NF-κB is an inflammatory transcription factor that is normally kept inactive in the cytoplasm by IκB proteins. When a stimulus (IKK) arrives, it triggers degradation of IκB, freeing NF-κB to switch on and enter the nucleus. But NF-κB also drives synthesis of new IκB — a negative feedback loop that eventually shuts it back off. Because making new IκB (transcription, translation, and folding) takes time — a delay τ — the system overshoots before the brakes kick in, producing sustained oscillations rather than settling smoothly. This model includes two such feedback loops: a fast one (IκBα) and a slow one (IκBε). Drag τ and the IKK stimulus strength, or click the amplitude map, to see how the oscillations change.
Reaction network ( R1 ) NF κ B + IkB α → k a NF κ B : IkB α ( R2 ) NF κ B : IkB α → k d NF κ B + IkB α ( R3 ) IKK + NF κ B : IkB α → r IKK + NF κ B ( R4 ) IKK + IkB α → r IKK ( R5 ) IkB α → g ∅ ( R6 ) NF κ B + prIkB α → f a NF κ B : prIkB α ( R7 ) NF κ B : prIkB α → f d NF κ B + prIkB α ( R8 ) prIkB α → a 1 prIkB α + IkB α ( R9 ) NF κ B : prIkB α → b 1 NF κ B : prIkB α + IkB α ( R10 ) NF κ B + IkB ε → k a NF κ B : IkB ε ( R11 ) NF κ B : IkB ε → k d NF κ B + IkB ε ( R12 ) IKK + NF κ B : IkB ε → r IKK + NF κ B ( R13 ) IKK + IkB ε → r IKK ( R14 ) IkB ε → g ∅ ( R15 ) NF κ B + prIkB ε → f a NF κ B : prIkB ε ( R16 ) NF κ B : prIkB ε → f d NF κ B + prIkB ε ( R17 ) prIkB ε → a 2 prIkB ε + IkB ε ( R18 ) NF κ B : prIkB ε → b 2 NF κ B : prIkB ε + IkB ε \begin{array}{rlcl}
(\text{R1}) & \text{NF}\kappa\text{B} + \text{IkB}_\alpha & \xrightarrow{k_a} & \text{NF}\kappa\text{B}{:}\text{IkB}_\alpha \\[3pt]
(\text{R2}) & \text{NF}\kappa\text{B}{:}\text{IkB}_\alpha & \xrightarrow{k_d} & \text{NF}\kappa\text{B} + \text{IkB}_\alpha \\[3pt]
(\text{R3}) & \text{IKK} + \text{NF}\kappa\text{B}{:}\text{IkB}_\alpha & \xrightarrow{r} & \text{IKK} + \text{NF}\kappa\text{B} \\[3pt]
(\text{R4}) & \text{IKK} + \text{IkB}_\alpha & \xrightarrow{r} & \text{IKK} \\[3pt]
(\text{R5}) & \text{IkB}_\alpha & \xrightarrow{g} & \varnothing \\[3pt]
(\text{R6}) & \text{NF}\kappa\text{B} + \text{prIkB}_\alpha & \xrightarrow{f_a} & \text{NF}\kappa\text{B}{:}\text{prIkB}_\alpha \\[3pt]
(\text{R7}) & \text{NF}\kappa\text{B}{:}\text{prIkB}_\alpha & \xrightarrow{f_d} & \text{NF}\kappa\text{B} + \text{prIkB}_\alpha \\[3pt]
(\text{R8}) & \text{prIkB}_\alpha & \xrightarrow{a_1} & \text{prIkB}_\alpha + \text{IkB}_\alpha \\[3pt]
(\text{R9}) & \text{NF}\kappa\text{B}{:}\text{prIkB}_\alpha & \xrightarrow{b_1} & \text{NF}\kappa\text{B}{:}\text{prIkB}_\alpha + \text{IkB}_\alpha \\[3pt]
(\text{R10}) & \text{NF}\kappa\text{B} + \text{IkB}_\varepsilon & \xrightarrow{k_a} & \text{NF}\kappa\text{B}{:}\text{IkB}_\varepsilon \\[3pt]
(\text{R11}) & \text{NF}\kappa\text{B}{:}\text{IkB}_\varepsilon & \xrightarrow{k_d} & \text{NF}\kappa\text{B} + \text{IkB}_\varepsilon \\[3pt]
