Do Protein Cascades Increase Protein Concentration

Abstract

Signaling pathways consisting of phosphorylation/dephosphorylation cycles with no explicit feedback allow signals to propagate not but from upstream to downstream but besides from downstream to upstream due to retroactivity at the interconnection between phosphorylation/dephosphorylation cycles. Nevertheless, the extent to which a downstream perturbation can propagate upstream in a signaling cascade and the parameters that affect this propagation are presently unknown. Hither, we determine the downstream-to-upstream steady-state proceeds at each phase of the signaling cascade as a function of the cascade parameters. This gain can be fabricated smaller than 1 (attenuation) by sufficiently fast kinase rates compared to the phosphatase rates and/or by sufficiently large Michaelis-Menten constants and sufficiently low amounts of full stage protein. Numerical studies performed on sets of biologically relevant parameters indicated that ∼50% of these parameters could give rise to amplification of the downstream perturbation at some stage in a three-phase cascade. In an north-phase cascade, the pct of parameters that lead to an overall attenuation from the last phase to the first phase monotonically increases with the cascade length n and reaches 100% for cascades of length at to the lowest degree 6.

Introduction

Signaling pathways are ubiquitous in living systems and comprehend a central role in a cell'south ability to sense and respond to both external and internal input stimuli (

one

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,

). Numerous signaling pathways consist of cycles of reversible poly peptide modification, such as phosphorylation/dephosphorylation (PD) cycles, wherein a poly peptide is converted, reversibly, between 2 forms (

). Multiple PD cycles ofttimes appear continued in a cascade mode, such as in the MAPK cascades (

,

), and the length of the pour has been shown to have of import furnishings, for example, on signal amplification, point duration, and signaling fourth dimension (

,

,

). In particular, a wealth of work has been employing metabolic control analysis (MCA) approaches to determine analytically the distension gains across the cascade as a small perturbation applied at the top of the cascade propagates toward the bottom stages (

,

,

). To our knowledge, no study has been performed on how perturbations at the bottom of a pour propagate toward the superlative of the cascade.

Because cascades often intersect each other by sharing common components, such every bit protein substrates or kinases (

,

), perturbations at bottom or intermediate stages in a cascade tin frequently occur. These intersections are already known to cause unwanted crosstalk between the signaling stages downstream of the intersection point (

,

,

,

). Still, no attention was given to crosstalk between the stages upstream of the intersection point. Several of these works, in fact, viewed a signaling cascade as the modular composition of PD cycles, resulting in a system where the signal travels simply from upstream to downstream. Theoretical work, however, has shown that PD cycles (every bit several other biomolecular systems) cannot exist modularly continued with each other because of retroactivity effects at interconnections (

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,

,

). Initial experimental validation of these effects on the steady-state response of a PD cycle take besides appeared (

,

,

). These effects modify the behavior of an upstream organization when it is continued to its downstream clients and are relevant especially in signaling cascades, in which each PD cycle has several downstream targets. Equally a result of retroactivity, signaling cascades allow signals to also travel from downstream to upstream, that is, they permit bidirectional signal propagation (

,

). Every bit a result, a perturbation at the bottom of the cascade can propagate to the upstream stages and have repercussions on the overall signaling.

A perturbation at the bottom of a cascade tin exist due to a number of factors. For example, when a downstream target or a substrate is shared with other signaling pathways, its free concentration is perturbed by these other pathways. Hence, the amount of target/substrate available to the pour under report can suddenly modify. Similarly, the introduction of an inhibitor of an active enzyme, equally performed in targeted drug design, creates a perturbation at the targeted phase of the cascade.

How large is the effect of such perturbations on the upstream stages? How does the length of a cascade bear on backward signal transfer?

Answering these questions will reveal the extent to which aberrant signaling in the upstream stages of a cascade tin be acquired past retroactivity from sharing downstream targets/substrates. It will also provide tools for targeted drug design by quantifying the off-target furnishings of inhibitors on the upstream stages.

In this article, we address these questions in cascades with a single phosphorylation cycle per stage past explicitly incorporating retroactivity in the PD cycle model. Specifically, we consider pocket-sized perturbations at the bottom of the pour and explicitly quantify, to our knowledge, for the first time how such perturbations propagate from downstream to upstream. Our principal results are as follows. We provide analytical expressions for the downstream-to-upstream transmission gains. These establish the extent to which a perturbation at the bottom of the pour can propagate upstream and provide sufficient atmospheric condition for attenuation. Through all-encompassing numerical simulation, we discovered that, surprisingly, natural cascades tin can dilate a perturbation equally it propagates upstream, but the probability of attenuation is substantially higher than that of amplification. In addition, the probability of attenuation increases with the number of stages in the pour.

Methods

Nosotros consider a signaling pour composed of n phosphorylation/dephosphorylation (PD) cycles as depicted in Fig. i. The sensitivity of response to perturbations occurring at the peak of the cascade, for example in W ^∗ ₀, has been extensively studied employing MCA approaches (

,

,

). Past dissimilarity, here we investigate the sensitivity of response of each wheel to a perturbation at the bottom of the pour. This perturbation can be due, for case, to an inhibitor of the active enzyme West ^∗ n, as it is employed in targeted drug design (

) or to the signaling from some other pathway sharing a substrate with W ^∗ due north. Our method is based on assuming a small perturbation, on linearizing the system dynamics about the steady state, and on determining the corresponding change of each bicycle phosphorylated poly peptide. Because our approach is based on linearization, it is similar in spirit to MCA approaches, which also assume small perturbations and linearize the organization dynamics. Here, nosotros are interested in determining how finer the perturbation propagates upstream. We thus explicitly compute the sensitivity gain from i phase to the side by side upstream as a role of the pour parameters.

Figure 1 A signaling cascade with northward stages of PD cycles. The phosphorylated protein Westward^∗ i _–1 of phase i–1 functions equally a kinase for protein Wi of the next phase downstream. Dephosphorylation is brought about past the phosphatase Easti. A downstream perturbation in the concentration of D, in which D can be a substrate shared with other signaling pathways or an inhibitor of the active enzyme Due west^∗ n, results in a perturbation of protein concentration in all upstream stages.

