on fiber Ca²⁺ levels in our experiments. Thus, in studies on NFAT and HDACs, the initial N/C remains quite stable after change from culture to recording solution and conditions, providing no other interventions are carried out. In contrast, Foxo1-GFP displays a characteristic exponential approach to an increased N/C after change from culture to recording solution. This gradual increase in N/C was originally viewed as an annoying time-dependent baseline change from which to look at changes in the slope upon activation or inhibition of Akt activity (24). However, with our present analysis, the initial time-dependent increase in Foxo1-GFP is interpreted as the approach to a new control steady state and is characterized to extract the apparent rate constants kI′ and kE′ for Foxo1-GFP nuclear influx and efflux under control conditions. These control values can then be compared with the values of the same apparent rate constants under conditions of modified activity of Akt and, thus, can be used to determine changes in relative phosphorylation status of nuclear and cytoplasmic Foxo1 in the test compared with control conditions (see below).

Interpretation of effects of experimental modulation of Akt activity.

Our reduced model was used to determine changes in the apparent rate constants kI′ and kE′ for nuclear influx and efflux of Foxo1-GFP under experimental conditions of suppressed or activated activity of the IGF-I/PI3K/Akt pathway compared with the control experimental conditions. In interpreting the basis for the observed changes in kI′ or kE′ under various experimental conditions, we assume that application of direct or indirect inhibitors or activators of Akt phosphorylation (and activity) does not alter the intrinsic properties of the nuclear import or export transport systems. We also assume that the direct or indirect inhibitors and activators of Akt do not alter the concentrations of any of the various cofactors for nuclear import or export. Instead, any detected changes in the apparent rate constant kI′ for Foxo1 nuclear influx are attributed exclusively to changes in cytoplasmic Foxo1 phosphorylation/dephosphorylation levels, and any detected changes in the apparent rate constant kE′ for nuclear efflux are attributed exclusively to changes in Foxo1 phosphorylation/dephosphorylation in the nuclei.

Using the reduced model, we found that direct or indirect inhibition of Akt activity caused an almost threefold increase in kI′ (Fig. 4, B and D), which we attribute to a corresponding almost threefold increase in the cytoplasmic Foxo1-GFP dephosphorylation fraction. This increased cytoplasmic Foxo1 dephosphorylation fraction is in the anticipated direction for inhibition of the enzyme responsible for Foxo1 phosphorylation. In contrast, in the same fibers, the model shows that the Akt inhibitors did not decrease kE′, which would be expected if inhibition of Akt decreased the Foxo1 phosphorylation fraction in the nuclei, since phosphorylated Foxo1 is the form of Foxo1 that is transported out of the nuclei.

On the basis of the high-time-resolution Foxo1-GFP measurements during application of IGF-I, which should provide our most reliable estimate of the rapid Foxo1-GFP fluxes in response to IGF-I, application of IGF-I caused a nearly fivefold decrease in kI′ (Fig. 5, B and D). This indicates that IGF-I application decreased Foxo1-GFP fractional dephosphorylation (i.e., increased fractional phosphorylation) in the cytoplasm, which decreased Foxo1 nuclear influx proportionally. Interestingly, IGF-I application also caused a sevenfold increase in kE′ (Fig. 5, B and D), indicating that IGF-I increased Foxo1-GFP fractional phosphorylation in nuclei and cytoplasm to a similar extent. Both changes would contribute to the marked decline in Foxo1-GFP N/C after addition of IGF-I. The fact that kE′ could increase sevenfold on exposure to IGF-I shows that, under our control conditions, the fractional phosphorylation of Foxo1-GFP in fiber nuclei prior to IGF-I addition must have been ≤1:7, which would, in turn, allow IGF-I to increase kE′ by more than sevenfold. These considerations indicate a large potential dynamic range for regulation of Foxo1 relative phosphorylation and nuclear-cytoplasmic distributions.