(\text{R12}) & \text{IKK} + \text{NF}\kappa\text{B}{:}\text{IkB}_\varepsilon & \xrightarrow{r} & \text{IKK} + \text{NF}\kappa\text{B} \\[3pt]
(\text{R13}) & \text{IKK} + \text{IkB}_\varepsilon & \xrightarrow{r} & \text{IKK} \\[3pt]
(\text{R14}) & \text{IkB}_\varepsilon & \xrightarrow{g} & \varnothing \\[3pt]
(\text{R15}) & \text{NF}\kappa\text{B} + \text{prIkB}_\varepsilon & \xrightarrow{f_a} & \text{NF}\kappa\text{B}{:}\text{prIkB}_\varepsilon \\[3pt]
(\text{R16}) & \text{NF}\kappa\text{B}{:}\text{prIkB}_\varepsilon & \xrightarrow{f_d} & \text{NF}\kappa\text{B} + \text{prIkB}_\varepsilon \\[3pt]
(\text{R17}) & \text{prIkB}_\varepsilon & \xrightarrow{a_2} & \text{prIkB}_\varepsilon + \text{IkB}_\varepsilon \\[3pt]
(\text{R18}) & \text{NF}\kappa\text{B}{:}\text{prIkB}_\varepsilon & \xrightarrow{b_2} & \text{NF}\kappa\text{B}{:}\text{prIkB}_\varepsilon + \text{IkB}_\varepsilon
\end{array} ( R1 ) ( R2 ) ( R3 ) ( R4 ) ( R5 ) ( R6 ) ( R7 ) ( R8 ) ( R9 ) ( R10 ) ( R11 ) ( R12 ) ( R13 ) ( R14 ) ( R15 ) ( R16 ) ( R17 ) ( R18 ) NF κ B + IkB α NF κ B : IkB α IKK + NF κ B : IkB α IKK + IkB α IkB α NF κ B + prIkB α NF κ B : prIkB α prIkB α NF κ B : prIkB α NF κ B + IkB ε NF κ B : IkB ε IKK + NF κ B : IkB ε IKK + IkB ε IkB ε NF κ B + prIkB ε NF κ B : prIkB ε prIkB ε NF κ B : prIkB ε k a k d r r g f a f d a 1 b 1 k a k d r r g f a f d a 2 b 2 NF κ B : IkB α NF κ B + IkB α IKK + NF κ B IKK ∅ NF κ B : prIkB α NF κ B + prIkB α prIkB α + IkB α NF κ B : prIkB α + IkB α NF κ B : IkB ε NF κ B + IkB ε IKK + NF κ B IKK ∅ NF κ B : prIkB ε NF κ B + prIkB ε prIkB ε + IkB ε NF κ B : prIkB ε + IkB ε These 18 mass-action reactions define the system, and each becomes a term in the 9 coupled ODEs. The two feedback loops are identical in form — α (R1–R9) and ε (R10–R18) — but differ in speed. The IκB synthesis reactions (R8, R9 and R17, R18) are the ones that carry the transcriptional delay τ: new IκB appears only τ minutes after NF-κB binds the promoter, and that lag is what turns steady signaling into oscillations. The stiff system is integrated with an adaptive Runge–Kutta (Dormand–Prince) solver.
Ordinary differential equations d [ NF κ B ] d t = − k a NF κ B I κ B α + k d NF κ B : I κ B α + r IKK NF κ B : I κ B α − k a NF κ B I κ B ε + k d NF κ B : I κ B ε + r IKK NF κ B : I κ B ε − f a NF κ B prI κ B α + f d NF κ B : prI κ B α − f a NF κ B prI κ B ε + f d NF κ B : prI κ B ε d [ I κ B α ] d t = − k a NF κ B I κ B α + k d NF κ B : I κ B α + a 1 prI κ B α ( t − τ ) + b 1 NF κ B : prI κ B α ( t − τ ) − g I κ B α − r IKK I κ B α d [ I κ B ε ] d t = − k a NF κ B I κ B ε + k d NF κ B : I κ B ε + a 2 prI κ B ε ( t − τ ) + b 2 NF κ B : prI κ B ε ( t − τ ) − g I κ B ε − r IKK I κ B ε d [ NF κ B : I κ B α ] d t = k a NF κ B I κ B α − k d NF κ B : I κ B α − r IKK NF κ B : I κ B α d [ NF κ B : I κ B ε ] d t = k a NF κ B I κ B ε − k d NF κ B : I κ B ε − r IKK NF κ B : I κ B ε d [ prI κ B α ] d t = − f a NF κ B prI κ B α + f d NF κ B : prI κ B α d [ prI κ B ε ] d t = − f a NF κ B prI κ B ε + f d NF κ B : prI κ B ε d [ NF κ B : prI κ B α ] d t = f a NF κ B prI κ B α − f d NF κ B : prI κ B α d [ NF κ B : prI κ B ε ] d t = f a NF κ B prI κ B ε − f d NF κ B : prI κ B ε \small