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Cascade model

At each stage i, for i ∈{1,…,n}, we denote by Westward^∗ i _−one the kinase, by Easti the phosphatase, by Wi the protein substrate, and by Westward∗i the phosphorylated form of Westwardi. The kinase Due west^∗ i _−ane binds to Westi to grade the substrate-kinase complex Teni. This circuitous then turns into Westward∗i. The phosphorylated protein W∗i is, in turn, a kinase for the side by side cycle and binds to downstream substrates, forming the circuitous 10i ₊₁. The phosphatase Ei activates the dephosphorylation of the poly peptide W∗i by binding to Due west∗i and forming the complex Yi. This complex is in turn converted to Wi. We use the following two-step reaction model for each phosphorylation and dephosphorylation reaction (

,

) at stage i ∈{1,…,due north} of the pour:

${\text{W}}_i+{\text{Due west}}_{i-1}^{\ast }\underset{{\overline{a}}_i}{\overset{a_i}{\rightleftharpoons }}{\text{X}}_i\stackrel{g_i}{\to }{\text{West}}_i^{\ast }+{\text{W}}_{i-1}^{\ast },$

We assume that protein Wi and phosphatase Ei are conserved at every phase, and are in total amounts WiT and EiT, respectively. Therefore, we accept the conservation relations

$\begin{array}{l}{W}_i+W_i^{\ast }+{\mathrm{10}}_i+Y_i+X_{i+1}={\mathrm{Due\; west}}_{iT},\hfill \\ E_i+Y_i=E_{iT},\hfill \end{array}$

(ane)

in which, for a species X, we have denoted past X its concentration. Nosotros presume that the input kinase to the showtime phase, W∗ ₀, is produced at rate 1000(t) and decays at rate δ, that is,

Finally, we assume that the output poly peptide of the last stage, Westward∗n, reacts with species D downstream of the cascade. These species D tin model, for case, a signaling molecule or an inhibitor of the active enzyme W^∗ n (a drug), such equally considered in targeted drug design (

), in which the total concentration of D tin be perturbed, for case, by adding more drug. Species D tin can also model a substrate that is shared with other signaling pathways. In this instance, D is a substrate for another active enzyme, say Due south, whose concentration is controlled past another signaling cascade. Hence, the amount of complimentary D plus the amount of D bound to W^∗ due north, which nosotros call DT, can exist perturbed (it tin increase or decrease) by a modify in the concentration of the agile enzyme S. Denoting by Tenn ₊₁ the complex formed past Due west^∗ n and D, we have that

${\text{W}}_{\mathrm{due\; north}}^{\ast }+\text{D}\underset{{\overline{a}}_{\mathrm{north}+1}}{\overset{a_{\mathrm{northward}+1}}{\rightleftharpoons }}{\text{X}}_{n+\mathrm{ane}}\text{with}{D}_T\triangleq D+X_{n+1}.$

In this written report, we consider DT as the parameter to exist perturbed and summate the sensitivity of the steady-state response of each bicycle's active poly peptide to small perturbations in DT _.

The differential equations that describe the dynamics of the cascade are given, for i ∈{1,…,n}, by

$\begin{array}{l}{\dot{\mathrm{Westward}}}_0^{\ast }=-\delta W_0^{\ast }+k\left(t\right)-\begin{array}{c}\left(a_1{\mathrm{Westward}}_0^{\ast }{W}_{\mathrm{ane}}-\left({\overline{a}}_{\mathrm{i}}+{\mathrm{thou}}_{\mathrm{i}}\right)X_{\mathrm{i}}\right)\end{array}\hfill \\ {\dot{X}}_i=a_i{\mathrm{West}}_{i-1}^{\ast }{\mathrm{West}}_i-\left({\overline{a}}_i+{\mathrm{grand}}_i\right)X_i\hfill \\ {\dot{\mathrm{Due\; west}}}_i^{\ast }=k_i{X}_i-b_i{W}_i^{\ast }{\mathrm{Due\; east}}_i+{\overline{b}}_i{Y}_i-\begin{array}{c}\left(a_{i+\mathrm{ane}}{W}_i^{\ast }{W}_{i+1}-\left({\overline{a}}_{i+1}+g_{i+1}\right)X_{i+1}\right)\end{array}\hfill \\ {\dot{Y}}_i=b_i{\mathrm{Due\; west}}_i^{\ast }{E}_i-\left({\overline{b}}_i+{\overline{k}}_i\right)Y_i\hfill \\ {\dot{W}}_n^{\ast }=k_{\mathrm{due\; north}}{\mathrm{Ten}}_n-b_n{\mathrm{Due\; west}}_n^{\ast }{E}_{\mathrm{due\; north}}+{\overline{b}}_{\mathrm{north}}{Y}_n-\begin{array}{c}\left(a_{n+1}D{\mathrm{Westward}}_{\mathrm{north}}^{\ast }-{\overline{a}}_{n+\mathrm{one}}{\mathrm{10}}_{n+\mathrm{i}}\right)\end{array}\hfill \\ {\dot{X}}_{n+\mathrm{ane}}=a_{n+1}D{W}_{\mathrm{north}}^{\ast }-{\overline{a}}_{n+\mathrm{one}}{X}_{\mathrm{due\; north}+1}.\hfill \end{array}$

Recognizing that the terms in the boxes correspond to

${\dot{\mathrm{Ten}}}_1$

,

${\dot{X}}_{i+1}$

, and

${\dot{\mathrm{10}}}_{n+1}$

, respectively, and employing the conservation law (Eq. 1), nosotros obtain for i ∈{1,…,n} that

$\begin{array}{l}{\dot{\mathrm{Due\; west}}}_0^{\ast }=-\delta {\mathrm{Due\; west}}_0^{\ast }+m\left(t\right)-{\dot{\mathrm{10}}}_1\hfill \\ {\dot{\mathrm{10}}}_i=a_i{W}_{i-\mathrm{i}}^{\ast }\left(W_{iT}-W_i^{\ast }-{\mathrm{10}}_i-Y_i-X_{i+1}\right)-\left({\overline{a}}_i+{\mathrm{chiliad}}_i\right){\mathrm{Ten}}_i\hfill \\ {\dot{W}}_i^{\ast }=k_i{X}_i-b_i{W}_i^{\ast }\left({\mathrm{East}}_{iT}-Y_i\right)+{\overline{b}}_i{Y}_i-{\dot{\mathrm{Ten}}}_{i+\mathrm{i}}\hfill \\ {\dot{Y}}_i=b_i{W}_i^{\ast }\left(E_{iT}-Y_i\right)-\left({\overline{b}}_i+{\overline{k}}_i\right)Y_i\hfill \\ {\dot{X}}_{n+1}=a_{n+1}\left(D_T-X_{n+1}\right)W_n^{\ast }-{\overline{a}}_{\mathrm{northward}+1}{\mathrm{Ten}}_{\mathrm{north}+\mathrm{one}}.\hfill \end{array}$

