# Modeling the signaling endosome hypothesis: Why a drive to the nucleus is better than a (random) walk

- Charles L Howe
^{1}Email author

**2**:43

**DOI: **10.1186/1742-4682-2-43

© Howe; licensee BioMed Central Ltd. 2005

**Received: **01 September 2005

**Accepted: **19 October 2005

**Published: **19 October 2005

## Abstract

### Background

Information transfer from the plasma membrane to the nucleus is a universal cell biological property. Such information is generally encoded in the form of post-translationally modified protein messengers. Textbook signaling models typically depend upon the diffusion of molecular signals from the site of initiation at the plasma membrane to the site of effector function within the nucleus. However, such models fail to consider several critical constraints placed upon diffusion by the cellular milieu, including the likelihood of signal termination by dephosphorylation. In contrast, signaling associated with retrogradely transported membrane-bounded organelles such as endosomes provides a dephosphorylation-resistant mechanism for the vectorial transmission of molecular signals. We explore the relative efficiencies of signal diffusion versus retrograde transport of signaling endosomes.

### Results

Using large-scale Monte Carlo simulations of diffusing STAT-3 molecules coupled with probabilistic modeling of dephosphorylation kinetics we found that predicted theoretical measures of STAT-3 diffusion likely overestimate the effective range of this signal. Compared to the inherently nucleus-directed movement of retrogradely transported signaling endosomes, diffusion of STAT-3 becomes less efficient at information transfer in spatial domains greater than 200 nanometers from the plasma membrane.

### Conclusion

Our model suggests that cells might utilize two distinct information transmission paradigms: 1) fast local signaling via diffusion over spatial domains on the order of less than 200 nanometers; 2) long-distance signaling via information packets associated with the cytoskeletal transport apparatus. Our model supports previous observations suggesting that the signaling endosome hypothesis is a subset of a more general hypothesis that the most efficient mechanism for intracellular signaling-at-a-distance involves the association of signaling molecules with molecular motors that move along the cytoskeleton. Importantly, however, cytoskeletal association of membrane-bounded complexes containing ligand-occupied transmembrane receptors and downstream effector molecules provides the ability to regenerate signals at any point along the transmission path. We conclude that signaling endosomes provide unique information transmission properties relevant to all cell architectures, and we propose that the majority of relevant information transmitted from the plasma membrane to the nucleus will be found in association with organelles of endocytic origin.

## Background

A number of findings support the concept that signaling from internal cellular membranes is a general phenomenon that is relevant to understanding receptor tyrosine kinase signaling in many cellular systems. For example, EGFR, as discussed above, is internalized via clathrin-coated vesicles following EGF-binding and receptor activation [13–15]. In the past, trafficking through this compartment was considered part of a normal degradative process that removes activated receptors from the plasma membrane and thereby truncates and controls downstream signaling [16]. But while this certainly remains a critical function of endocytosis, recent experiments demonstrate that EGFR remains phosphorylated and active following internalization [17], and that downstream signaling partners such as Ras colocalize with these internalized, endosome-associated receptors [18–23]. Moreover, the signals emanating from these internalized EGFR are biologically meaningful, as cell survival is directly supported by such signaling [24]. Likewise, Bild and colleagues recently observed that STAT-3 signaling initiated by EGFR activation localized to endocytic vesicles that moved from the plasma membrane to the nucleus, and they found that inhibition of EGFR endocytosis prevented STAT-3 nuclear translocation and abrogated STAT-3-mediated gene transcription [25]. However, while evidence supports the existence of signaling endosomes, it does not rule out simultaneous diffusion-based signal transduction.

## Results and discussion

### Assumptions – Transport Velocity

For modeling, a dynein-based transport rate of 5 microns per second is assumed, based on a report by Kikushima and colleagues [27]. This value was used for ease of calculation: with a cell radius of 7.5 microns and a nuclear radius of 2.5 microns, a 5 μm per second transport rate moves the signaling endosome from the plasma membrane to the nucleus in one second. Actual transport rates likely range from 1–10 μm per second in cytosol or axoplasm [7].

### Assumptions – Diffusion Coefficient

The crystal structure of STAT-3B [28], deposited in the Protein Data Bank as PDB 1BG1 [29], indicates unit cell dimensions of 17.4 × 17.4 × 7.9 nm. With the caveat that this structure is bound to an 18-base nucleic acid, the volume of a STAT-3B molecule is 2400 nm^{3}. Assuming a spherical molecule, STAT-3B therefore has a molecular radius of approximately 8 nm. Likewise, the molecular weight of STAT-3 is 100000 Daltons, and therefore one molecule of STAT-3 weighs 1.7 × 10^{-19} g. The Einstein-Stokes equation for the coefficient of diffusion is:

