Analytic Methods for Modeling Stochastic Regulatory Networks

Abstract

Recent single-cell experiments have revived interest in the unavoidable or intrinsic noise in biochemical and genetic networks arising from the small number of molecules of the participating species. That is, rather than modeling regulatory networks in terms of the deterministic dynamics of concentrations, we model the dynamics of the probability of a given copy number of the reactants in single cells. Most of the modeling activity of the last decade has centered on stochastic simulation, i.e., Monte Carlo methods for generating stochastic time series. Here we review the mathematical description in terms of probability distributions, introducing the relevant derivations and illustrating several cases for which analytic progress can be made either instead of or before turning to numerical computation. Analytic progress can be useful both for suggesting more efficient numerical methods and for obviating the computational expense of, for example, exploring parametric dependence.

Appendices

Appendix A. Deriving the Master Equation

Here we derive the master equation in one dimension from the laws of probability. The probability of having n proteins at a time $t + \tau$ (where $\tau$ is small) is equal to the probability of having had n′ proteins at time t multiplied by the (possibly time dependent) probability of transitioning from n′ to n in time τ, summed over all n′:

$$ {p_n}(t + \tau ) = \sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}(t){p_{{n|n{^\prime}}}}} (t,\tau ). $$

(181)

The transition probability ${p_{{n|n{^\prime}}}}(t,\tau )$ has two contributions: (1) the probability of transitioning from state $n{^\prime} \ne n$ to state n, equal to a transition rate ${w_{{nn{^\prime}}}}(t)$ times the transition time τ, and (2) the probability of starting and remaining in state $n{^\prime} = n$, which we call ${\pi_n}(t)$:

$$ {p_{{n|n{^\prime}}}}(t,\tau ) = (1 - {\delta_{{nn{^\prime}}}}){w_{{nn{^\prime}}}}(t)\tau + {\delta_{{nn{^\prime}}}}{\pi_n}(t). $$

(182)

Applying the normalization condition $\sum\limits_n {{p_{{n|n{^\prime}}}}(t,\tau ) = 1}$ to Eq. 182, we obtain

$$ {\pi_n}(t) = 1 - \tau \sum\limits_{{n{^\prime} \ne n}} {{w_{{n{^\prime}n}}}(t),} $$

(183)

making Eq. 181

$$ {p_n}(t + \tau ) = \sum\limits_{{n{^\prime}}}{{p_{{n{^\prime}}}}} (t)\left[ {(1 -{\delta_{{nn{^\prime}}}}){w_{{nn{^\prime}}}}(t)\tau +{\delta_{{nn{^\prime}}}}\left( {1 - \tau\sum\limits_{{n{^\prime\prime} \ne n}} {{w_{{n{^\prime\prime}n}}}(t)} } \right)}\right],$$

(184)

$$ = \tau \sum\limits_{{n{^\prime}}}{{p_{{n{^\prime}}}}(t){w_{{nn{^\prime}}}}(t) - \tau {p_n}(t){w_{{nn}}} + {p_n}(t) - \tau{p_n}(t)\sum\limits_{{n{^\prime\prime} \ne n}} {{w_{{n{^\prime\prime}n}}}(t)} },$$

(185)

$$ \hskip 3pc {= {p_n}(t) + \tau \left[ {\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}(t){w_{{nn{^\prime}}}}(t) - {p_n}(t)\sum\limits_{{n{^\prime}}} {{w_{{n{^\prime}n}}}(t)} } } \right].} $$

(186)

Taking the limit $\tau \to 0$, we obtain

$$ \frac{{{\text{d}}{p_n}}}{{{\hbox{d}}t}} = \sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}{w_{{nn{^\prime}}}}(t) - {p_n}} \sum\limits_{{n{^\prime}}} {{w_{{n{^\prime}n}}}(t),} $$

(187)

as in Eq. 4.

Appendix B.Summation of the Birth–Death Master Equation

Here we sum the one-dimensional birth–death master equation over n against (1) n, to derive the mean dynamics, (2) ${x^n}$, to derive the equation of motion in generating function space, and (3) $|n\rangle$, to find the equation of motion in operator notation.

First we sum the master equation (Eq. 9) over n against n to derive the dynamics of the mean $\langle n\rangle = \sum\nolimits_n {n{p_n}}$:

$$ {\partial_t}\sum\limits_n {n{p_n}} = - g\sum\limits_n {n{p_n}} - \sum\limits_n {{n^2}{p_n}} + g\sum\limits_n {n{p_{{n - 1}}}} + \sum\limits_n {n(n + 1){p_{{n + 1}}}}. $$

(188)

The left-hand side (LHS) is the time derivative of the mean. On the RHS, recalling that n runs from 0 to ∞, we may define for the third term $n{^\prime} \equiv n - 1$ and for the fourth term $n{^\prime}{^\prime} \equiv n + 1$, giving

$$ {\partial_t}\langle n\rangle = - g\sum\limits_n {n{p_n}} - \sum\limits_n {{n^2}{p_n}} + g\sum\limits_{{n{^\prime} = - 1}}^{\infty } {(n{^\prime} + 1){p_{{n{^\prime}}}}} + \sum\limits_{{n{^\prime}{^\prime} = 1}}^{\infty } {(n{^\prime}{^\prime} - 1)n{^\prime}{^\prime}{p_{{n{^\prime}{^\prime}}}}}. $$

(189)

The lower limits on all sums can be set to 0 once more without changing the expression (in the third term we impose ${p_{{ - 1}}} \equiv 0$ since protein number cannot be negative); simplifying gives

$$ {\partial_t}\langle n\rangle = - g\sum\limits_n {n{p_n}} - \sum\limits_n {{n^2}{p_n}} + g\sum\limits_{{n{^\prime}}} {n{^\prime}{p_{{n{^\prime}}}}} + g\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}} + \sum\limits_{{n{^\prime}{^\prime}}} {{{n{^\prime}{^\prime}}^2}{p_{{n{^\prime}{^\prime}}}}} - \sum\limits_{{n{^\prime}{^\prime}}} {n{^\prime}{^\prime}{p_{{n{^\prime}{^\prime}}}}}, $$

(190)

$$ = g\sum\limits_{{n{^\prime}}}{{p_{{n{^\prime}}}}} -\sum\limits_{{n{^\prime}{^\prime}}}{n{^\prime}{^\prime}{p_{{n{^\prime}{^\prime}}}}}$$

(191)

$$ = g - \langle n\rangle, $$

(192)

as in Eq. 8.