(two)

Perturbation analysis

In this section, we consider the cascade to be at the steady country and investigate how a minor perturbation in the concentration DT perturbs the steady-land concentrations at every stage of the cascade. We denote the steady-land value of the upstream input k(t) past

$\overline{k}$

and that of DT by

${\overline{D}}_T$

. The corresponding equilibrium values of the poly peptide concentrations

for i ∈{1,…,n}, and Xn ₊₁ are denoted by

for i ∈{one,…,n}, and

${\overline{X}}_{n+1}$

, respectively. We represent the perturbation of DT with respect to its steady-state value past

$d_T=D_T-{\overline{D}}_T$

. Note that if dT > 0, the downstream perturbation is positive, that is, the concentration DT increases. If instead dT < 0, the downstream perturbation is negative, that is, the concentration DT decreases. Hence, both positive and negative perturbations are considered. The corresponding perturbations of the states of the cascade about the equilibrium values

for i ∈{1,…,n}, and

${\overline{\mathrm{Ten}}}_{n+1}$

are denoted by

for i ∈{ane,…,n}, and x _n+one, respectively. Similarly, denote past Zi for i ∈{1,…,northward} the concentration of the total phosphorylated poly peptide at stage i, that is, Zi =W ^∗ i + Yi +Xi ₊₁. Denote the corresponding perturbation about the steady state

past zi, which tin can be written as

for all i ∈{1,…,northward}.

The linearization of the organisation in Eq. 2 almost the equilibrium

${\overline{W}}_0^{\ast },{\overline{W}}_i^{\ast },{\overline{\mathrm{Ten}}}_i,{\overline{Y}}_i,\text{and}{\overline{\mathrm{10}}}_{n+\mathrm{ane}}$

for i ∈{1,…,n} is given past

$\begin{array}{l}{\dot{w}}_0^{\ast }=-\delta {\mathrm{due\; west}}_0^{\ast }-{\dot{\mathrm{10}}}_1\hfill \\ {\dot{x}}_i=a_i{\overline{\mathrm{Due\; west}}}_i{w}_{i-\mathrm{i}}^{\ast }+a_i{\overline{W}}_{i-\mathrm{one}}^{\ast }\left(-w_i^{\ast }-x_i-y_i-x_{i+1}\right)-\left(\overline{a_i}+k_i\right)x_i\hfill \\ {\dot{w}}_i^{\ast }=g_i{x}_i+b_i{\overline{W}}_i^{\ast }{y}_i-b_i{\overline{E}}_i{w}_i^{\ast }+\overline{b_i}{y}_i-{\dot{\mathrm{10}}}_{i+\mathrm{i}}\hfill \\ {\dot{y}}_i=-b_i{\overline{W}}_i^{\ast }{y}_i+b_i{\overline{E}}_i{w}_i^{\ast }-\left(\overline{b_i}+\overline{{\mathrm{chiliad}}_i}\right)y_i\hfill \\ {\dot{x}}_{\mathrm{due\; north}+\mathrm{ane}}=a_{n+\mathrm{one}}\overline{D}{\mathrm{due\; west}}_n^{\ast }-a_{n+1}{\overline{\mathrm{West}}}_n^{\ast }{x}_{\mathrm{northward}+\mathrm{ane}}-\overline{a_{n+1}}{\mathrm{10}}_{\mathrm{due\; north}+1}+a_{\mathrm{due\; north}+1}{\overline{W}}_n^{\ast }{d}_T,\hfill \end{array}$

(3)

in which we have for i ∈{1,…,n} that (from setting the time derivatives in the expressions in Eq. 2 equal to zero)

${\overline{W}}_i=K_i\frac{{\overline{k}}_i}{{\mathrm{thou}}_i}\frac{\overline{E_i}}{{\overline{W}}_i^{\ast }+{\overline{K}}_i}\left(1+\frac{{\overline{\mathrm{Westward}}}_i^{\ast }}{{\overline{M}}_i}\right)\frac{{\overline{\mathrm{West}}}_i^{\ast }}{{\overline{W}}_{i-1}^{\ast }},$

(5)

in which

is the Michaelis-Menten constant of the dephosphorylation reaction, while

is the Michaelis-Menten constant of the phosphorylation reaction.

Because nosotros are interested in the steady-state values of w∗i, we set the time derivatives to nil in organization in Eq. 3 to obtain

$w_i^{\ast }=T_i\left({\overline{W}}_i{w}_{i-1}^{\ast }-{\overline{W}}_{i-\mathrm{i}}^{\ast }{x}_{i+1}\right),$

(9)

for i ∈{one,…,n}, in which

${\tilde{E}}_i\triangleq \frac{{\overline{K}}_i{\mathrm{East}}_{iT}}{{\left({\overline{W}}_i^{\ast }+{\overline{M}}_i\right)}^2},$

(10)

$T_i\triangleq \frac{1}{{\overline{\mathrm{Due\; west}}}_{i-1}^{\ast }+{\tilde{E}}_i\left({\overline{W}}_{i-1}^{\ast }+\frac{\overline{k_i}}{{\mathrm{thousand}}_i}\left({\overline{\mathrm{Due\; west}}}_{i-1}^{\ast }+{\mathrm{Thou}}_i\right)\right)}.$

(xi)

Fig. 2 represents (8), (ix) in a block diagram form, which highlights the directionality of bespeak propagation through the stages in the cascade. Basically, the perturbation dT propagates upstream in the cascade through perturbations in the concentrations 11 _. Hence, in this steady-country response model, retroactivity is due to the complex Ten_i of the active poly peptide with its downstream substrate.