*D* = (1/8)(*k*·*T*)/(π·γ·η)

where *k* is Boltzmann's constant, *T* is absolute temperature in degrees Kelvin, γ is the radius of the molecule, and η is the viscosity of an isotropic medium. The viscosity of axoplasm is approximately 5 centipoise [30], a value that also approximates cytoplasm [31, 32]. Hence,

*k* = 1.3805 × 10^{-20} m^{2}·g·(1/(s^{2}·K))

*T* = 310 K

γ = 8 × 10^{-9} m

*m* = 1.7 × 10^{-19} g

η = 5 g/(m·s)

Therefore, the coefficient of diffusion for a molecule of STAT-3 is:

*D* = 4.3 μm^{2} per second

Likewise, the instantaneous velocity *v*_{
x
}, the step length δ, and the step rate τ, were derived as:

*v*_{
x
} = ((*k*·*T*)/*m*)^{0.5} = 5 m/s

δ = (1/4)(*k*·*T*)/(*v*_{
x
}·π·γ·η) = 1.7 × 10^{-12} m

τ = *v*_{
x
}/δ = 2.9 × 10^{12} sec^{-1}

It is important to note that our mass estimation may substantially underestimate the actual mass of the functional STAT-3 molecular complex, described by Sehgal and colleagues as two populations with masses ranging from 200–400 kDa ("Statosome I") to 1–2 MDa ("Statosome II") [33, 34]. Such a massive molecular complex certainly has important biological implications for STAT-3 diffusion. However, because no crystal structure exists for these higher molecular weight statosomes from which to calculate the molecular radius, and in order to calculate the "best-case scenario" for effective diffusion distance, we have calculated the STAT-3 diffusion coefficient on the basis of a 100 kDa monomeric molecule. The actual diffusion coefficient for STAT-3 may be 30% of the value calculated above (assuming 2 MDa mass and a four-fold increase in molecular radius to account for molecular packing of the statosome) and the root-mean-square displacement may be 50% of the value calculated below. The impact of these variables awaits further investigation.

### Assumptions – Diffusion Modeling

We modeled diffusion using a random walk algorithm in two dimensions. The choice of dimensionality was constrained by the intensive computational burden associated with three-dimensional algorithms, as discussed below (see Methods). At every iteration of the random walk two pseudo-random numbers (see Methods) were generated and used to determine the direction of movement in the x-y plane. Using the instantaneous velocity *v*_{
x
}, the step length δ, and the step rate τ, defined above, we conclude that a diffusing molecule of STAT-3 will randomly walk 3 × 10^{12} steps per second, and each step will be 1.7 × 10^{-12} meters long. Thus, the root-mean-square displacement for STAT-3 diffusion in one second is 2.9 μm. The random walk was modeled on one second of biological time using a loop of 3 × 10^{12} iterations. During each iteration the molecule randomly moved ± 1.7 × 10^{-12} meters in the x-plane and ± 1.7 × 10^{-12} meters in the y-plane.

### Assumptions – Dephosphorylation Kinetics

The decay of a phospho-protein is an exponential function mapped between the plasma membrane and the nucleus [5, 35]:

α^{2} = (*K*_{
p
})(*L*^{2}/*D*)

And the probability function for dephosphorylation is:

*p*(*x*)/*p*(*m*) = (e^{αx}– e^{-αx})/*x*(e^{α} – e^{-α})

Where α is a dimensionless measure of dephosphorylation probability, *K*_{
p
} is the first-order rate constant for the activity of the relevant phosphatase, *L* is the cell diameter, *D* is the diffusion coefficient, *x* is the distance from the cell center, and *m* is the distance from cell center to plasma membrane normalized to a value of one. α scales such that for α = 10, half of all phospho-molecules become dephosphorylated within approximately 0.075 units of distance from the plasma membrane to the cell center (e.g. 750 nm for a cell with 10 μm radius) [5]. In general, *K*_{
p
}, the first-order rate constant of phosphatase activity, varies between 0.1 per second and 10 per second [4, 35–37]. For our model *K*_{
p
} = 5 was assumed, yielding α = 8.1.

With regard to an estimate of enzymatic activity relevant to dephosphorylation of STAT-3, Todd and colleagues report a second-order rate constant of 40000/M·s for dephosphorylation of Erk1/2 [38], which gives:

*k*_{
cat
}/*k*_{
m
} = 40000/M·s

Furthermore, Denu and colleagues report that diphosphosphorylated Erk1/2 peptides exhibit *k*_{
m
} values of approximately 100 μM in vitro [39]. Therefore:

*k*_{
cat
} = 4/s

Since *k*_{
cat
} measures the number of substrate molecules turned over per enzyme per second, a *k*_{
cat
} of 4 per second means that, on average, each molecule of enzyme (phosphatase) converts (dephosphorylates) 4 substrate molecules every second. Assuming a degree of molecular similarity between Erk dephosphorylation and STAT-3 dephosphorylation, and for ease of calculation, we set *k*_{
cat
} = 5 per second. It is important to note that this assumption may not be valid, but has been necessarily adopted in the absence of better biophysical data in order to illustrate the potential circumscription of diffusion by dephosphorylation.