Next we sum the master equation (Eq. 9) over n against ${x^n}$ to derive the equation of motion of the generating function $G(x,t) = \sum\nolimits_n {{p_n}} (t){x^n}$:

$$ {\partial_t}\sum\limits_n {{p_n}{x^n}} = - g\sum\limits_n {{p_n}{x^n}} - \sum\limits_n {n{p_n}{x^n}} + g\sum\limits_n {{p_{{n - 1}}}{x^n}} + \sum\limits_n {(n + 1)} {p_{{n + 1}}}{x^n}. $$

(193)

The LHS and the first term on the RHS reduce directly to functions of G. The third and fourth terms on the RHS benefit from the same index shifts applied to obtain Eq. 189, giving

$$ {\partial_t}G = - gG - \sum\limits_n {{p_n}n{x^n}} + g\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}{x^{{n{^\prime} + 1}}}} + \sum\limits_{{n{^\prime}{^\prime}}} {n{^\prime}{^\prime}{p_{{n{^\prime}{^\prime}}}}{x^{{n{^\prime}{^\prime} - 1}}}}. $$

(194)

We may now eliminate bare appearances of n in the second and fourth terms on the RHS using $n{x^n} = x{\partial_x}{x^n}$ and $n{x^{{n - 1}}} = {\partial_x}{x^n}$ respectively, giving

$$ \eqalign{{\partial_t}G = - gG - \sum\limits_n {{p_n}x{\partial_x}{x^n}} + g\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}{x^{{n{^\prime} + 1}}}} + \sum\limits_{{n{^\prime}{^\prime}}} {{p_{{n{^\prime}{^\prime}}}}{\partial_x}{x^{{n{^\prime}{^\prime}}}}}, }$$

(195)

$$ = - gG - x{\partial_x}G + gxG + {\partial_x}G, $$

(196)

$$ = - (x - 1)({\partial_x} - g)G. $$

(197)

as in Eq. 23.

Finally, we sum the master equation (Eq. 9) over n against $|n\rangle$ to derive the equation of motion of the generating function in operator notation $|G\rangle = \sum\nolimits_n {{p_n}|n\rangle }$:

$$ {\partial_t}\sum\limits_n {{p_n}|n\rangle } = - g\sum\limits_n {{p_n}|n\rangle } - \sum\limits_n {n{p_n}|n\rangle } + g\sum\limits_n {{p_{{n - 1}}}|n\rangle } + \sum\limits_n {(n + 1){p_{{n + 1}}}|n\rangle }. $$

(198)

Again we may reduce the LHS and the first term on the RHS directly, and we may apply index shifts to the third and fourth terms on the RHS:

$$ {\partial_t}|G\rangle = - g|G\rangle - \sum\limits_n {n{p_n}|n\rangle } + g\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}} |n{^\prime} + 1\rangle + \sum\limits_{{n{^\prime}{^\prime}}} {n{^\prime}{^\prime}{p_{{n{^\prime}{^\prime}}}}|n{^\prime}{^\prime} - 1\rangle }. $$

(199)

Now defining operators ${\hat{a}^{ + }}$ and ${\hat{a}^{ - }}$ that raise and lower the protein number by 1, respectively, i.e.,

$$ {\hat{a}^{ + }}|n\rangle \equiv \left| {n + 1} \right\rangle, $$

(200)

$$ {\hat{a}^{ - }}|n\rangle \equiv n|n - 1\rangle, $$

(201)

Equation 198 can be written

$$ {\partial_t}|G\rangle = - g|G\rangle - \sum\limits_n {{p_n}{{\hat{a}}^{ + }}{{\hat{a}}^{ - }}|n\rangle } + g\sum\limits_{{n{^\prime}}} {{p_{{n{^\prime}}}}{{\hat{a}}^{ + }}|n{^\prime}\rangle } + \sum\limits_{{n{^\prime}{^\prime}}} {{p_{{n{^\prime}{^\prime}}}}{{\hat{a}}^{ - }}|n{^\prime}{^\prime}\rangle }, $$

(202)

$$ = - g|G\rangle - {\hat{a}^{ + }}{\hat{a}^{ - }}|G\rangle + g{\hat{a}^{ + }}|G\rangle + {\hat{a}^{ - }}|G\rangle, $$

(203)

$$ = - ({\hat{a}^{ + }} - 1)({\hat{a}^{ - }} - g)|G\rangle $$

(204)

as in Eq. 50.

Appendix C.Statistics of the Birth–Death Distribution

In this appendix we calculate the means of a birth–death distribution. In the simplest case, starting directly from the solution in Eq. 16 we obtain:

$$ \langle n\rangle = \sum\limits_n {n{p_n}} = \sum\limits_n n \frac{{{g^n}}}{{n!}}{{\hbox{e}}^{{ - g}}} = {{\hbox{e}}^{{ - g}}}g{\partial_g}\sum\limits_n {\frac{{{g^n}}}{{n!}}} = g. $$

(205)

Similarly we can calculate

$$ \langle n(n - 1)\rangle = \sum\limits_n n (n - 1)\frac{{{g^n}}}{{n!}}{{\hbox{e}}^{{ - g}}} = {{\hbox{e}}^{{ - g}}}{g^2}\partial_g^2\sum\limits_n {\frac{{{g^n}}}{{n!}}} = {g^2}, $$

(206)

which gives us the variance

$$ \langle {n^2}\rangle - {\langle n\rangle^2} = {g^2} + g - {g^2} = g. $$

(207)

These results can also be obtained in the operator formalism, and the fact that for steady state $|G\rangle = \left| {j = 0} \right\rangle$. For example:

$$ \langle n\rangle = {\left. {\langle x|{{\hat{a}}^{ + }}{{\hat{a}}^{ - }}|j = 0\rangle } \right|_{{x = 1}}} = {\left. {\langle x|{{\hat{a}}^{ + }}{{\hat{a}}^{ - }}|G\rangle } \right|_{{x = 1}}} = \langle x|{\hat{a}^{ + }}{\hat{a}^{ - }}\sum\limits_n {{{\left. {{p_n}|n\rangle } \right|}_{{x = 1}}}} = \sum\limits_n {{{\left. {n{p_n}{x^n}} \right|}_{{x = 1}}}} = \sum\limits_n {n{p_n}}. $$

(208)

To use the number operator on the $|j = 0\rangle$ state we need to rewrite it in terms of ${\hat{a}^{ + }} = {\hat{b}^{ + }} + 1$ and ${\hat{a}^{ - }} = {\hat{b}^{ - }} + g$. Acting to the left we obtain:

$$ \langle n\rangle = {\langle x|({\hat{b}^{ + }}{\hat{b}^{ - }} + g + {\hat{b}^{ - }} + g{\hat{b}^{ + }})|0\rangle_{{x = 1}}} = {\langle x|g|0\rangle_{{x = 1}}} + {\langle x|g|1\rangle_{{x = 1}}} = g. $$

(209)

We can also calculate the mean of the bursty process. Multiplying Eq. 148 by n and summing over n, we use,

$$ \sum\limits_{{n = 0}}^{\infty } {n{p_{{n - B}}}} = \sum\limits_{{n = 0}}^{\infty } {(n + B)} {p_n} = \langle n\rangle + B, $$

(210)

to obtain:

$$ g(\langle n\rangle + B) - g\langle n\rangle + \langle {n^2}\rangle - \langle n\rangle - \langle {n^2}\rangle = 0, $$

(211)

which is solved by:

$$ \langle n\rangle = gB. $$

(212)

The second moment $\langle {n^2}\rangle = \sum {{n^2}} {p_n}$ is calculated the same way:

$$ \langle {n^2}\rangle = \frac{1}{2}\left( {2g\langle n\rangle B + \langle n\rangle + g{B^2}} \right) = {(gB)^2} + \frac{{g{B^2} + gB}}{2}, $$

(213)

which yields the variance as

$$ \delta {n^2} = \langle {n^2}\rangle - {\langle n\rangle^2} = \frac{{gB}}{2}(1 + B). $$

(214)

Appendix D.Asymptotic Distributions for Large Protein Number

Here we show that in the limit of large protein number n, the steady state of the birth–death process (the Poisson distribution, Eq. 16) approaches a Gaussian distribution with mean and variance equal to g, the ratio of production to degradation rates. We then show that in the same limit, the steady state solution to the birth–death Fokker–Planck equation (Eq. 84) also approaches a Gaussian distribution with mean and variance equal to g.

