An Uncertainty Quantification Approach to the Study of Gene Expression Robustness

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An Uncertainty Quantification Approach to the Study of Gene Expression Robustness

Pierre Degond, Shi Jin, Yuhua Zhu

To cite this version:

Pierre Degond, Shi Jin, Yuhua Zhu. An Uncertainty Quantification Approach to the Study of Gene

Expression Robustness. Methods and Applications of Analysis, International Press, In press. �hal-

02499522�

An Uncertainty Quantification Approach to the Study of Gene Expression Robustness

Pierre Degond

Shi Jin

Yuhua Zhu

Dedicated to Professor Ling Hsaios 80th birthday

Abstract

We study a chemical kinetic system with uncertainty modeling a gene regulatory network in biology. Specifically, we consider a system of two equations for the messenger RNA and micro RNA content of a cell. Our target is to provide a simple framework for noise buffering in gene expression through micro RNA production. Here the uncertainty, modeled by random variables, enters the system through the initial data and the source term. We obtain a sharp decay rate of the solution to the steady state, which reveals that the biology system is not sensitive to the initial perturbation around the steady state. The sharp regularity estimate leads to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy. Numerical experiments are conducted to verify the theoretical findings.

Key words. Gene Expression, generalized Polynomial Chaos, sensitivity analysis, spec- tral accuracy

AMS subject classifications. 35Q92, 92C37, 65M70, 65M12

1 Introduction

In this paper, we are interested in a model of a simple gene regulatory network describing the regulation of the transcription of nuclear DNA by microRNAs (further denoted byµRNA).

The synthesis of a protein from its DNA sequence involves several steps: the binding of a transcription factor (which can be a protein or another type of molecule) on the gene promotor sequence initiates the transcription of DNA into messenger RNA (further denoted by mRNA).

mRNA is later translated into proteins in the ribosomes. Here, we are specifically interested in the first step, i.e. the transcription of DNA into mRNA. This transcription is subject to a high level of noise due for instance to noise in the availability of transcription factors. Yet, cells have to perform functions with a high level of precision and some noise buffering systems must

∗Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom. ([email protected])

†School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China. ([email protected])

‡Department of Mathematics, Stanford University, Stanford, CA 94305, United States.

([email protected])

arXiv:1910.07188v1 [math.NA] 16 Oct 2019

be at play. In recent years, the role ofµRNA has attracted focus. µRNAs are very short RNA sequences which are coded by non protein-coding sequences of the nuclear DNA. They seem to have (among other roles) a role in the regulation of transcription. Indeed, in many cases, the transcription factor initiates transcription of DNA into both the mRNA and a regulatory µRNA. The synthetized µRNA binds on the mRNA and prevents its translation into proteins.

It has been argued that the main function of this regulation is to reduce the effect of noise in the transcription process (see [2, 3] and the review [10]).

Our model involves a pair of chemical kinetic equations for the mRNA andµRNA content, with source terms modeling the action of the transcription factor. The effect of the noise is taken into account by adding some uncertainty in the noise term and the initial data. We are interested in looking at how this uncertainty propagates to the mRNA content and in comparing this uncertainty between situations including µRNA production or not. The uncertainty is modeled by random variables with given probability density functions.

A classical approach to the study of noise in gene regulatory networks is through the chemical master equation [18] which is solved numerically by means of Gillespie’s algorithm [9], see e.g.

[4, 16]. Here, we use the chemical kinetic approach, which is a valid approximation of the chemical master equation when ther number of molecules is large. However, this approximation is far from being valid in a cell. This is why we mitigate this discrepancy by assuming a random availability of transcription factors. The advantage is a considerably simpler treatment than with the chemical master equation while preserving the important features of the system. An alternate approach presented in [8] considers Brownian perturbations in the chemical kinetic equations. Introducing the joint probability density for mRNA and µRNA leads to a Fokker- Planck equation which can be analytically solved under some time-scale separation hypotheses.

Underlying this approach is the idea that random perturbations do not only affect the initial condition and the source term, but are present at all times. In the present work, we restrict to random perturbation of the source term and initial data which allows us to use the simpler framework of uncertainty quantification.

We will mainly focus on two aspects of this problem. First, we study how a random pertur- bation near the steady state will affect the system by analyzing the long-time behavior of the perturbative solution in the random space in terms of the weighted Sobolev norm H_πⁿ, where π is the probability density function of the random variable. We also study the stability and the convergence rate of a numerical method to the system with uncertainty, specifically, the generalized Polynomial Chaos approximation based stochastic Galerkin (gPC-SG) method.

There are plenty of developments regarding the sensitivity analysis and convergence analysis in uncertainty quantification. For example, the solution to elliptic equations, parabolic equations, [1, 6, 7], and kinetic equations [11, 14, 13, 12, 17, 15, 21, 20]. To our knowledge, there has been no such analysis done to a system of chemical kinetic equations describing a gene regulatory network.

There are mainly two difficulties in the analysis. The first one is in the sensitivity analysis in the random space. When we do estimates on the Sobolev normH_πⁿ, the size of the nonlinear terms will increase to O(2ⁿ). This will result in a strong assumption on the initial data, that is, the initial randomness is required to be as small as O(1/2ⁿ) to get an exponential decay.

Similarly, when we do the stability analysis of the gPC-SG method, if we approximate the solution byK-th order polynomial chaos bases, the size of the nonlinear terms in the resulting

deterministic system will be O(K!). If we directly do energy estimates on the approximate solution, then we can only prove stability when the initial randomness is as small as O(1/K!).