### Assumptions – Dephosphorylation Modeling

The random walk employed for modeling STAT-3 diffusion depends upon the massively iterative generation of random numbers to describe the movement of the walking molecule in two-dimensional space. Since significant computational time was already invested in our diffusion calculations for the generation of extremely long period pseudo-random numbers, we opted to model STAT-3 dephosphorylation as a stochastic event using the following logic: for any given randomly walking molecule, the probability of encountering a phosphatase is independent of both all other molecules and all other steps in the walk. Therefore, during one second of biological time, equivalent to 3 × 10^{12} steps in the random walk, and assuming that *k*_{
cat
} = 5 dephosphorylations per second, there will be 1.67 × 10^{-12} dephosphorylation events per step. This can be effectively modeled as a probability test by generating a pseudo-random number on (0,1) at each step of the random walk and asking whether this number is less than 1.67 × 10^{-12}. If the test is positive, the molecule is considered to be "dephosphorylated" and the random walk is truncated. High-speed modeling of time to dephosphorylation for a large number of molecules (i.e. in the absence of the random walk) led to a probability function that matched the equations described by Kholodenko [5].

### Results – Diffusion-only Model

### Results – Diffusion and Dephosphorylation Model

### Results – Endpoint Analysis of Both Models

### Predictions

### Caveats and Future Directions

The signaling endosome retrograde transport rate utilized in our model may overestimate the actual transport velocity, especially as an average across the entire lifetime of the endosome-associated signal. The rate we modeled did not account for the kinetics of endocytosis or of vesicle loading onto the microtubule network. Our previous observations suggested transport velocities that ranged from 5.6 μm per second to 0.56 μm per second [7], but experiments addressing real transport rates for a variety of signaling molecules are required to improve our model. On the other hand, while we potentially overestimated the retrograde transport rate for the signaling endosome, we also very likely overestimated the size of the effective diffusion domain due to the two-dimensional restrictions of our current model. While the cytoskeletal transport of the signaling endosome is inherently a dimensionally-restricted vectorial event, diffusion within the cell most certainly occurs in three dimensions. Our current model predicts a three-fold reduction in the actual root-mean-square displacement for STAT-3 as compared to the predicted displacement using a two-dimensional random walk model, and we predict that a model incorporating three dimensions will exhibit even greater curtailment of the effective spatial domain for diffusion. However, the addition of a third dimension to the random walk simulations substantially increases computational demand, and therefore this analysis awaits either a more efficient algorithm or more computer time. Our current and future goals are to parallelize the random walk algorithm in order to perform massively parallel diffusion simulations in three dimensions.

## Conclusion

Molecular diffusion obviously benefits from the extremely high molecular velocities of single particles moving in a vacuum. For gases and other very small molecules and under conditions of low viscosity or high temperature, diffusion is extremely fast and far-ranging. However, within the context of biological molecules and biological viscosities, diffusion is vastly circumscribed [1–3, 40]. Despite the limitations imposed by biological parameters, diffusion at first glance still appears to be a viable mechanism for the transmission of information through cytoplasm. In fact, the "textbook" conception of signal transduction depends upon the free diffusion of signaling molecules. However, closer scrutiny finds several faults in the diffusion model [1]. For example, diffusion is certainly directionless – even within the context of a bounded space such as the cell, the majority of molecular motions taken by a diffusing molecule are non-productive with regard to movement of signals toward a target (such as the nucleus). Likewise, a diffusing molecular signal is a ready target for interaction with and truncation by cytoplasmic phosphatases. Certainly, the effective range over which a diffusing signal maintains informational integrity depends upon the concentration and activity of equally randomly diffusing phosphatases, but it also seems likely that cells maintain levels of phosphatase sufficient to prevent run-away signal transduction [41, 42]. Thus, diffusion of information is limited by both lack of direction and inevitable signal elimination. In distinct contrast, the retrograde movement of quantal signaling units capable of regenerating the information content of the original stimulus is inherently vectorial. Therefore, signaling endosomes, despite an overall lower transport velocity compared to diffusion velocities, exhibit characteristics of an optimized information transmission system. We previously sought to determine the effective range over which Erk1/2 signaling endosomes exhibited greater efficiency than diffusing Erk1/2 molecules [7]. This work relied upon the direct comparison of the root-mean-square displacement for phosphorylated Erk1/2 with the retrograde transport velocity of neurotrophin-induced signaling endosomes. In an effort to refine this model we incorporated in our present study the additional element of dephosphorylation kinetics. Thus our current model addresses both the non-vectorial nature of diffusion and the inherent susceptibility to signal truncation by interaction with cellular phosphatases. Using an iterative random walk modeling scheme we determined that the root-mean-square displacement predicted by the coefficient of diffusion for STAT-3 overestimated the root-mean-square displacement observed in our simulations by a factor of 3. Incorporating this scaling factor into the equation for root-mean-square displacement through time, we found that signaling endosomes become more effective at the transmission of information when the distance from the plasma membrane exceeds 200 nanometers. This observation suggests that any cellular situation that requires the transmission of information in the form of phosphorylated signaling molecules over distances in excess of 200 nanometers would benefit from the packaging of such signals into quantal, cytoskeleton-associated signaling packets such as signaling endosomes.