The large-n limit of the Poisson distribution is most conveniently evaluated by first taking the log, which allows one to make use of Stirling’s approximation, $\log n! \approx n\log n - n + \log \sqrt {{2\pi n}}$ for large n:

$$ \log p(n) \approx - g + n\log g - n\log n + n - \log \sqrt {{2\pi n}}. $$

(215)

The derivative

$$ {\partial_n}\log p(n) = \log \frac{g}{n} + \mathcal{O}(1/n) $$

(216)

vanishes at the maximum

$$ n = g $$

(217)

about which we Taylor expand to second order:

$$ \log p(n) \approx \log p(g) + \frac{1}{2}{(n - g)^2}\partial_n^2{[\log p(n)]_g}, $$

(218)

$$ = - \log \sqrt {{2\pi g}} - \frac{{{{(n - g)}^2}}}{{2g}}. $$

(219)

The result is a Gaussian distribution with mean and variance equal to g:

$$ p(n) = \frac{1}{{\sqrt {{2\pi g}} }}{{\hbox{e}}^{{ - {{(n - g)}^2}/(2g)}}}. $$

(220)

The large-n limit of the Fokker–Planck distribution is similarly evaluated: the log and its derivatives are

$$ \log p(n) = \log \frac{1}{{\mathcal{Z}g}} + (4g - 1)\log \left( {1 + \frac{n}{g}} \right), $$

(221)

$$ {\partial_n}\log p(n) = \frac{{4g - 1}}{{n + g}} - 2, $$

(222)

$$ \partial_n^2\log p(n) = \frac{{1 - 4g}}{{{{(n + g)}^2}}}. $$

(223)

The first derivative vanishes at $n = g - 1/2 \approx g$ for large g, at which the second derivative evaluates to $\partial_n^2{[\log p(n)]_g} = - 1/g + 1/(4{g^2}) \approx - 1/g$. Taylor expanding to second order and exponentiating then give

$$ p(n) = \frac{1}{{\sqrt {{2\pi g}} }}{{\hbox{e}}^{{ - {{(n - g)}^2}/(2g)}}}, $$

(224)

where $\mathcal{Z}$ is eliminated by normalization.

Appendix E.Orthonormality and the Inner Product

Here we show that interpreting the generating function as a Fourier transform motivates a particular choice of inner product and conjugate state in the protein number basis.

We have defined the generating function in terms of a continuous variable x as $G(x) = \sum\nolimits_n {{p_n}{x^n}}$. We could equivalently write $x \equiv {{\hbox{e}}^{{i\varphi }}}$ and define

$$ G(\varphi ) = \sum\limits_n {{p_n}{{\hbox{e}}^{{in\varphi }}}}, $$

(225)

which makes clear that the generating function is simply the Fourier transform of the probability distribution in protein number n. In the state notation commonly used in quantum mechanics (42), Eq. 225 is written $\left\langle {\varphi |G} \right\rangle = \sum\nolimits_n {{p_n}} \left\langle {\varphi |n} \right\rangle$, where $\left\langle {\varphi |G} \right\rangle \equiv G(\varphi )$ and

$$ \left\langle {\varphi |n} \right\rangle = {{\hbox{e}}^{{in\varphi }}}. $$

(226)

Equation 226 is the representation of $|n\rangle$ in $\varphi$ space. In order to compute projections of $|n\rangle$ onto other states or itself using this representation, we must define a conjugate state $\left\langle {n|\varphi } \right\rangle$ and a consistent inner product. With complex exponentials, it is common to make the conjugate state the complex conjugate,

$$ \left\langle {n|\varphi } \right\rangle \equiv \left\langle {\varphi |n} \right\rangle ^* = {{\hbox{e}}^{{ - in\varphi }}}, $$

(227)

and the inner product the integral

$$ \left\langle {f|h} \right\rangle \equiv \int_0^{{2\pi }}{\frac{{{\text{d}}\varphi }}{{2\pi }}} \left\langle {f|\varphi }\right\rangle \left\langle {\varphi |h} \right\rangle, $$

(228)

for any f and h. Given the definition of conjugate state, this choice of inner product ensures the orthonormality of the $|n\rangle$ states:

$$ \left\langle {n|n{^\prime}} \right\rangle = \int_0^{{2\pi }} {\frac{{{\text{d}}\varphi }}{{2\pi }}} \left\langle {n|\varphi } \right\rangle \left\langle {\varphi |n{^\prime}} \right\rangle = \int_0^{{2\pi }} {\frac{{{\text{d}}\varphi }}{{2\pi }}} {{\text{e}}^{{i(n{^\prime} - n)\varphi }}} = {\delta_{{nn{^\prime}}}}. $$

(229)

We may now reinterpret these definitions in terms of our original variable x. Since $x = {{\hbox{e}}^{{i\varphi }}}$ we have ${\hbox{d}}x = i{{\hbox{e}}^{{i\varphi }}}{\hbox{d}}\varphi = ix{\hbox{d}}\varphi$, and in the orthonormality condition (Eq. 229) the integration from 0 to 2π along the real $\varphi$ line segment becomes contour integration along the unit circle in the complex x plane:

$$ \begin{array}{lll}\left\langle {n|n{^\prime}} \right\rangle =\int_0^{{2\pi }} {\frac{{{\text{d}}\varphi }}{{2\pi }}{{\text{e}}^{{- in\varphi }}}{{\text{e}}^{{in{^\prime}\varphi }}}} = \oint{\frac{{{\text{d}}x}}{{2\pi ix}}{x^{{ - n}}}{x^{{n{^\prime}}}}} &\\= \oint{\frac{{{\text{d}}x}}{{2\pi i}}\frac{1}{{{x^{{n + 1}}}}}{x^{{n{^\prime}}}}}= {\delta_{{nn{^\prime}}}}.\end{array} $$

(230)

Since we have defined

$$ \left\langle {x|n} \right\rangle = {x^n}, $$

(231)

Equation 230 suggests the definition of conjugate state

$$ \left\langle {n|x} \right\rangle = \frac{1}{{{x^{{n + 1}}}}} $$

(232)

and inner product

$$ \left\langle {f|h} \right\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \left\langle {f|x} \right\rangle \left\langle {x|h} \right\rangle $$

(233)

in x space.