An alternative case, with the assumption that flux rate constants are much greater than phosphorylation/dephosphorylation rate constants, is not consistent with our observations. In deriving our model, we have assumed a limiting case that the rate constants for phosphorylation/dephosphorylation are much faster than those for nuclear influx and efflux. However, it is also instructive to examine the opposite limiting case, namely, that nuclear flux rate constants are much faster than those for phosphorylation/dephosphorylation. Under the conditions that k_E >> k_Un and k_I >> k_Pc, essentially all nuclear Foxo that becomes phosphorylated will leave the nucleus and essentially all cytoplasmic Foxo that becomes dephosphorylated will enter the nucleus. In this case, nuclear phosphorylated Foxo (P_n) and cytoplasmic dephosphorylated Foxo (U_c) will be negligible, so that N = U_n and C = P_c.

In this case

$$\text{dN}/\text{d}t=k_{\text{Uc}}\text{C}\hspace{0.17em}-k_{\text{Pn}}\text{N}$$ (6)

Note that Eq. 6 has exactly the same form as Eq. 3 used above for the assumption of rapid phosphorylation/dephosphorylation rate constants compared with flux rate constants, but now kI′ = k_Uc (instead of k_IfU_c), and kE′ = k_Pn (instead of k_EfP_n). Thus Eq. 6 will fit the data exactly as well as the assumption of fast phosphorylation/dephosphorylation compared with flux rate constants used above. However, conclusions reached using this version with our experimental results seem to be inconsistent with the accepted action of IGF-I activation of Akt. In particular, we found that application of IGF-I causes a two- or fivefold decrease in kI′ (Fig. 5, B and D, respectively). This would indicate a two- or fivefold decrease in k_Cu (the rate of Foxo1 dephosphorylation in the cytoplasm). However, IGF-I is well known to activate Akt but is not known to activate the phosphatase(s) that dephosphorylate(s) Foxo1. Thus the assumption of fast nuclear flux rate constants compared with phosphorylation/dephosphorylation rate constants seems to be unreasonable on the basis of the conclusions resulting from applying that form of the model to our experimental results, even though formally it fits the data exactly as well as when the original assumptions are used. In contrast, assuming fast rate constants for phosphorylation/dephosphorylation compared with flux rate constants (original model) is perfectly consistent with increased Akt activity leading to a decrease in fractional dephosphorylation (increased fractional phosphorylation) of Foxo1 in the cytoplasm (and, thus, decreased rate of Foxo1 entry). These considerations indicate that the assumption of fast phosphorylation/dephosphorylation compared with nuclear influx and efflux is more reasonable than the alternative limiting case. Future modeling and experimental studies are needed to examine intermediate situations where all rate constants may be similar in magnitude and, thus, cannot be determined from a single observed experimental time course.

Future expanded potential application of our model.

As noted above, the values of kI′ and kE′ obtained from fitting our model to observed N/C time course data are each equal to the product of k_I or k_E and the corresponding Foxo1 dephosphorylation or phosphorylation fraction in the cytoplasm or nucleus, respectively. Here we utilized measured changes in kI′ or kE′ to detect corresponding changes in cytoplasmic dephosphorylation fraction or nuclear phosphorylation fraction of Foxo1 during application of modulators of Akt assuming k_I or k_E to be constant (i.e., assuming constant effectiveness of the nuclear import or export apparatus and constant concentrations of the corresponding nuclear transport cofactors). Alternatively, it is conceivable that the same experimental and analysis approach could be utilized under conditions of constant phosphorylation status of Foxo1 (or another molecule that shuttles into and out of muscle fiber nuclei), but under conditions that alter some aspect(s) of the nuclear import or export systems. In such cases, the observed changes in kI′ or kE′ would not be attributed to changes in dephosphorylation or phosphorylation fraction of Foxo1 but would, instead, be used to monitor relative effectiveness of the corresponding nuclear import or export system. Interestingly, in that case, two different cargos that are carried by the same nuclear import or export system should exhibit the same change in k_I or k_E when the activity of that transport system is modulated.

Role of initial conditions and apparent rate constants in determining the N/C time course.