Our model suggests that cells utilize two distinct information transmission paradigms: 1) fast local signaling via diffusion over spatial domains on the order of less than 200 nanometers; 2) long-distance (>200 nanometers) signaling via information packets associated with the cytoskeletal transport apparatus. Moreover, while we have focused explicitly on the role of signaling endosomes derived from the internalization of plasma membrane receptor tyrosine kinases and associated downstream signaling partners, our model suggests that any signal that must move from the outer reaches of the cytoplasm to the perinuclear region would benefit from an association with the retrograde transport machine. For example, transcription factors may associate directly with molecular motors and chaperone proteins that protect them from dephosphorylation in a nonvesiculated manner that takes advantage of directional retrograde transport in the absence of a plasma-membrane-derived organelle. Such a mechanism was recently proposed for the transport of soluble (i.e. non-membrane-associated) activated Erk1/2 within injured axons [43]. Thus, our model supports previous observations suggesting that the signaling endosome hypothesis is a subset of a more general hypothesis that the most efficient mechanism for intracellular signaling-at-a-distance involves the association of signaling molecules with molecular motors that move along the cytoskeleton [4]. The additional benefit provided by the cytoskeletal association of membrane-bounded complexes that package a ligand-bound transmembrane receptor with downstream effector molecules is the ability to regenerate the signal at any point along the transmission path [7]. We conclude that signaling endosomes provide unique information transmission properties relevant to all cell architectures, and we propose that the majority of relevant information transmitted from the plasma membrane to the nucleus will be found in association with organelles of endocytic origin.

## Methods

### Pseudo-Random Number Generation

It should be self-evident that "built-in" pseudo-random number generators (RNGs) available in the majority of operating systems and programming languages are essentially useless for large-scale Monte Carlo simulations [44]. However, during our initial efforts to optimize the processing time for the one-second simulations we experimented with several common RNGs; all failed to exhibit sufficiently long periods, a failure that was manifested in an initial period of random walking followed by capture in a continuously repeating cyclical path. We also experimented with an implementation of the Mersenne Twister algorithm, which exhibited a robust period (theoretically 2^{19937}-1) and computational demand comparable to many other standard RNGs [45]. However, our final optimized diffusion-only code utilized a multiply-with-carry RNG (MWC) described by George Marsaglia [44, 46, 47]. The MWC algorithm generates extremely long-period pseudo-random numbers on [0,1], and we utilized this very efficient RNG for Boolean testing of step direction in two dimensions. For the combined diffusion and dephosphorylation models, we used the Mersenne Twister modified to generate pseudo-random numbers on (0,1) for the probabilistic determination of a dephosphorylation event and the MWC algorithm for step direction determination.

### Hardware

We utilized a variety of platforms for development, testing, and implementation of the diffusion models, including the IBM Power4 p690 supercomputer (running AIX 5.2) and the SGI Altix 3700 supercomputer (running SGI Advanced Linux 3.4) at the University of Minnesota Supercomputing Institute. The serial models described above were primarily implemented on a single processor Intel P4 3.0 GHz machine running Red Hat Linux 9.0. The IBM Power4, the SGI Altix 3700, and a dual processor Xeon 3.0 GHz Nocona box running Red Hat Enterprise Linux 3.0 were used for development and testing of parallel implementations. Total wallclock time on all platforms currently exceeds 10000 hours.

### Software

All algorithms were coded in C and compiled with gcc or xlc (serial implementations) or with pgcc, xlc, or icc (OpenMP parallel implementations). Our first diffusion model efforts required more than one week of dedicated processing time per walk; after several rounds of code optimization we could obtain one second of simulated time in approximately 48 hours on the Power4 architecture and the Pentium 4 architecture described above.

## Declarations

### Acknowledgements

The author thanks the University of Minnesota Supercomputing Institute (MSI) http://www.msi.umn.edu for access to the IBM Power4 pSeries 690 and to the SGI Altix supercomputers. The author also thanks Dr. Birali Runesha of the MSI for technical assistance. This work was supported by Donald and Frances Herdrich and by grant RG3636 from the National Multiple Sclerosis Society.

## Authors’ Affiliations

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