With these definitions, we will find Cauchy’s theorem,

$$ \oint {\frac{{{\text{d}}x}}{{2\pi i}}\frac{{f(x)}}{{{{(x - a)}^{{n + 1}}}}} = \frac{1}{{n!}}} \partial_x^n{[f(x)]_{{x = a}}}\theta (n), $$

(234)

(with the convention $\theta (0) = 1$ for the Heaviside function) very useful in evaluating projections. For example, we may immediately use it to confirm the orthonormality condition,

$$ \left\langle {n|n{^\prime}} \right\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \frac{1}{{{x^{{n + 1}}}}}{x^{{n{^\prime}}}} = \frac{1}{{n!}}\partial_x^n{[{x^{{n{^\prime}}}}]_{{x = 0}}}\theta (n) = {\delta_{{nn{^\prime}}}}, $$

(235)

and in Appendix H we use it to compute analytic expressions for the projections between protein number states $|n\rangle$ and birth–death eigenstates $|j\rangle$.

Appendix F.Properties of the Raising and Lowering Operators

Just as in the operator treatment of the quantum harmonic oscillator (42), in this paper we have defined operators ${\hat{a}^{ + }}$ and ${\hat{a}^{ - }}$ that act on $|n\rangle$ states by raising and lowering the protein number n by, 1 respectively, i.e.,

$$ {\hat{a}^{ + }}|n\rangle = |n + 1\rangle, $$

(236)

$$ {\hat{a}^{ - }}|n\rangle = n|n - 1\rangle. $$

(237)

(note, however, that the prefactors here, 1 and n, are different than those conventionally used for the harmonic oscillator, $\sqrt {{n + 1}}$ and $\sqrt {n}$, respectively). In this appendix, we derive the actions of these operators when acting to the left, as well as their commutation relation.

The actions of ${\hat{a}^{ + }}$ and ${\hat{a}^{ - }}$ to the left can be found by projecting onto Eqs. 236 and 237 the conjugate state $\langle n{^\prime}|$:

$$ \left\langle {n{^\prime}|{{\hat{a}}^{ + }}|n} \right\rangle = \left\langle {n{^\prime}|n + 1} \right\rangle = {\delta_{{n{^\prime},n + 1}}} = {\delta_{{n{^\prime} - 1,n}}} = \left\langle {n{^\prime} - 1|n} \right\rangle, $$

(238)

$$ \left\langle {n{^\prime}|{{\hat{a}}^{ - }}|n} \right\rangle = \left\langle {n{^\prime}|n|n - 1} \right\rangle = n{\delta_{{n{^\prime},n - 1}}} = (n{^\prime} + 1){\delta_{{n{^\prime} + 1,n}}} = (n{^\prime} + 1)\left\langle {n{^\prime} + 1|n} \right\rangle. $$

(239)

The first and last expressions in each case imply

$$ \langle n{^\prime}|{\hat{a}^{ + }} = \langle n{^\prime} - 1|, $$

(240)

$$ \langle n{^\prime}|{\hat{a}^{ - }} = (n{^\prime} + 1)\langle n{^\prime} + 1|, $$

(241)

as in Eqs. 56–57.

The commutation relation between ${\hat{a}^{ - }}$ and ${\hat{a}^{ + }}$ is defined as $[{\hat{a}^{ - }},{\hat{a}^{ + }}] \equiv {\hat{a}^{ - }}{\hat{a}^{ + }} - {\hat{a}^{ + }}{\hat{a}^{ - }}$. It is evaluated by considering its action on a state $|n\rangle$:

$$ [{\hat{a}^{ - }},{\hat{a}^{ + }}]|n\rangle = {\hat{a}^{ - }}{\hat{a}^{ + }}|n\rangle - {\hat{a}^{ + }}{\hat{a}^{ - }}|n\rangle = {\hat{a}^{ - }}|n + 1\rangle - {\hat{a}^{ + }}n|n - 1\rangle = (n + 1)|n\rangle - n|n\rangle = |n\rangle. $$

(242)

The first and last expressions imply

$$ [{\hat{a}^{ - }},{\hat{a}^{ + }}] = 1, $$

(243)

just as with the quantum harmonic oscillator.

Appendix G. Eigenvalues and Eigenfunctions in the Operator Representation

Here we derive the eigenfunctions $|{\lambda_j}\rangle$ (also called eigenstates in the operator representation) and eigenvalues ${\lambda_j}$ of the operator ${\hat{L}} = {\hat{b}^{ + }}{\hat{b}^{ - }}$, for which

$$ {\hat{L}}|{\lambda_j}\rangle = {\lambda_j}|{\lambda_j}\rangle, $$

(244)

as well as the actions of the individual operators ${\hat{b}^{ + }}$ and ${\hat{b}^{ - }}$ on the eigenstates. We will find that the commutation relation

$$ \left[ {{{\hat{b}}^{ - }},{{\hat{b}}^{ + }}} \right] = 1 $$

(245)

and the existence of the steady state solution $\langle x|G\rangle = {{\hbox{e}}^{{q(x - 1)}}}$, for which

$$ {\hat{L}}|G\rangle = 0, $$

(246)

are all that are necessary to completely define the eigenvalues and eigenstates of ${\hat{L}}$. The treatment here finds many parallels with the derivation of the eigenvalue spectrum of the quantum harmonic oscillator (42).

First it is useful to compute the commutation relation of ${\hat{L}}$ with each of its components ${\hat{b}^{ + }}$ and ${\hat{b}^{ - }}$:

$$ \left[ {{{\hat{b}}^{ + }},{\hat{L}}} \right] = \left[ {{{\hat{b}}^{ + }},{{\hat{b}}^{ + }}{{\hat{b}}^{ - }}} \right] = {\hat{b}^{ + }}\left[ {{{\hat{b}}^{ + }},{{\hat{b}}^{ - }}} \right] + \left[ {{{\hat{b}}^{ + }},{{\hat{b}}^{ + }}} \right]{\hat{b}^{ - }} = - {\hat{b}^{ + }}, $$

(247)

$$ \left[ {{{\hat{b}}^{ - }},{\hat{L}}} \right] = \left[ {{{\hat{b}}^{ - }},{{\hat{b}}^{ + }}{{\hat{b}}^{ - }}} \right] = {\hat{b}^{ + }}\left[ {{{\hat{b}}^{ - }},{{\hat{b}}^{ - }}} \right] + \left[ {{{\hat{b}}^{ - }},{{\hat{b}}^{ + }}} \right]{\hat{b}^{ - }} = - {\hat{b}^{ - }}. $$

(248)