\def\N{\text{NF}\kappa\text{B}}
\def\Aa{\text{I}\kappa\text{B}_\alpha}
\def\Ae{\text{I}\kappa\text{B}_\varepsilon}
\def\Ca{\N{:}\Aa}
\def\Ce{\N{:}\Ae}
\def\Pa{\text{prI}\kappa\text{B}_\alpha}
\def\Pe{\text{prI}\kappa\text{B}_\varepsilon}
\def\Qa{\N{:}\Pa}
\def\Qe{\N{:}\Pe}
\def\IKK{\text{IKK}}
\begin{aligned}
\frac{d[\N]}{dt} ={}& -k_a\,\N\,\Aa + k_d\,\Ca + r\,\IKK\,\Ca - k_a\,\N\,\Ae + k_d\,\Ce + r\,\IKK\,\Ce \\
&{} - f_a\,\N\,\Pa + f_d\,\Qa - f_a\,\N\,\Pe + f_d\,\Qe \\[6pt]
\frac{d[\Aa]}{dt} ={}& -k_a\,\N\,\Aa + k_d\,\Ca + \textcolor{#d97706}{a_1\,\Pa(t-\tau) + b_1\,\Qa(t-\tau)} - g\,\Aa - r\,\IKK\,\Aa \\[6pt]
\frac{d[\Ae]}{dt} ={}& -k_a\,\N\,\Ae + k_d\,\Ce + \textcolor{#10b981}{a_2\,\Pe(t-\tau) + b_2\,\Qe(t-\tau)} - g\,\Ae - r\,\IKK\,\Ae \\[6pt]
\frac{d[\Ca]}{dt} ={}& k_a\,\N\,\Aa - k_d\,\Ca - r\,\IKK\,\Ca \\[6pt]
\frac{d[\Ce]}{dt} ={}& k_a\,\N\,\Ae - k_d\,\Ce - r\,\IKK\,\Ce \\[6pt]
\frac{d[\Pa]}{dt} ={}& -f_a\,\N\,\Pa + f_d\,\Qa \\[6pt]
\frac{d[\Pe]}{dt} ={}& -f_a\,\N\,\Pe + f_d\,\Qe \\[6pt]
\frac{d[\Qa]}{dt} ={}& f_a\,\N\,\Pa - f_d\,\Qa \\[6pt]
\frac{d[\Qe]}{dt} ={}& f_a\,\N\,\Pe - f_d\,\Qe
\end{aligned} d t d [ NF κ B ] = d t d [ I κ B α ] = d t d [ I κ B ε ] = d t d [ NF κ B : I κ B α ] = d t d [ NF κ B : I κ B ε ] = d t d [ prI κ B α ] = d t d [ prI κ B ε ] = d t d [ NF κ B : prI κ B α ] = d t d [ NF κ B : prI κ B ε ] = − k a NF κ B I κ B α + k d NF κ B : I κ B α + r IKK NF κ B : I κ B α − k a NF κ B I κ B ε + k d NF κ B : I κ B ε + r IKK NF κ B : I κ B ε − f a NF κ B prI κ B α + f d NF κ B : prI κ B α − f a NF κ B prI κ B ε + f d NF κ B : prI κ B ε − k a NF κ B I κ B α + k d NF κ B : I κ B α + a 1 prI κ B α ( t − τ ) + b 1 NF κ B : prI κ B α ( t − τ ) − g I κ B α − r IKK I κ B α − k a NF κ B I κ B ε + k d NF κ B : I κ B ε + a 2 prI κ B ε ( t − τ ) + b 2 NF κ B : prI κ B ε ( t − τ ) − g I κ B ε − r IKK I κ B ε k a NF κ B I κ B α − k d NF κ B : I κ B α − r IKK NF κ B : I κ B α k a NF κ B I κ B ε − k d NF κ B : I κ B ε − r IKK NF κ B : I κ B ε − f a NF κ B prI κ B α + f d NF κ B : prI κ B α − f a NF κ B prI κ B ε + f d NF κ B : prI κ B ε f a NF κ B prI κ B α − f d NF κ B : prI κ B α f a NF κ B prI κ B ε − f d NF κ B : prI κ B ε The colored terms are the delayed IκB synthesis (α and ε loops) that carry the τ delay.
Full species list NF-κB (free, active) IκBα (free) NF-κB:IκBα complex IκBα promoter (transcribing) NF-κB:IκBα-promoter complex IκBε (free) NF-κB:IκBε complex IκBε promoter (transcribing) NF-κB:IκBε-promoter complex