It is important to note that the observed time course of N/C over a given interval during which the fractional phosphorylation of nuclear and cytoplasmic Foxo1 remains constant is determined by the apparent rate constants kI′ and kE′ corresponding to the fractional phosphorylation during the measurements and by the initial N/C in the fiber. The two simulated N/C time courses in Fig. 9 illustrate the role of the starting N/C. In both cases, kI′ and kE′ were identical, resulting in the same steady-state N/C (N/C_ss) and the same time constant for approach to that steady state. However, there was a fourfold difference in the initial values between the two curves, which would be due to different conditions for the two curves prior to the start of the simulation interval, and the simulated time courses are correspondingly different, even though the apparent rate constants for Foxo1 nuclear influx and efflux are identical for the simulated interval. Although the two simulated time courses appear different in Fig. 9, if the lower curve is examined from the same initial condition as the upper curve by starting when the lower curve reaches N/C = 6.0 (which occurs at ∼65 min), the extrapolated lower curve would be identical to the upper curve.

Fig. 9.

Simulated time courses for Foxo1 N/C using the same apparent rate constants for nuclear influx and efflux, but with different initial values of N/C. For both simulations, kI′ = 0.126 and kE′ = 0.014 (N/C_ss = 9.0, indicated by dashed line). These parameter values are very similar to those obtained from fits to the observed control records (Fig. 3). Initial values of N/C were arbitrarily set to 1.5 and 6 for the lower and higher curve, respectively. Each simulated N/C time course was calculated for an interval of 110 min, the duration of the observed control records in Fig. 3.

Caveats on the models.

It must be noted that the four-state model, which was simplified to the two-state model used here, already constitutes a great simplification of the actual biochemical system. For example, there are three Akt phosphorylation sites on Foxo1, and thus multiple partially phosphorylated species of Foxo1 may be present in the nucleus and the cytoplasm. In addition, several other posttranslational modifications, including Foxo phosphorylation at other sites by other enzymes (8, 24, 29), Foxo acetylation (4, 10, 21, 25), and Foxo ubiquitination (14), also influence Foxo1 nuclear import or export (8). However, without the ability to discriminate between the various phosphorylation or other posttranslational modification states, we consider a single effective “phosphorylated” state in the cytoplasm that cannot enter the nuclei and a single effective “dephosphorylated” state that can enter the nuclei from the cytoplasm. In the nuclei, we consider only a single effective phosphorylated state that can exit the nuclei and a single effective dephosphorylated state that cannot exit the nuclei. These limitations are not serious for the four-state model, because it is not possible to determine all the various partially phosphorylated species or the combinations of other posttranslational modification states of nuclear or cytoplasmic Foxo. A full model including the large number of intermediate states (26) has too many parameters to achieve a practical preliminary understanding of general aspects of Foxo function and regulation for the present application. For simplicity, many different molecular species are lumped into single “states” that can or cannot cross the nuclear envelope, with a single pair of effective rate constants governing the overall interconversion between these states in the nucleus and another pair in the cytoplasm. The further reduced two-state model already closely reproduces the experimental data (Figs. 3–5), so additional parameter values are unlikely to be specified by the present data without additional restrictions established by other experimental measurements. The present two-state model is intuitive, is easy to manipulate, and provides conclusions regarding changes in the cytoplasm vs. changes in the nuclei during various experimental situations. In the future, when advanced molecular and imaging techniques allow the various molecular species to be experimentally distinguished, the model can be appropriately expanded to explicitly include these corresponding additional states.

Nuclear-cytoplasmic flux kinetics under various nuclear-to-cytoplasmic volume ratios.

Finally, although we recognize and applied the condition that in muscle fibers the cytoplasmic volume far exceeds the nuclear volume and, thus, serves as an effective constant reservoir for Foxo1, we present a general mathematical analysis of nuclear-cytoplasmic movements, including the possibility of any ratio of nuclear to cytoplasmic volume.

Conclusion.