(here we have used Eq. 245 and the properties $[\,f,gh] =g[\,f,h] + [\,f,g]h$ and $[\,f,f] = 0$ for any f, g, and h). We now consider a particular eigenvalue λ and evaluate the action of ${\hat{L}}{\hat{b}^{ + }}$ on $|\lambda \rangle$:

$$ {\hat{L}}{\hat{b}^{ + }}|\lambda \rangle = {\hat{b}^{ + }}{\hat{L}}|\lambda \rangle - \left[ {{{\hat{b}}^{ + }},{\hat{L}}} \right]|\lambda \rangle = {\hat{b}^{ + }}\lambda |\lambda \rangle + {\hat{b}^{ + }}|\lambda \rangle = (\lambda + 1){\hat{b}^{ + }}|\lambda \rangle. $$

(249)

The equality of the first and last expressions,

$$ {\hat{L}}\left( {{{\hat{b}}^{ + }}|\lambda \rangle } \right) = (\lambda + 1)\left( {{{\hat{b}}^{ + }}|\lambda \rangle } \right) $$

(250)

reveals two results: (1) the existence of a state with eigenvalue $\lambda$ implies the existence of a state with eigenvalue $\lambda + 1$, and (2) the state with eigenvalue $\lambda + 1$ is proportional to ${\hat{b}^{ + }}|\lambda \rangle$. By induction, the first result means that the eigenvalues are spaced by 1; moreover, the existence of a state with eigenvalue zero (the steady state solution; Eq. 246) anchors the eigenvalues to the integers, i.e.,

$$ {\lambda_j} = j $$

(251)

for integer j. The second result can now be written explicitly

$$ {\hat{b}^{ + }}|j\rangle = \left| {j + 1} \right\rangle, $$

(252)

where we are free by normalization to set the proportionality constant to 1. Equation 252 demonstrates that ${\hat{b}^{ + }}$ is a raising operator for the eigenstates $|j\rangle$.

Evaluating the action of ${\hat{L}}{\hat{b}^{ - }}$ on eigenstate $|j\rangle$ similarly reveals that ${\hat{b}^{ - }}|j\rangle$ is proportional to $|j - 1\rangle$ (cf. Eq. 249); consistency with the eigenvalue equation (Eq. 244) requires that the proportionality constant be j, giving

$$ {\hat{b}^{ - }}|\,j\rangle = j|\,j - 1\rangle. $$

(253)

Equation 253 demonstrates that ${\hat{b}^{ - }}$ is a lowering operator for the eigenstates $|j\rangle$. The operators ${\hat{b}^{ + }}$ and ${\hat{b}^{ - }}$ raise and lower $|j\rangle$ as ${\hat{a}^{ + }}$ and ${\hat{a}^{ - }}$ do $|n\rangle$; therefore they act to the left as (cf. Eqs. 238–239)

$$ \langle \,j|{\hat{b}^{ + }} = \langle \,j - 1|, $$

(254)

$$ \langle \,j|{\hat{b}^{ - }} = (\,j + 1)\langle \,j + 1|. $$

(255)

Equation 253 imposes a floor on the eigenvalue spectrum, since ${\hat{b}^{ - }}|0\rangle = 0$, and no lower eigenstates can be generated. Therefore the eigenvalues are limited to nonnegative integers:

$$ j \in \{ 0,1,2,3, \ldots \}. $$

(256)

Equation 252 describes how any state can be obtained from the $j = 0$ state:

$$ |j\rangle = {({\hat{b}^{ + }})^j}|0\rangle. $$

(257)

Recalling that ${\hat{b}^{ + }} = {\hat{a}^{ + }} - 1$ and that $|0\rangle$ is the steady state solution, Eq. 257 can be used to derive the representation of the eigenfunctions in x space, $\langle x|j\rangle$. Projecting $\langle x|$ onto Eq. 257 and recalling that ${\hat{a}^{ + }}$ corresponds to x (Eq. 54) and that $\langle x|0\rangle = {{\hbox{e}}^{{g(x - 1)}}}$ (Eq. 24), we obtain

$$ \langle x|j\rangle = {(x - 1)^j}{{\hbox{e}}^g}^{{(x - 1)}}. $$

(258)

We may now define the conjugate state $\langle j|x\rangle$ such that the $|j\rangle$ are orthonormal under the inner product defined in Eq. 233:

$$ {\delta_{{jj{^\prime}}}} = \langle j|j{^\prime}\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \langle j|x\rangle \langle x|j{^\prime}\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \langle j|x\rangle {(x - 1)^{{j{^\prime}}}}{{\hbox{e}}^{{g(x - 1)}}} = \oint {\frac{{{\text{d}}y}}{{2\pi i}}} y^{j{^\prime}}f_{j(y)}, $$

(259)

where $y \equiv x - 1$ and ${f_j}(x - 1) \equiv {{\hbox{e}}^{{g(x - 1)}}}\left\langle {j|x} \right\rangle$. Equation 259 is equivalent to Eq. 230, from which we identify ${f_j}(y) = 1/{y^{{j + 1}}}$ and thus

$$ \left\langle {j|x} \right\rangle = \frac{{{{\text{e}}^{{ - g(x - 1)}}}}}{{{{(x - 1)}^{{j + 1}}}}}. $$

(260)

We have shown that the operator ${\hat{L}} = {\hat{b}^{ + }}{\hat{b}^{ - }}$ has nonnegative integer eigenvalues j and eigenfunctions (in x space) given by Eqs. 258 and 260.

Appendix H. Overlaps Between Protein Number States and Eigenstates

Here we describe two methods for computing the overlaps $\left\langle {n|j} \right\rangle$ and $\left\langle {j|n} \right\rangle$ between the protein number states $|n\rangle$ and the eigenstates $|j\rangle$: by contour integration and by recursive updating.

The first method evaluates the overlaps using the inner product defined in Eq. 233 and Cauchy’s theorem (Eq. 234). Recalling the representations in x space of the protein number states and eigenstates (Eqs. 46, 48, and 65–66), the first overlap becomes

$$ \left\langle {n|j} \right\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \left\langle {n|x} \right\rangle \left\langle {x|j} \right\rangle = \oint {\frac{{{\text{d}}x}}{{2\pi i}}} \,\,\,\frac{{{{\text{e}}^{{g(x - 1)}}}{{(x - 1)}^j}}}{{{x^{{n + 1}}}}} = \frac{1}{{n!}}\partial_x^n{[{{\hbox{e}}^{{g(x - 1)}}}{(x - 1)^j}]_{{x = 0}}}. $$

(261)

Repeated derivatives of a product follow a binomial expansion, i.e.,

$$ \left\langle {n|j} \right\rangle = \frac{1}{{n!}}\sum\limits_{{\ell = 0}}^n {\frac{{n!}}{{\ell !(n - \ell )!}}} \partial_x^{{n - \ell }}{[{{\hbox{e}}^{{g(x - 1)}}}]_{{x = 0}}}\partial_x^{\ell }{[{(x - 1)^j}]_{{x = 0}}}, $$

(262)

$$ \hskip 2pc= \sum\limits_{{\ell = 0}}^n {\frac{{n!}}{{\ell !(n - \ell )!}}} [{g^{{n - \ell }}}{{\hbox{e}}^{{ - g}}}]\left[ {\frac{{j!}}{{(j - \ell )}}{{( - 1)}^{{j - \ell }}}\theta (j - \ell )} \right], $$

(263)

$$ = {( - 1)^j}{{\text{e}}^{{ - g}}}{g^n}j!{\xi_{{nj}}},$$

(264)

where we define

$$ {\xi_{{nj}}} \equiv \sum\limits_{{\ell = 0}}^{{\min (n,j)}} {\frac{1}{{\ell !(n - \ell )!(j - \ell )!{{( - g)}^{\ell }}}}}. $$

(265)

Following similar steps, the conjugate overlap evaluates to

$$ \left\langle {j|n} \right\rangle = n!{( - g)^j}{\xi_{{nj}}}, $$

(266)

with ${\xi_{{nj}}}$ as in Eq. 265. For the special case $j = 0$, Eqs. 264 and 266 reduce to Eqs. 77–78.