Effigy 2 A cake diagram representation of the steady-country response of stage i to a minor downstream perturbation in DT _. The downstream perturbation propagates upstream through perturbations 11 in the complexes of active proteins with their downstream substrates.

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Results

Analytical results

Referring to Fig. 2, the perturbation dT propagates upstream through perturbations eleven and causes perturbations zi and w ^∗ i in the full and free phosphorylated protein concentrations, respectively, at every stage. How do these perturbations transfer from one stage of the cascade to the next one upstream?

To reply this question, we calculate the gains

${\Phi }_i=\frac{\left|z_i\right|}{\left|z_{i+1}\right|}\text{and}{\Psi }_i=\frac{\left|{\mathrm{due\; west}}_i^{\ast }\right|}{\left|{\mathrm{westward}}_{i+1}^{\ast }\right|}$

at every stage i. A gain >ane means that pocket-sized perturbations are amplified as they transfer from downstream to upstream, while a gain <1 means that small-scale perturbations are attenuated as they transfer from downstream to upstream.

Because |zi| = Φi|zi _+ane|, nosotros accept that

where ∏ denotes multiplication. We thus define the total gain Φtot from stage n to phase 1 as

Similarly, the full gain Ψtot from stage n to stage 1 is divers equally

Having a total gain <1 ways that, overall, the cascade attenuates downstream perturbations, even if some stages may amplify the perturbation.

We first focus on the gains Φi of total agile protein concentration. The total active protein concentration can exist experimentally determined by measuring protein activity through phosphospecific antibodies (

). By contrast, the free active protein may be more than difficult to measure out. When it is an active transcription factor, it can exist measured indirectly, for case, by placing a reporter cistron under the control of the promoter that information technology regulates. The expression of the gain Φi at each stage i tin be explicitly calculated as a office of the pour parameters from the relations in the block diagram of Fig. 2 (encounter the Supporting Cloth). This expression is given by

${\Phi }_i=\left(\frac{{\tilde{E}}_i\frac{{\overline{k}}_i}{k_i}+F_i}{1+{\tilde{E}}_i+{\tilde{E}}_i\frac{{\overline{\mathrm{one\; thousand}}}_i}{{\mathrm{chiliad}}_i}+F_i}\right)\left(\frac{\frac{{\overline{k}}_{i+1}}{{\mathrm{grand}}_{i+\mathrm{i}}}{\widetilde{\mathrm{Eastward}}}_{i+1}}{\frac{{\overline{k}}_{i+1}}{{\mathrm{thou}}_{i+1}}{\widetilde{\mathrm{Due\; east}}}_{i+1}+F_{i+1}}\right)\text{for}\text{all}i\in \left\{1,\dots ,n-\mathrm{i}\right\},$

in which Fi and Fi ₊₁ are positive quantities. Because

$\frac{{\tilde{E}}_i\frac{{\overline{k}}_i}{k_i}+F_i}{1+{\tilde{E}}_i+{\tilde{E}}_i\frac{{\overline{k}}_i}{k_i}+F_i}<\mathrm{ane}$

and

$\frac{\frac{{\overline{k}}_{i+1}}{k_{i+1}}{\tilde{E}}_{i+1}}{{\tilde{E}}_{i+1}\frac{{\overline{\mathrm{thousand}}}_{i+1}}{{\mathrm{yard}}_{i+1}}+F_{i+1}}<\mathrm{i},$

we have that

Furthermore, we have that (meet the Supporting Material)

$\mathrm{south}ig\mathrm{due\; north}\left(z_i\right)=-\mathrm{southward}ign\left(z_{i+\mathrm{ane}}\right)\text{for}\text{all}i\in \left\{1,\dots ,n-1\right\},$

that is, an increase of Zi ₊₁ implies a decrease of Zi _. Therefore, there is a sign reversal of the perturbation on the total phosphorylated protein concentration beyond the stages and the magnitude of the perturbation at every stage is e'er attenuated as it propagates upstream in the cascade. That is, |z _i| < |z ₂| <…< |zn ₋₁| < |zn| for all parameter values. Furthermore, this implies too that we take overall attenuation from downstream to upstream in the pour, that is, Φtot < ane. Because these facts do not depend on the specific parameter values or the length of the cascade, they highlight a new structural property of signaling cascades.

For the perturbation on the complimentary active protein concentration, we also have that (see the Supporting Material)

$sign\left(w_i^{\ast }\right)=-si\mathrm{chiliad}n\left(w_{i+1}^{\ast }\right)\text{for}\text{all}i\in \left\{1,\dots ,n-1\right\},$

that is, when the perturbation w ^∗ i _+i is positive the adjacent upstream phase has a perturbation w ^∗ i with negative sign. Hence, if the downstream perturbation causes a subtract of the agile protein concentration at one phase, it causes an increase of the agile protein concentration in the next upstream stage. An expression of the stage gain Ψi tin can exist calculated as a function of the pour parameters starting from the relations of the block diagram of Fig. 2. The exact expression is calculated in the Supporting Material and it is such that

${\Psi }_i\le \frac{\frac{{\overline{g}}_{i+\mathrm{i}}}{{\mathrm{one\; thousand}}_{i+\mathrm{one}}}\frac{E_{\left(i+\mathrm{one}\right)T}}{{\overline{K}}_{i+1}}}{1+\frac{E_{iT}}{\left({\overline{K}}_i+{\mathrm{West}}_{iT}\right)\left(\mathrm{i}+\frac{W_{iT}}{{\overline{K}}_i}\right)}\left(\mathrm{one}+\frac{\overline{{\mathrm{thousand}}_i}}{k_i}\left(1+\frac{{\mathrm{1000}}_i}{W_{\left(i-\mathrm{one}\right)T}}\right)\right)}.$

(12)

Therefore, ane can control the amount of attenuation/distension through the cascade parameters as follows. The smaller the Due west _(i−1)T, the more the attenuation from stage i + 1 to i (i.east., the smaller the upper bound on Ψi in Eq. 12). Moreover, sufficiently large values of Ki and

${\overline{K}}_i$

for all i lead to an increased attenuation at every phase. In turn, large Ki and

${\overline{K}}_i$

and small WiT are responsible for a decreased sensitivity of the response of stage i to upstream stimuli (

). Equally a consequence, a more graded upstream-to-downstream response at all stages is associated with an increased attenuation of downstream perturbations.