The quantitative mathematical modeling of nuclear-cytoplasmic movements of transcriptional regulators or other molecules, in general, and of Foxo1-GFP, in particular, as studied here, provides values for unidirectional nuclear influx and efflux under various experimental conditions, as well as control conditions, and, thereby, reveals previously undetermined properties of Foxo1 phosphorylation/dephosphorylation status in nuclei and cytoplasm of skeletal muscle fibers. Similar analysis and interpretations could be applied to other cell types with appropriate consideration of their nuclear/cytoplasmic volumes.

GRANTS

This work was supported by National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS) Grant R01-AR-056477 and by a University of Maryland Baltimore (UMB)/University of Maryland Baltimore County (UMBC) (Research and Innovation partnership) Seed Grant (Program). R. J. Wimmer was partially supported by NIAMS Training Grant T32 AR-007592 to the Interdisciplinary Program in Muscle Biology National Institute of General Medical Sciences Training Grant T32 GM-008181 to the Training Program in Integrative Membrane Biology, University of Maryland School of Medicine. T. N. Schachter was partially supported by NIAMS Training Grant T32 AR-007592 and National Heart, Lung, and Blood Institute Training Grant T32 HL-072751 to the Program in Cardiac and Vascular Cell Biology, University of Maryland School of Medicine. D. P. Stonko was partially supported by the UMB/UMBC Seed Grant.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

M.F.S., R.J.W., and Y.L. conceived the project; R.J.W., M.F.S., and B.E.P. developed the model; D.P.S., R.J.W., M.F.S., and B.E.P. carried out the sensitivity analysis; T.N.S., R.J.W., and Y.L. carried out the experiments; R.J.W., Y.L., B.E.P., and M.F.S. analyzed data and applied the model; M.F.S., R.J.W., Y.L., and B.E.P. drafted and edited the paper; all authors approved the final version of the manuscript. All experiments were performed in the laboratory of M.F.S. at the University of Maryland, Baltimore.

Appendix

Four-state model.

We derive the rate of change of the four Foxo1-GFP populations: nuclear phosphorylated (P_n), nuclear unphosphorylated (U_n), cytoplasmic phosphorylated (P_c), and cytoplasmic unphosphorylated (U_c). Dephosphorylation/phosphorylation is taken to be reversible, with rate constants k_Pn, k_Un, k_Pc, and k_Uc, while the influx and efflux rate constants are k_I and k_E, respectively, as in Fig. 2A. Furthermore, we keep track of the volume fraction of the nucleus (V_n) and cytosol (V_c) when converting between nucleus and cytosol. From these, we generate the four-variable system

$$\begin{array}{c}\frac{{\text{dU}}_{\text{c}}}{\text{d}t}=-k_{\text{Pc}}{\text{U}}_{\text{c}}+k_{\text{Uc}}{\text{P}}_{\text{c}}-k_{\text{I}}{\text{U}}_{\text{c}}\\ \frac{{\text{dP}}_{\text{c}}}{\text{d}t}=k_{\text{Pc}}{\text{U}}_{\text{c}}-k_{\text{Uc}}{\text{P}}_{\text{c}}+\frac{{\text{V}}_{\text{n}}}{{\text{V}}_{\text{c}}}{k}_{\text{E}}{\text{P}}_{\text{n}}\\ \frac{{\text{dU}}_{\text{n}}}{\text{d}t}=-k_{\text{Pn}}{\text{U}}_{\text{n}}+k_{\text{Un}}{\text{P}}_{\text{n}}+\frac{{\text{V}}_{\text{c}}}{{\text{V}}_{\text{n}}}{k}_{\text{I}}{\text{U}}_{\text{c}},\\ \frac{{\text{dP}}_{\text{n}}}{\text{d}t}=k_{\text{Pn}}{\text{U}}_{\text{n}}-k_{\text{Un}}{\text{P}}_{\text{n}}-k_{\text{E}}{\text{P}}_{\text{n}}\end{array}$$ (A1)

Two-state model.