It is more computationally efficient to compute the overlaps recursively using rules that can be derived from the raising and lowering operations (Eqs. 52–53, 56–57, and 67–70). For example, using the raising operators,

$$ \left\langle {n|j + 1} \right\rangle = \left\langle {n|{{\hat{b}}^{ + }}|j} \right\rangle = \left\langle {n|({{\hat{a}}^{ + }} - 1)|j} \right\rangle = \left\langle {n - 1|j} \right\rangle - \left\langle {n|j} \right\rangle, $$

(267)

which can be initialized using $\left\langle {n|0} \right\rangle = {{\hbox{e}}^{{ - g}}}{g^n}/n!$ (Eq. 264) and updated recursively in j. Equation 267 makes clear that in n space the $(j + 1)$th mode is simply the (negative of the) discrete derivative of the jth mode. The lowering operators give an alternative update rule,

$$ (n + 1)\left\langle {n + 1|j} \right\rangle = \left\langle {n|{{\hat{a}}^{ - }}|j} \right\rangle = \left\langle {n|({{\hat{b}}^{ - }} + g)|j} \right\rangle = j\left\langle {n|j - 1} \right\rangle + g\left\langle {n|j} \right\rangle, $$

(268)

which can be initialized using $\left\langle {0|j} \right\rangle = {( - 1)^j}{{\hbox{e}}^{{ - g}}}$ (Eq. 264) and updated recursively in n. One may similarly derive recursion relations for $\left\langle {j|n} \right\rangle$, i.e.,

$$ \left\langle \,{j|n + 1} \right\rangle = \left\langle \,{j - 1|n} \right\rangle + \left\langle \,{j|n} \right\rangle, $$

(269)

$$ (\,j + 1)\left\langle \,{j + 1|n} \right\rangle = n\left\langle \,{j|n - 1} \right\rangle - g\left\langle \,{j|n} \right\rangle, $$

(270)

initialized with $\left\langle {j|0} \right\rangle = {( - g)^j}/j!$ or $\left\langle {0|n} \right\rangle = 1$, respectively (Eq. 266) and updated recursively in n or j respectively. Two-term recursion relations can be similarly derived from the full operator ${\hat{b}^{ + }}{\hat{b}^{ - }}$ (29).

Appendix I. The Equivalence of the Fokker–Planck and Langevin Descriptions

In this appendix, we start from the Langevin equation defined in Eqs. 93 and 94 and derive the Fokker–Planck equation in Eq. 81. As described in Subheading 4, the observed trajectory is one realization ${r_{\eta }}(t)$ of the random stochastic process with Gaussian noise as presented in Eq. 92:

$$ {p_n}(t) = \int {\mathcal{D}\eta \delta (n - {r_{\eta}}(t))P[\eta ] \equiv \left\langle {\delta (n - {r_{\eta }}(t))}\right\rangle } . $$

(271)

Consider the evolution of the probability distribution:

$$ {p_n}(t + \Delta t) - {p_n}(t) = \left\langle {\delta (n - {r_{\eta }}(t + \Delta t)) - \delta (n - {r_{\eta }}(t))} \right\rangle, $$

(272)

and expand the increment in the trajectory as $r(t + \Delta t) = r(t) + \Delta r(t)$. We can now Taylor expand the difference in delta functions:

$$ {p_n}(t + \Delta t) - {p_n}(t) = \left\langle {( -\Delta r(t))\delta {^\prime}(n - {r_{\eta }}(t)) + \frac{1}{2}{{( -\Delta r(t))}^2}\delta {^\prime\prime}(n - {r_{\eta }}(t))}\right\rangle,$$

(273)

$$ = - {\partial_n}\left\langle {\Delta r(t)\delta (n -{r_{\eta }}(t))} \right\rangle + \frac{1}{2}\partial_n^2\left\langle{{{(\Delta r(t))}^2}\delta (n - {r_{\eta }}(t))} \right\rangle,$$

(274)

where the primes denote derivatives in n. Using the Langevin equation to calculate the increments in the trajectories:

$$ \Delta r(t) = v[r(t)]\Delta t + \eta (t)\Delta t, $$

(275)

we obtain:

$$ \left\langle {\Delta r(t)\delta (n - {r_{\eta }}(t))}\right\rangle = \left\langle {v[r(t)]\Delta t\delta (n - {r_{\eta}}(t))} \right\rangle + \left\langle {\eta (t)\Delta t\delta (n -{r_{\eta }}(t))} \right\rangle, $$

(276)

$$ = v[r(t)]\Delta t\left\langle {\delta (n - {r_{\eta}}(t))} \right\rangle = v(v)\Delta t{p_n}(t). $$

(277)

$$ \left\langle {{{(\Delta r(t))}^2}\delta (n - {r_{\eta }}(t))} \right\rangle = {(\Delta t)^2}\left\langle {{{(v[r(t)])}^2}\delta (n - {r_{\eta }}(t))} \right\rangle + 2\left\langle {v[r(t)]\eta (t)\delta (n - {r_{\eta }}(t))} \right\rangle + \left\langle {{\eta^2}(t)\delta (n - {r_{\eta }}(t))} \right\rangle, $$

(278)

$$ = {(\Delta t)^2}{(v(n))^2}{p_n}(t) + \left\langle {{\eta^2}(t)} \right\rangle {p_n}(t), $$

(279)

where we have used (assuming a discretized process):

$$ \left\langle {\eta (t)\delta (n - {r_{\eta }}(t))} \right\rangle = \left\langle {\eta (t)} \right\rangle \left\langle {\delta (n - {r_{\eta }}(t))} \right\rangle = 0. $$

(280)

We identify

$$ D(n) = \frac{{\Delta t}}{2}\left\langle {{\eta^2}(t)} \right\rangle. $$

(281)

Putting together all elements and keeping only leading terms in $\Delta t$:

$$ \frac{{{p_n}(t + \Delta t) - {p_n}(t)}}{{\Delta t}} = - {\partial_n}[u(n)\Delta t{p_n}(t)] + \partial_n^2[D(n){p_n}(t)]. $$

