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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.


Simulation of the feed forward serial network Equation (5) in response to a retrograde signal from Equation (2). (A) Retrograde signal being fed into gene network, τ = 5. (B) Convergence of the solutions of Equation (5) to a T-periodic solution post transience. h[u] is taken to be the same function as f[u] defined in Equation (3) multiplied by a factor of 1000, and we set λ = 0.01. Other parameter values are as in Figure 2.
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Figure 3: Simulation of the feed forward serial network Equation (5) in response to a retrograde signal from Equation (2). (A) Retrograde signal being fed into gene network, τ = 5. (B) Convergence of the solutions of Equation (5) to a T-periodic solution post transience. h[u] is taken to be the same function as f[u] defined in Equation (3) multiplied by a factor of 1000, and we set λ = 0.01. Other parameter values are as in Figure 2.

Mentions: Hence, c(t) converges to a T-periodic solution following any transient dynamics, as shown in Figure 3. More significantly, there is now a strong DC component to the signal so that the relative amplitude of the oscillatory part has been suppressed. Indeed it is possible to find parameter values for which , where is the time-averaged protein output (Krakauer et al., 2002). Therefore, in order to characterize the protein output in terms of the frequency ω of uI(t), we find the time average of c(t) post transience. This can be done by simply integrating Equation (5) over a period of uI(t):


A frequency-dependent decoding mechanism for axonal length sensing.

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

Simulation of the feed forward serial network Equation (5) in response to a retrograde signal from Equation (2). (A) Retrograde signal being fed into gene network, τ = 5. (B) Convergence of the solutions of Equation (5) to a T-periodic solution post transience. h[u] is taken to be the same function as f[u] defined in Equation (3) multiplied by a factor of 1000, and we set λ = 0.01. Other parameter values are as in Figure 2.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4508512&req=5

Figure 3: Simulation of the feed forward serial network Equation (5) in response to a retrograde signal from Equation (2). (A) Retrograde signal being fed into gene network, τ = 5. (B) Convergence of the solutions of Equation (5) to a T-periodic solution post transience. h[u] is taken to be the same function as f[u] defined in Equation (3) multiplied by a factor of 1000, and we set λ = 0.01. Other parameter values are as in Figure 2.
Mentions: Hence, c(t) converges to a T-periodic solution following any transient dynamics, as shown in Figure 3. More significantly, there is now a strong DC component to the signal so that the relative amplitude of the oscillatory part has been suppressed. Indeed it is possible to find parameter values for which , where is the time-averaged protein output (Krakauer et al., 2002). Therefore, in order to characterize the protein output in terms of the frequency ω of uI(t), we find the time average of c(t) post transience. This can be done by simply integrating Equation (5) over a period of uI(t):

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.