From the expressions in Eq. 12, information technology too follows that a sufficient status for having attenuation at stage i of the downstream perturbation is that

This condition is valid for general PD cascades. However, information technology has a particularly simple pregnant in the case in which the signaling pathway is weakly activated as explained in what follows. In Heinrich et al. (

), information technology was constitute that a requirement for upstream-to-downstream signal amplification is that the phosphorylation rate constant should be larger than the dephosphorylation rate constant. For a weakly activated pathway with Ki ≫ Due west _(i−1)T, the phosphorylation charge per unit abiding is well approximated by αi = kiWiT/Ki (see the Supporting Cloth). In the case in which

${\overline{\mathrm{Grand}}}_i\gg W_{iT}$

, the dephosphorylation rate abiding is well approximated past

${\beta }_i={\overline{k}}_i{E}_{iT}/{\overline{K}}_i$

(see the Supporting Material). Every bit a consequence, to accept upstream-to-downstream signal amplification, it is required that αi > βi, which, when Ki ≥ WiT, implies that

This, in turn, implies that Ψi ₋₁ < i and hence that the downstream perturbation is adulterate equally information technology transfers from stage i to phase i−ane. Hence, in weakly activated pathways in which Ki ≥ WiT,

${\overline{\mathrm{Chiliad}}}_i\gg {\mathrm{Westward}}_{iT}$

, and Ki ≫ W _(i−one)T, upstream-to-downstream signal amplification is associated with attenuation of downstream perturbations every bit they transfer upstream. This, in turn, implies unidirectional signal propagation from upstream to downstream.

From the expressions in Eq. 12, it as well follows that a necessary condition for having Ψi > i, that is, for amplifying a downstream perturbation as it transfers from stage

$i+1$

to stage i, is that

This condition, in turn, in the instance in which

${\overline{M}}_{i+\mathrm{one}}\gg W_{\left(i+1\right)T}$

, Westward _(i+1)T ≤ Ki ₊₁, and Ki _+ane ≫ WiT implies that the phosphorylation rate constant αi ₊₁ is smaller than the dephosphorylation charge per unit abiding βi ₊₁. As a issue, there is no amplification at stage i + one of the signal traveling from upstream to downstream every bit the required condition for distension as determined by Heinrich et al. (

) is violated. Hence, in weakly activated pathways in which Ki _+one ≥ West _(i+ane)T,

${\overline{M}}_{i+\mathrm{i}}\gg {\mathrm{Westward}}_{\left(i+1\right)T}$

, and Ki _+one ≫ WiT if a downstream perturbation is amplified as it propagates from phase

$i+1$

to stage i, then there is no distension from phase i to stage

$i+\mathrm{i}$

for the signal traveling from upstream to downstream in response to a stimulus at the top of the pour.

From the expressions of Ψi, we can besides derive a necessary condition for attenuation (see the Supporting Material). Specifically, to take Ψi < 1 at stage i, information technology is necessary that

$\frac{\frac{{\overline{k}}_{i+\mathrm{ane}}}{k_{i+1}}\frac{{\overline{\mathrm{One\; thousand}}}_{i+\mathrm{ane}}{E}_{\left(i+1\right)T}}{{\left({\overline{W}}_{i+\mathrm{i}}^{\ast }+{\overline{\mathrm{1000}}}_{i+1}\right)}^{\mathrm{ii}}}}{1+\frac{{\overline{K}}_i{\mathrm{East}}_{iT}}{{\left({\overline{W}}_i^{\ast }+{\overline{K}}_i\right)}^2}\left[1+\frac{\overline{k_i}}{k_i}\left(1+\frac{{\mathrm{One\; thousand}}_i}{{\overline{W}}_{i-\mathrm{one}}^{\ast }}\left(1+\left(1+\frac{{\overline{W}}_i^{\ast }}{{\overline{K}}_i}\right)\frac{{\overline{\mathrm{West}}}_i^{\ast }}{{\overline{W}}_{i-1}^{\ast }}\right)\right)\right]}<1.$

(13)

If the necessary condition is violated at phase i, then either stage

$i-1$

or phase i amplify the downstream perturbation. This expression can be employed to determine parameter values for which distension of the downstream perturbation tin can result at any given stage and can be useful to make up one's mind the efficacy of the off-target effects of an inhibitor.

To conclude the analytical study, we investigate how dT affects w ^∗ n and zn _. Information technology can be shown (encounter the Supporting Textile) that |w ^∗ n| < |dT| and that |zn| < |dT|. That is, the perturbation dT induces changes westward ^∗ northward and zn almost

${\overline{\mathrm{Due\; west}}}_n^{\ast }$

and

${\overline{Z}}_{\mathrm{northward}}$

, respectively, that are less than dT in magnitude, regardless of the parameters. Also, nosotros have that sign(dT) = −sign(westward ^∗ n) and sign(dT) = sign(zn).

Numerical results

In this section, we commencement illustrate the results on a three-stage cascade example. We then employ the analytically computed expressions Ψi to determine the probability that natural cascades attenuate a downstream perturbation every bit it transfers upstream in the cascade. Nosotros finally study the issue of the length of the pour on the overall gain Ψtot _. All simulations are performed on the full nonlinear model of the system in Eq. two in MATLAB (The MathWorks, Natick, MA) using the built-in ODE23s solver.

Fig. 3 shows how the perturbation propagates upstream in a three-stage pour for the parameter values of Huang and Ferrell (

). This figure illustrates that, surprisingly, the relationship between due west ^∗ i and dT is approximately linear even for big perturbations dT (up to 400 nM). Hence, the theoretical results must hold. In particular, the values of w ^∗ _i and w ^∗ _iii are negative whereas the value of due west ^∗ ₂ is positive. That is, the perturbation on W ^∗ i switches sign from ane stage to the next upstream. The gains Ψi calculated from the expression in the Supporting Material for the parameter values of Huang and Ferrell (

) are given past Ψ₁ = two.45 × 10⁻⁵ and Ψ₂ = 2.xiv × 10⁻ⁱⁱ. Because Ψ_i and Ψ_two are both <1, the cascade should attenuate the downstream perturbation at every stage. This is confirmed by Fig. three in which for the same value of dT _, we have that |west ^∗ i| becomes smaller and smaller as the stage i decreases (i.e., as the perturbation propagates upstream). Because the values of Ψi are ≪ane, this three-stage cascade practically enforces unidirectional signal propagation from upstream to downstream. Note that as long as the applied perturbation dT is small enough, the relationship betwixt dT and w ^∗ i is linear and hence all our results concur independently of the parameter values. Additional examples for different parameter values are provided in the Supporting Material.