(282)

Taking the limit $\Delta t \to 0$ we recover the Fokker–Planck equation:

$$ {\partial_t}{p_n}(t) = - {\partial_n}[u(n)\Delta t{p_n}(t)] + \partial_n^2[D(n){p_n}(t)]. $$

(283)

Appendix J. Derivation of the Hill Function

The Hill function can be derived as an effective production rate for an autoregulating gene with two production states (Subheading 4.1.4). The gene is found either in an inactive state (−), in which the production rate is a constant ${g_{ - }}$, or in an active state (+), in which the production rate ${w_{ + }}{n^h}$ depends on the number of proteins n, which incorporates the autoregulation; h describes the cooperativity, with $h > 0$ corresponding to activation and $h < 0$ corresponding to repression. The probability $p_n^{\pm }$ of the gene being in a given state + or − and there being n proteins evolves in time according to the master equation in Eq. 118 with ${\Omega_{{zz{^\prime}}}}$ as in Eq. 140; in the steady state:

$$ 0 = - {\mathcal{L}^{\pm }}p_n^{\pm }\pm \omega_{+} {n^h}p_n^{ - } \mp \omega - p_n^{ + }, $$

(284)

where ${\mathcal{L}^{\pm }}$ describes simple birth–death terms with constant production rates ${g_{\pm }}$ in each of the two states. We define moments of the master equation as

$$ {\pi^{\pm }} \equiv \sum\limits_n {p_n^{\pm }}, $$

(285)

$$ {\pi^{\pm }}\mu_{\ell }^{\pm } \equiv \sum\limits_n {p_n^{\pm }} {n^{\ell }}\quad {\hbox{for}}\;\ell \geqslant 1. $$

(286)

with

$$ {\pi^{ + }} + {\pi^{ - }} = 1 $$

(287)

by normalization.

Summing Eq. 284 (top signs) over n (and recalling that the birth–death terms sum to zero) gives

$$ 0 = {\omega_{ + }}\sum\limits_n {{n^h}p_n^{ - }} - \omega - \sum\limits_n {p_n^{ + }}, $$

(288)

$$ = {\omega_{ + }}{\pi^{ - }}\mu_h^{ - } - {\omega_{ -}}{\pi^{ + }} $$

(289)

which, with Eq. 287, becomes an expression for ${\pi^{ + }}$, the probability of being in the active state:

$$ {\pi^{ + }} = \frac{{\mu_h^{ - }}}{{\mu_h^{ - } + {\omega_{ + }}/{\omega_{ - }}}}. $$

(290)

We solve for $\mu_h^{ - }$ using two approximations. The first is that higher moments can be decoupled, i.e.,

$$ \mu_{{h + 1}}^{ - } \approx \mu_h^{ - }\mu_1^{ - }\quad {\hbox{for}}\;h \geqslant 1, $$

(291)

which implies that

$$ \mu_h^{ - } \approx {(\mu_1^{ - })^h}. $$

(292)

This approximation allows one to simplify the mean equation in the + state (obtained by summing the Eq. 284, top signs, against n over n):

$$ 0 = {\pi^{ + }}{g_{ + }} - \sum\limits_n {np_n^{ + } + {\omega_{ + }}} \sum\limits_n {{n^{{h + 1}}}p_n^{ - }} - \omega - \sum\limits_n {np_n^{ + }}, $$

(293)

$$ = {\pi^{ + }}{g_{ + }} - {\pi^{ + }}\mu_1^{ + } + \omega + {\pi^{ - }}\mu_{{h + 1}}^{ - } - \omega - {\pi^{ + }}\mu_1^{ + },$$

(294)

$$ \approx {\pi^{ + }}{g_{ + }} - {\pi^{ + }}\mu_1^{ + } +{\omega_{ + }}{\pi^{ - }}\mu_h^{ - }\mu_1^{ - } - \omega - {\pi^{ +}}\mu_1^{ + }, $$

(295)

$$ = {\pi^{ + }}{g_{ + }} - {\pi^{ + }}\mu_1^{ + } + \omega - {\pi^{ + }}(\mu_1^{ - } - \mu_1^{ + }), $$

(296)

where the last step uses Eq. 289. The second approximation is that transitions between states are fast, i.e., ${\omega_{ + }}\sim {\omega_{ - }} \gg 1$. This approximation allows us to neglect the first two terms of Eq. 296 compared to the third term, which implies that $\mu_1^{ - } \approx \mu_1^{ + }$. Summing Eq. 286 over ± for ℓ = 1 then gives

$$ {\pi^{ + }}\mu_1^{ + } + {\pi_{ - }}\mu_1^{ - } \approx ({\pi^{ + }} + {\pi^{ - }})\mu_1^{ - } = \mu_1^{ - } = \sum\limits_n {{p_n}n} = \bar{n}, $$

(297)

which shows that $\mu_1^{ - }$ approximates the mean of the distribution. Therefore with Eq. 292 the probability that the gene is active (Eq. 290) can be written

$$ {\pi^{ + }} = \frac{{{{\bar{n}}^h}}}{{{{\bar{n}}^h} + K}}, $$

(298)

with equilibrium constant $K \equiv {\omega_{ + }}/{\omega_{ - }}$. The effective production rate is the sum of the production rates in each state times the corresponding probabilities of being in each state:

$$ g(\bar{n}) = {g_{ - }}{\pi^{ - }} + {g_{ + }}{\pi^{ + }}, $$

(299)

$$ = {g_{ - }}\left( {1 - \frac{{{{\bar{n}}^h}}}{{{{\bar{n}}^h} + K}}} \right) + {g_{ + }}\frac{{{{\bar{n}}^h}}}{{{{\bar{n}}^h} + K}}, $$

(300)

$$ = \frac{{{g_{ - }}K + {g_{ + }}{{\bar{n}}^h}}}{{{{\bar{n}}^h} + K}}, $$

(301)

which is the Hill function (Eq. 109).

Appendix K.Limiting Case of the Two-State Gene

Here we show that the spectral solution to transcriptional bursting (Eq. 138) reduces to the hypergeometric form (Eq. 128) in the limit of Z = 2 production states. We also derive the slightly simpler expression for the special case of zero production in the inactive state.