We make a series of reductive assumptions to obtain a lower-dimensional system commensurate with the experimental data. Combining the cytosolic (C = U_c + P_c) and nuclear (N = U_n + P_n) species, we obtain from Eq. A1 the system

$$\begin{array}{c}\frac{\text{dC}}{\text{d}t}={\widehat{k}}_{\mathrm{E}}{\text{P}}_{\text{n}}-k_{\text{I}}{\mathrm{\text{U}}}_{\text{c}}\\ \frac{\text{dN}}{\text{d}t}=-\frac{{\text{V}}_{\text{c}}}{{\text{V}}_{\text{n}}}{\widehat{k}}_{\text{E}}{\text{P}}_{\text{n}}+\frac{{\text{V}}_{\text{c}}}{{\text{V}}_{\text{n}}}{k}_{\text{I}}{U}_{\text{c}}\end{array}$$ (A2)

where k̂_E = V_n/V_ck_E. We assume that the dephosphorylation/phosphorylation steps occur more rapidly than the nuclear-cytoplasmic flux (i.e., k_Pn, k_Un, k_Pc, and k_Uc >> k_I, k_E), implying a rapid equilibration that yields

$$\begin{array}{c}0=k_{\text{Pc}}{\text{U}}_{\text{c}}-k_{\text{Uc}}{\text{P}}_{\text{c}}\\ 0=k_{\text{Pn}}{\text{U}}_{\text{n}}-k_{\text{Un}}{\text{P}}_{\text{n}}\end{array}$$

so that P_n = [k_Pn/(k_Pn + k_Un)]N and U_c = [k_Uc/(k_Uc + k_Pc)]C, and then we can write the previous system (Eq. A2) as a two-variable system

$$\begin{array}{c}\frac{\text{dC}}{\text{d}t}=\epsilon (k_E^{\prime }\text{N}\hspace{0.17em}-k_I^{\prime }\text{C})\\ \frac{\text{dN}}{\text{d}t}=-k_E^{\prime }\text{N}\hspace{0.17em}+k_I^{\prime }\text{C}\end{array}$$ (A3)

where kE′ = [k_Pn/(k_Pn + k_Un)]k̂_E, kI′ = [k_Uc/(k_Uc + k_Pc)]k_I, and ε = V_n/V_c is a small parameter in our system on the order of 0.01–0.05. Conservation has C_T = C + εN, so we can reduce Eq. A3 to the single equation

$$\frac{\text{dN}}{\text{d}t}=-k_E^{\prime }\text{N}\hspace{0.17em}+k_I^{\prime }{\text{(C}}_{\text{T}}-\epsilon \text{N})=k_I^{\prime }{\text{C}}_{\text{T}}-(k_E^{\prime }+\epsilon k_I^{\prime })\text{N}$$ (A4)

which we can solve to obtain

$$\text{N}(t)={\text{C}}_{\text{T}}\left[\left(\frac{{\text{N}}_0}{{\text{C}}_{\text{T}}}-\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right)e^{-({k^{\prime }}_{\text{E}}+\epsilon k_I^{\prime })t}+\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right]$$

where N₀ is the initial condition for nuclear fluorescence. The nuclear-to-cytoplasmic ratio can then be written

$$\frac{\text{N}}{\text{C}}=\frac{\left[\left(\frac{{\text{N}}_0}{{\text{C}}_{\text{T}}}-\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right)e^{-(k_E^{\prime }+\epsilon k_I^{\prime })t}+\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right]}{\text{1}-\epsilon \left[\left(\frac{{\text{N}}_0}{{\text{C}}_{\text{T}}}-\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right)e^{-(k_E^{\prime }+\epsilon k_I^{\prime })t}+\frac{k_I^{\prime }}{k_E^{\prime }+\epsilon k_I^{\prime }}\right]}$$

In the limit that ε → 0 (cytosolic volume is much larger than nuclear volume), we obtain the form used in Eq. 5

$$\frac{\text{N}}{\text{C}}=\left(\frac{{\text{N}}_0}{{\text{C}}_{\text{T}}}-\frac{k_I^{\prime }}{k_E^{\prime }}\right)e^{-(k_E^{\prime })t}+\frac{k_I^{\prime }}{k_E^{\prime }}$$

Note that in this limit the constant cytosolic fluorescence need not be the total C_T but may simply be an initial constant C₀.