Effigy three Attenuation and sign-reversal in a three-stage cascade. The x axis shows the value of the perturbation dT and the y axis shows the steady-state value of the resulting perturbations w ^∗ _ane, w ^∗ ₂, and westward ^∗ ₃. Simulation is performed on the total nonlinear ODE model given past Eq. 2. The parameters of each stage i are taken from Huang and Ferrell () and are given by ki = 150 (min)⁻ⁱ, ${\overline{\mathrm{thousand}}}_i=150{\left(\min \right)}^{-1}$, αi = 2.5 (nM min)⁻¹, ${\overline{a}}_i=600{\left(\min \right)}^{-\mathrm{ane}}$, bi = 2.5 (nM min)⁻¹, ${\overline{b}}_i=600{\left(\min \right)}^{-1}$, East _3T = 120 nM, Due east _2T = 0.3 nM, East _1T = 0.3 nM, Westward _3T = 1200 nM, Westward _2T = 1200 nM, W _1T = 3 nM, ${\overline{\mathrm{Westward}}}_0^{\ast }=0.3\text{nM}$, and ${\overline{D}}_T=0\text{nM}$. As a result, Ki = 300 nM and ${\overline{K}}_i=300\text{nM}$.

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To validate the necessary condition for attenuation at stage i, we constructed a parameter gear up that violates the necessary condition for attenuation (see Eq. 13). In this instance, we should await that at the stage i for which Ψi > 1, the downstream perturbation is amplified, that is, |westward ^∗ i| > |west ^∗ i ₊₁|. The necessary condition in Eq. xiii tin be violated by choosing phosphatase amounts that increase with the stage number, that is, Eastward _1T ≪ Due east _2T ≪ E _3T and substrate amounts that decrease with the phase number, that is, W _1T ≫ W _2T ≫ W _3T. We utilized these conditions and constructed a cascade that amplifies downstream perturbations. The consequence is shown in Fig. iv. The resulting parameter values are still biologically meaningful equally they are contained in the parameter intervals estimated in Huang and Ferrell (

). Therefore, these cascades are capable of as well transmitting a perturbation from downstream to upstream by amplifying its amplitude.

Effigy 4 Amplification in a 3-phase cascade. Numerical simulation of system in Eq. ii: value of |w ^∗ i| for i ∈{1,…,n} in response to a unit perturbation dT = ane. This plot shows that violation of the necessary condition leads to amplification of the downstream perturbation equally it transfers upstream in the cascade. Parameters of stage i are given past: ki = 150 (min)^−one, ${\overline{\mathrm{thousand}}}_i=150{\left(\min \right)}^{-1}$, ai = 2500 (nM min)⁻¹, ${\overline{a}}_i=600{\left(\min \right)}^{-1}$, bi = 2500 (nM min)^−one, ${\overline{b}}_i=600{\left(\min \right)}^{-1}$, E _3T = 120 nM, E _2T = 30 nM, E _1T = 0.three nM, W _3T = 3 nM, W _2T = 30 nM, W _1T = 1200 nM, ${\overline{\mathrm{West}}}_0^{\ast }=0.3\text{nM}$, and ${\overline{D}}_T=\mathrm{0.nine}\text{nM}$.

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Practise natural signaling cascades attenuate downstream perturbations?

To make up one's mind the probability that a natural signaling pour attenuates or amplifies downstream perturbations, we evaluated the expression of the gains Ψi on parameters extracted with uniform probability distribution from intervals taken from the literature (

,

,

32

• Levchenko A.
• Bruck J.
• Sternberg P.Due west.

Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold properties.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

,

). We present the results first for a iii-phase cascade starting from conservative intervals and we progressively reduce the size of the intervals. In all cases, each parameter has a range and a uniform probability distribution is used to sample parameters for each range. Also, even though the range of parameters for each cycle is the same, in the simulations each cycle has different parameters (randomly picked from the given range).

Conservative intervals

In this case, nosotros randomly chose parameters through a uniform probability distribution from the intervals given in Table 1. The maximum and minimum values of the intervals were called to be the maximum and minimum of the union of the intervals defined in Huang and Ferrel (

) and Bhalla and Iyengar (

). This is a conservative way of choosing the intervals equally the parameters of Huang and Ferrell (

) and Bhalla and Iyengar (

) are taken from dissimilar organisms. In selecting the range for DT, we causeless that D is a downstream protein substrate and thus its interval of variation was chosen to be the aforementioned as that for WiT _.

Table 1 Conservative intervals

Parameter	Interval for simulation	Interval from Huang and Ferrell ( 28 Huang C.Y. Ferrell Jr., J.E. Ultrasensitivity in the mitogen-activated poly peptide kinase cascade. Crossref PubMed Scopus (914) Google Scholar )	Interval from Bhalla and Iyengar ( 31 Bhalla U.S. Iyengar R. Emergent properties of networks of biological signaling pathways. Crossref PubMed Scopus (1225) Google Scholar )
ki, ${\overline{k}}_i$	[6.3, 600]	[150, 150]	[6.3, 600]
ai, bi	[xviii.018, 4545.45]	[2500, 2500]	[eighteen.018, 4545.45]
${\overline{a}}_i$, ${\overline{b}}_i$	[25.2, 2400]	[600, 600]	[25.two, 2400]
EiT	[0.3, 224]	[0.3, 120]	[3.ii, 224]
WiT	[3, 1200]	[iii, 1200]	[180, 360]
${\overline{W}}_0^{\ast }$	[0.3, 100]	[0.iii, 0.3]	[100, 100]
${\overline{D}}_T$	[0, 1200]	—	—

For each of the parameters of the cascade, we indicate the interval considered for simulation and the intervals given in Huang and Ferrel (

) and Bhalla and Iyengar (

). For simulation, a uniform probability distribution over each interval is called to sample parameter values. Also, each stage has different parameters even though all were extracted from a uniform probability distribution.