In the case of two states (Eqs. 121–122; z = ±), Eq. 138 becomes

$$ jG_j^{\pm } + {\omega_{ \mp }}G_j^{\pm } - {\omega_{\pm }}G_j^{ \mp } = {\omega_{\pm }}\sum\limits_{{j{^\prime} < j}} {G_{{j{^\prime}}}^{ \mp }} \frac{{{{( \mp \Delta )}^{{j - j{^\prime}}}}}}{{(j - j{^\prime})!}}, $$

(302)

where $\Delta = {\Delta_{{ + - }}} = - {\Delta_{{ - + }}}$. Initializing with $G_0^{\pm } = {\omega_{\pm }}/({\omega_{ + }} + {\omega_{ - }})$ and computing the first few terms reveals the pattern

$$ G_j^{\pm } = \frac{{{\omega_{\pm }}}}{{{\omega_{ + }} +{\omega_{ - }}}}\frac{{{{( \mp \Delta)}^j}}}{{j!}}\frac{{\prod\nolimits_{{j{^\prime} = 0}}^{{j - 1}} {(j{^\prime} +{\omega_{ \mp }})} }}{{\prod\nolimits_{{j{^\prime\prime} = 0}}^{{j - 1}} {(j{^\prime\prime} +{\omega_{ + }} + {\omega_{ - }} + 1)} }}, $$

(303)

$$ \hskip 1pc= \frac{{{\omega_{\pm }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\frac{{{{( \mp \Delta )}^j}}}{{j!}}\frac{{\Gamma (j + {\omega_{ \mp }})}}{{\Gamma ({\omega_{ \mp }})}}\frac{{\Gamma ({\omega_{ + }} + {\omega_{ - }} + 1)}}{{\Gamma (j + {\omega_{ + }} + {\omega_{ - }} + 1)}}, $$

(304)

where in the second line the products are written in terms of the Gamma function. Writing the total generating function $|G\rangle = \sum\nolimits_{\pm } {|{G_{\pm }}\rangle }$ in position space recovers the hypergeometric form (Eq. 128):

$$ G(x) = \sum\limits_{\pm } {\left\langle {x|{G_{\pm }}}\right\rangle }, $$

(305)

$$ = \sum\limits_{\pm } {\sum\limits_j {\left\langle{x|{j_{\pm }}} \right\rangle \left\langle {{j_{\pm }}|{G_{\pm }}}\right\rangle } }, $$

(306)

$$ = \sum\limits_{\pm } {\sum\limits_j {{{(x - 1)}^j}{{\text{e}}^{{{g_{\pm }}(x - 1)}}}G_j^{\pm }} }, $$

(307)

$$ = \sum\limits_{\pm } {\frac{{{\omega_{\pm}}}}{{{\omega_{ + }} + {\omega_{ - }}}}} {{\text{e}}^{{{g_{\pm }}(x- 1)}}}\Phi [{\omega_{ \mp }},{\omega_{ + }} + {\omega_{ - }} + 1;\mp \Delta (x - 1)], $$

(308)

where

$$ \Phi [\alpha, \beta; u] = \sum\limits_{{j = 0}}^{\infty } {\frac{{\Gamma (j + \alpha )}}{{\Gamma (\alpha )}}} \frac{{\Gamma (\beta )}}{{\Gamma (j + \beta )}}\frac{{{u^j}}}{{j!}} $$

(309)

is the confluent hypergeometric function of the first kind.

In the limit ${g_{ - }} = 0$, Eq. 308 reads

$$ G(x) = \frac{{{\omega_{ + }}}}{{{\omega_{ + }} + {\omega_{ - }}}}{e^u}\Phi [{\omega_{ - }},{\omega_{ + }} + {\omega_{ - }} + 1; - u] + \frac{{{\omega_{ - }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\Phi [{\omega_{ + }},{\omega_{ + }} + {\omega_{ - }} + 1;u], $$

(310)

where $u \equiv g + (x - 1)$. Using the fact that (49)

$$ {{\hbox{e}}^u}\Phi [\alpha, \beta; - u] = \Phi [\beta - \alpha, \beta; u], $$

(311)

Equation 310 can be written

$$ G(x) = \frac{{{\omega_{ + }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\Phi [{\omega_{ + }} + 1,{\omega_{ + }} + {\omega_{ - }} + 1;u] + \frac{{{\omega_{ - }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\Phi [{\omega_{ + }},{\omega_{ + }} + {\omega_{ - }} + 1;u], $$

(312)

or, noting Eq. 309 and the fact that $\Gamma (s + 1) = s\Gamma (s)$ for any s,

$$ G = (x)\sum\limits_j {\left( {\frac{{{\omega_{ + }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\frac{{\Gamma (j + {\omega_{ + }} + 1)}}{{\Gamma ({\omega_{ + }} + 1)}} + \frac{{{\omega_{ - }}}}{{{\omega_{ + }} + {\omega_{ - }}}}\frac{{\Gamma (j + {\omega_{ + }})}}{{\Gamma ({\omega_{ + }})}}} \right)} \frac{{\Gamma ({\omega_{ + }} + {\omega_{ - }} + 1)}}{{\Gamma (j + {\omega_{ + }} + {\omega_{ - }} + 1)}}\frac{{{u^j}}}{{j!}}, $$

(313)

$$ = \sum\limits_j {\left( {\frac{{{\omega_{ +}}}}{{{\omega_{ + }} + {\omega_{ - }}}}\frac{{(j + {\omega_{ +}})\Gamma (j + {\omega_{ + }})}}{{{\omega_{ + }}\Gamma ({\omega_{ +}})}} + \frac{{{\omega_{ - }}}}{{{\omega_{ + }} + {\omega_{ -}}}}\frac{{\Gamma (j + {\omega_{ + }})}}{{\Gamma ({\omega_{ + }})}}}\right)} \frac{{({\omega_{ + }} + {\omega_{ - }})\Gamma ({\omega_{ +}} + {\omega_{ - }})}}{{(j + {\omega_{ + }} + {\omega_{ - }})\Gamma(j + {\omega_{ + }} + {\omega_{ - }})}}\frac{{{u^j}}}{{j!}}, $$

(314)

$$ = \sum\limits_j {\frac{{\Gamma (j + {\omega_{ +}})}}{{\Gamma ({\omega_{ + }})}}\frac{{\Gamma ({\omega_{ + }} +{\omega_{ - }})}}{{\Gamma (j + {\omega_{ + }} + {\omega_{ -}})}}\frac{{{u^j}}}{{j!}}}, $$

(315)

$$ = \Phi [{\omega_{ + }},{\omega_{ + }} + {\omega_{ -}};u], $$

(316)

as in Eq. 129. The marginal ${p_n}$ is obtained by ${p_n} = \partial_u^n{[G(x)]_0}/n!$; using Eq. 316 and the derivative of the confluent hypergeometric function,

$$ \partial_u^n\Phi [\alpha, \beta; u] = \frac{{\Gamma (n + \alpha )}}{{\Gamma (\alpha )}}\frac{{\Gamma (\beta )}}{{\Gamma (n + \beta )}}\Phi [\alpha + n,\beta + n;u], $$

(317)

one obtains

$$ {p_n} = \frac{{g_{ + }^n}}{{n!}}\frac{{\Gamma (n + {\omega_{ + }})}}{{\Gamma ({\omega_{ + }})}}\frac{{\Gamma ({\omega_{ + }} + {\omega_{ - }})}}{{\Gamma (n + {\omega_{ + }} + {\omega_{ - }})}}\Phi [{\omega_{ + }} + n,{\omega_{ + }} + {\omega_{ - }} + n; - {g_{ + }}], $$

(318)

as in Eq. 130.