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A frequency-dependent decoding mechanism for axonal length sensing.

Bressloff PC, Karamched BR - Front Cell Neurosci (2015)

Bottom Line: We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network.If the protein output were thresholded, then this could provide a mechanism for axonal length control.We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Utah Salt Lake City, UT, USA.

ABSTRACT
We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

No MeSH data available.


Related in: MedlinePlus

Plot of estimated errors in axonal length based on 100 simulations of the chemical master Equation (14) using the Gillespie algorithm with input s(t) = h[uI(t)]. (A) Plot of uncertainty in axonal length ΔL vs. threshold axonal lengths L0. (B) Relative error (ΔL ∕ L0) vs. axonal length. Same parameters as Figure 6. L0 was found by averaging over the mean protein outputs and determining what length that protein value corresponded to according to the curve shown in Figure 4. ΔL was determined by looking at what axonal length each individual mean protein output realization corresponded to according to Equation (4) and then finding the variance in this set of values.
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Figure 8: Plot of estimated errors in axonal length based on 100 simulations of the chemical master Equation (14) using the Gillespie algorithm with input s(t) = h[uI(t)]. (A) Plot of uncertainty in axonal length ΔL vs. threshold axonal lengths L0. (B) Relative error (ΔL ∕ L0) vs. axonal length. Same parameters as Figure 6. L0 was found by averaging over the mean protein outputs and determining what length that protein value corresponded to according to the curve shown in Figure 4. ΔL was determined by looking at what axonal length each individual mean protein output realization corresponded to according to Equation (4) and then finding the variance in this set of values.

Mentions: with T0 the oscillation period at the critical length L0, i.e., L(T0) = L0. Assuming that the length L increases at least linearly with T, we see that the relative error grows with the critical oscillation period T0 and, hence, the critical axonal length L0. Although this is a crude estimate, we find that the same qualitative behavior is observed in numerical simulations of the full stochastic model. This is shown in Figure 8, where we plot the relative error ΔL ∕ L0 vs. axonal length. Our analysis suggests that the frequency-encoded protein threshold mechanism could break down for long axons. An analogous result was shown to hold in Karamched and Bressloff (2015), where the robustness of the encoding of axonal length in the frequency of a pulsatile signal was investigated. There we found that the encoding of axonal length into frequency became less reliable at long axon lengths due to accumulation of white noise signified by a high coefficient of variation in the frequency of the retrograde signal. In this work, the retrograde signal is deterministic, and the error in protein output is accounted for strictly by the random variations in the activities of independent gene promoters. Hence the error in length sensing could be more devastating in real life situations, since noise would impact both the encoding and the decoding processes. Thus, wherever the sources of noise may be, their impact on this frequency-dependent mechanism is clear: large neurons would have a more difficult time sensing their own length when compared with smaller neurons.


A frequency-dependent decoding mechanism for axonal length sensing.

Bressloff PC, Karamched BR - Front Cell Neurosci (2015)

Plot of estimated errors in axonal length based on 100 simulations of the chemical master Equation (14) using the Gillespie algorithm with input s(t) = h[uI(t)]. (A) Plot of uncertainty in axonal length ΔL vs. threshold axonal lengths L0. (B) Relative error (ΔL ∕ L0) vs. axonal length. Same parameters as Figure 6. L0 was found by averaging over the mean protein outputs and determining what length that protein value corresponded to according to the curve shown in Figure 4. ΔL was determined by looking at what axonal length each individual mean protein output realization corresponded to according to Equation (4) and then finding the variance in this set of values.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4508512&req=5

Figure 8: Plot of estimated errors in axonal length based on 100 simulations of the chemical master Equation (14) using the Gillespie algorithm with input s(t) = h[uI(t)]. (A) Plot of uncertainty in axonal length ΔL vs. threshold axonal lengths L0. (B) Relative error (ΔL ∕ L0) vs. axonal length. Same parameters as Figure 6. L0 was found by averaging over the mean protein outputs and determining what length that protein value corresponded to according to the curve shown in Figure 4. ΔL was determined by looking at what axonal length each individual mean protein output realization corresponded to according to Equation (4) and then finding the variance in this set of values.
Mentions: with T0 the oscillation period at the critical length L0, i.e., L(T0) = L0. Assuming that the length L increases at least linearly with T, we see that the relative error grows with the critical oscillation period T0 and, hence, the critical axonal length L0. Although this is a crude estimate, we find that the same qualitative behavior is observed in numerical simulations of the full stochastic model. This is shown in Figure 8, where we plot the relative error ΔL ∕ L0 vs. axonal length. Our analysis suggests that the frequency-encoded protein threshold mechanism could break down for long axons. An analogous result was shown to hold in Karamched and Bressloff (2015), where the robustness of the encoding of axonal length in the frequency of a pulsatile signal was investigated. There we found that the encoding of axonal length into frequency became less reliable at long axon lengths due to accumulation of white noise signified by a high coefficient of variation in the frequency of the retrograde signal. In this work, the retrograde signal is deterministic, and the error in protein output is accounted for strictly by the random variations in the activities of independent gene promoters. Hence the error in length sensing could be more devastating in real life situations, since noise would impact both the encoding and the decoding processes. Thus, wherever the sources of noise may be, their impact on this frequency-dependent mechanism is clear: large neurons would have a more difficult time sensing their own length when compared with smaller neurons.

Bottom Line: We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network.If the protein output were thresholded, then this could provide a mechanism for axonal length control.We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Utah Salt Lake City, UT, USA.

ABSTRACT
We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

No MeSH data available.


Related in: MedlinePlus