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We faux the three-stage cascade ten,000 times and the results are reported in Table two. This table shows the percentage of simulations that resulted in Ψi > 1 for every i ∈ {1,2}, that is, that resulted in attenuation at stage i. The probability of phase one attenuating the downstream perturbation is 71.34% and the probability of stage 2 attenuating it is 55%. Moreover, because the probability that Ψtot < 1 is 79.4%, the probability of such cascades providing an overall attenuation of a downstream perturbation is quite loftier. To explore whether 10,000 simulations were enough to obtain meaningful probability figures, we calculated at each new simulation the percentage of all performed simulations that resulted in attenuation. The probabilities converge for every stage to the values given in Table 2; hence, performing more simulations volition non significantly modify the results (see the Supporting Material).

Table 2 Three-stage cascade attenuation percentage

	Ψ1	Ψ2	Ψtot
% of Ψi < 1	71.34	55	79.4

The parameters are taken randomly from Table 1.

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Intervals based on Bhalla and Iyengar ()

We considered the nominal parameter values given in Bhalla and Iyengar (

) and and then constructed intervals past varying these values by 20, 50, and 80%. Specifically, for every parameter with nominal value p, we considered a confidence interval of the course [(1 – 0.x) p, (1 + 0.x) p] for the three different cases in which x = 2, x = five, and 10 = eight. The results for these three different cases are shown in Table 3. Fifty-fifty when the parameters are allowed to vary by

$80\text{\%}$

from the nominal values, the probability that any given stage attenuates the perturbation is very loftier and the probability that the cascade provides overall attenuation (i.east., Ψtot < 1) is ane. As performed in the previous case, the results of Table 3 are obtained performing 10,000 numerical simulations. In the Supporting Textile, we show that this number is large enough to achieve convergence of the probabilities.

Tabular array iii Three-phase pour attenuation pct for different intervals near the nominal parameter values of Bhalla and Iyengar ()

	Ψ1	Ψ2	Ψtot
% of Ψi < 1 with 20% variation	100	100	100
% of Ψi < 1 with l% variation	99.98	100	100
% of Ψi < 1 with lxxx% variation	96.895	99.91	100

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Intervals based on Levchenko et al. (

32

• Levchenko A.
• Bruck J.
• Sternberg P.W.

Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold backdrop.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

)

We next considered the nominal parameter values given in Levchenko et al. (

32

• Levchenko A.
• Bruck J.
• Sternberg P.Westward.

Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold backdrop.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

) and constructed intervals past varying these values by 20, 50, and 80%. Specifically, for every parameter with nominal value p, we considered a confidence interval of the form [(1 – 0.x) p, (1 + 0.ten) p] for the three unlike cases in which x = 2, x = 5, and x = 8. The results for these iii different cases are shown in Table 4. When the parameters are allowed to alter by 50% with respect to the nominal values, the probability of attenuation at each stage is lower than the values obtained for the parameters of Bhalla and Iyengar (

) (Tabular array 3). With 80% parameter variation, there is a significant percentage of the possible parameters (10%) that allows the states to amplify, overall, the downstream perturbation from stage 3 to stage 1. Moreover, 50% of the parameters led to having Ψ_one > 1 or Ψ₂ > 1 and but 2.2% of the parameters led to having both Ψ_ane > 1 and Ψ₂ > i. Therefore, 50% of the possible parameter values pb to amplification in at least one stage in the cascade. The results of Table 4 are obtained performing 10,000 numerical simulations. The Supporting Material shows that, by the time the 10,000th simulation is performed, the probability has converged to its final value.

Table 4 Three-stage cascade attenuation percentage for unlike intervals near the nominal parameter values of Levchenko et al. (

32

• Levchenko A.
• Bruck J.
• Sternberg P.Westward.

Scaffold proteins may biphasically impact the levels of mitogen-activated protein kinase signaling and reduce its threshold properties.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

)

	Ψone	Ψ2	Ψtot
% of Ψi < one with 20% variation	77.49	100	100
% of Ψi < 1 with l% variation	65.85	93.32	97.07
% of Ψi < 1 with 80% variation	64.69	82.68	ninety.91

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We so analyzed how the length n of the pour affects the overall attenuation from phase north to stage one, that is, how it affects the proceeds Ψtot _. To perform this study, nosotros first fake a x-stage cascade ten,000 times with the aforementioned parameter ranges equally given in Table i. The result is shown in Table v. The probability of the final ii stages (i = 8,9) attenuating the perturbation has significantly increased compared to the iii-stage example (Table two). Furthermore, the probability of overall attenuation, that is, that Ψtot < 1, is 1. Hence, fifty-fifty when some stages amplify the downstream perturbation, the rest of the stages provide attenuation so that the overall attenuation in the cascade is much more than the overall amplification. To confirm that 10,000 simulations were enough to provide meaningful probability figures, we analyzed the convergence of the probability afterwards each simulation run in the Supporting Cloth.

Table 5 10-stage cascade attenuation percentage for the parameter values in Table ane

i	1	2	3	iv	5	half dozen	seven	8	nine	Ψtot
% of Ψi < 1	67.3	71.8	72.9	73.three	73.7	74.5	72.nine	76.2	59.8	100

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Finally, to written report how the number of stages in a pour impacts the probability of overall attenuation, that is, the probability that Ψtot < ane, nosotros performed a number of numerical simulations extracting parameters from the intervals of Tabular array one for cascades with increasing number of stages. The probability of overall attenuation monotonically increases equally the number of stages in the pour increases and it reaches 100% for cascades of length at least 6 (Fig. 5). For each number of stages, due north, nosotros performed a sufficiently large number of simulations for different values of the parameters sampled in the intervals of Table 1 (see the Supporting Material). This result implies that for a fixed range of parameters, adding more than stages contributes significantly to the probability of overall attenuation from stage northward to stage ane. For instance, the probability of a 3-stage cascade providing overall attenuation was found to exist 79.4% while, for the aforementioned range of parameters, the probability of a ten-stage cascade providing overall attenuation was found to exist 100%.

Discussion

Upstream-to-downstream point transfer in signaling cascades determines how external stimuli at the pinnacle of the cascade, such as growth factors, hormones, and neurotransmitters, affect downstream targets, such as factor expression. Several works focused on determining the sensitivity of each stage of a cascade to modest perturbations at the top of the pour. In these studies, it was institute that multiple stages in the pour can boost the overall cascade sensitivity to upstream input stimuli (

,

,

). Downstream-to-upstream signal transfer, on the other hand, determines how a perturbation at the bottom of the cascade due, for example, to a drug or to sharing a substrate with another signaling pathway, affects the upstream stages of the cascade. This has not been studied before.

Here, nosotros take studied for the first time (to our knowledge) the response of each stage of a cascade to small perturbations in a substrate or inhibitor at the bottom of the cascade. One of our results is that larger numbers of stages in the cascade lead to college overall attenuation of the point transfer from downstream to upstream. This provides another reason why natural signaling cascades are usually composed of multiple stages: more than stages enforce unidirectional signal propagation, which is certainly desirable in any natural or man-made bespeak transmission system.

We accept computed analytical expressions of the downstream-to-upstream gains at each phase of the cascade every bit a office of the cascade parameters. These expressions uncover ii main structural properties of signaling cascades, which are independent of the specific parameter values.

Starting time, the perturbation on the full or free active protein concentration switches sign at each stage of the pour as it propagates upstream. That is, if at one stage the amount of free or total active protein increases considering of the perturbation, it must decrease at the next upstream stage.

Second, the perturbation on the full amount of active protein is attenuated as it propagates from one stage to the next i upstream. By dissimilarity, the way the perturbation propagates on the gratis corporeality of active poly peptide depends on the specific parameter values. We take provided a sufficient status for attenuation, which applies to general PD cascades and has a particularly simple meaning in the special instance of weakly activated pathways. That is, for weakly activated pathways in which each cycle operates in the hyperbolic regime, amplification of a perturbation at the top of the cascade as it propagates downstream implies attenuation of a perturbation at the bottom of the cascade as it propagates upstream.

Although simulation studies performed in Ventura et al. (

) suggested that a perturbation is attenuated as it propagates upstream in the cascade, the analytical expressions of the gains establish in this commodity clearly bear witness that amplification of the perturbation on the free poly peptide concentration is also possible. To sympathise whether natural signaling cascades are more than probable to benumb or to amplify a downstream perturbation on the free agile protein concentration, we performed a numerical study. In this study, the gain Ψi at each stage was computed with parameter values randomly extracted from biologically meaningful sets obtained from the literature (

,

,

32

• Levchenko A.
• Bruck J.
• Sternberg P.West.

Scaffold proteins may biphasically bear upon the levels of mitogen-activated protein kinase signaling and reduce its threshold backdrop.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

,

). This numerical study reveals that signaling cascades are substantially more likely to benumb a downstream perturbation than to amplify it and that longer signaling cascades take a college probability of overall attenuation. Nonetheless, in signaling cascades of length iii, which is the near common length found in practice, ∼50% of the biologically meaningful parameters taken from Levchenko et al. (

32

• Levchenko A.
• Bruck J.
• Sternberg P.Westward.

Scaffold proteins may biphasically affect the levels of mitogen-activated poly peptide kinase signaling and reduce its threshold properties.

• Crossref
• PubMed
• Scopus (376)
• Google Scholar

) lead to amplification at least at i stage and ∼

$10\text{\%}$

of them resulted in overall amplification (from stage 3 to stage 1).

In summary, our findings suggest that the furnishings of crosstalk between signaling pathways sharing common components tin can be felt even upstream of the common component as opposed to merely downstream of information technology as previously believed. We believe this provides a new mechanism past which a pathway tin become overactivated as constitute in several pathological conditions such equally cancer (

,

,

,

). At the same time, our report provides tools to understand how the effects of a targeted drug (

,

) may propagate to obtain astray effects and how these furnishings depend on the cascade parameters.

This commodity addresses cascades in which, at each stage, at that place is a single phosphorylation bike. All the same, several natural cascades, such as the MAPK cascade, display double phosphorylation and experimental work performed in Drosophila embryos has demonstrated that a perturbation in one of the substrates at the bottom of the pour affects the phosphorylation level at the last cycle of the cascade (

). Whether such a perturbation can propagate on the higher levels of the cascade was not addressed. In future work, we thus plan to extend our gain calculations to cascades with double phosphorylation in order to establish the extent to which such perturbations propagate on the higher levels of the MAPK cascade. It was shown in previous piece of work that the presence of double phosphorylation tin pb to sustained oscillations even in the absence of explicit negative feedback (

). In such instances, our analysis will have to extend to dynamic perturbations as opposed to static perturbations in club to understand how these oscillations propagate upstream in the cascade.

Recently published experimental articles clearly prove that perturbations in the downstream targets of a signaling cascade crusade a perturbation in the immediate upstream signaling stage. Specifically, Kim et al. (

) showed, through in vivo experiments in the Drosophila embryo, that irresolute the level of one of the substrates of the MAPK cascade influences the level of MAPK phosphorylation. Additionally, Ventura et al. (

) showed, through experiments on a reconstituted covalent modification bike, that the add-on of a downstream target changes the steady-state value of the modified protein of the upstream wheel. These results are promising; notwithstanding, boosted experiments are required to validate the attenuation/amplification predictions of this article on the higher levels of a cascade. Specifically, validating the prediction that the perturbation on the total protein concentration is attenuated as it propagates upstream is particularly appealing, because it does not depend on the specific parameter values. Furthermore, it requires usa to measure the total phosphorylated poly peptide, which is a much easier task to accomplish than measuring the free phosphorylated protein. We program to validate experimentally this prediction in our futurity work.

D.D.V. and H.R.O. were in part supported past Air Force Office of Scientific Research grant No. FA9550-09-1-0211. A.C.5. and South.D.1000. were supported by grants from the Department of Defense Chest Cancer Enquiry Program and the Heart for Computational Medicine and Bioinformatics.

Supporting Material

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Article Info

Publication History

Editor: Andre Levchenko.

Accepted: Feb 11, 2022

Received: November 29, 2010

Identification

DOI: https://doi.org/x.1016/j.bpj.2011.02.014

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