To sum up, how to get a sharp estimate in terms ofnandKwithout strong assumption on the initial data or steady state is the main difficulty in this problem.

In this paper, we obtain a sharp decay of the random perturbation around the steady state in terms of its Sobolev norm H_πⁿ through a carefully designed weighted energy norm. Under some mild conditions on the initial data that is independent of n, we find that the random perturbation near the steady state will decay exponentially. Our results also reveal that the solution preserves the regularity in the random space. Moreover, with another weighted energy norm, we prove the stability of theK-th order gPC-SG method with an assumption on initial data independent ofK. The smoothness of the solution in the random space and the stability of the gPC-SG method allows us to prove the spectral convergence of the gPC-SG method. When approximating the numerical solution by theK-th order polynomial chaos basis, the error of the approximation solution inH_πⁿ isO(K⁻ⁿ).

This paper is organized as follows. Section 2 gives an introduction to the chemical kinetic system modeling the targeted gene regulatory network and its corresponding steady state. The main result and proof sketch about the sensitivity of the system under random perturbation near steady state is stated in Section 2.1. The proof of this result is in the following Section 3. In Section 4, the gPC-SG method is introduced and the stability and the convergence rate of this method are stated in Section 4.2. The proof of these two results are in Section 5 and 6 respectively. In Section 7 we numerically study how the presence of RNA inuences the noise in the concentration of unbound mRNA.

2 The model

Consider the following model,







∂tρ˜=S(z)−a˜ρ−cρ˜m,˜

∂tm˜ =S(z)−bm˜ −cρ˜m,˜

(2.1)

with initial data ˜ρ(0, z),m(0, z). a, b, c˜ are positive constants. Here ˜ρ(t, z),m(t, z) respectively˜ stand for the content of unbound mRNA andµRNA of a cell at timet. S(z) is the source term which models the production of mRNA and µRNA through DNA transcription. We assume that a molecule of µRNA is produced each time a molecule of mRNA is produced, hence the same source term arises in the two equations. The production of mRNA andµRNA is subject to the availability of the transcription factor, which is random. We model this randomness by assuming that the source term is a given function of a random variablez(modelling for instance the concentration of transcription factors) with probability density function π(z) on a compact setIz⊂R. The first equation of (2.1) models the decay of unbound mRNA through its binding to an unbound µRNA (the term −cρ˜m) or through other degradation mechanisms (the term˜

−a˜ρ). The second equation of (2.1) describes the decay of unboundµRNA through its binding to an unbound mRNA (the term−cρ˜m˜ again) and through other degradation mechanisms (the term −bm). Note that the binding of˜ µRNA to mRNA consumes one molecule of µRNA and one molecule of mRNA at the same time, which explains why the same loss term is involved in

the two equations. The remaining unbound mRNA is then supposed to enter the translation process into proteins through the actions of ribosomes. This step is supposed to occur later and is not included in the model.

If one sets∂tρ˜=∂tm˜ = 0, one can get the steady state ρ^∞(z),m˜^∞(z), ρ^∞=br^∞, m˜^∞=ar^∞, withr^∞(z) = 1

2c

−1 +√

∆

>0, ∆ = 1 +4cS(z)

ab >1. (2.2) Let (ρ, m) = ( ˜ρ−ρ^∞,m˜ −m˜^∞) be the random perturbative solution around the steady state, then (ρ(t, z), m(t, z)) satisfies

(∂tρ=−(a+acr^∞)ρ−bcr^∞m−cρm,

∂tm=−(b+bcr^∞)m−acr^∞ρ−cρm,

(2.3) with initial data,

ρ(0, z) = ˜ρ(0, z)−ρ^∞(z), m(0, z) = ˜m(0, z)−m˜^∞(z).

2.1 Main results and proof sketch

We are interested in the estimates for the solution (ρ, m) in the random space using the norm kρ(t)k²_Hn

π =

n

X

i=0

Z

(∂_zⁱρ)²π(z)dz, km(t)k²_Hn π =

n

X

i=0

Z

(∂_zⁱm)²π(z)dz. (2.4) There are two reasons why we are interested in this Sobolev norm. First, by studying this norm, we can understand how sensitive the system with respect to the random perturbation around the steady state is and how this perturbation evolves in time. Second, this norm gives the Sobolev regularity of the solution in the random space. We will approximate the solution by the gPC- SG method in the random space in Section 4. Such regularity allows us to prove the spectral convergence of the method.

The difficulty in the analysis is to get an estimate of kρk²_Hn

π,kmk²_Hn

π that is sharp for large n. Forn= 0, one can do standard energy estimates on akρk²+bkmk² to get an exponential decay of the random perturbation in time under a smallness assumption on initial data. We will show the result forn= 0 in the following lemma, and explain why it is not trivial to extend it ton >0 after the proof of the lemma.

Lemma 2.1. If initially, the random perturbations satisfy

kρ(0)k²_π≤ b²

4c², km(0)k²_π≤ a² 4c²,

then the perturbations(kρ(t)k²_π,km(t)k²_π)decay exponentially in time as follows, kρ(t)k²_π≤ 1

a

akρ(0)k²_π+bkm(0)k²_π

e^−at, km(t)k²_π≤ 1 b

akρ(0)k²_π+bkm(0)k²_π e^−bt. Proof. Multiplyingaρandbmto the two equations in (2.3) respectively, and then adding them

together gives, 1

2∂t aρ²+bm²

=−a²ρ²−b²m²− a²cr^∞ρ²+ 2abcr^∞mρ+b²cr^∞m²

| {z }

linear part

−acρ²m−bcm²ρ

| {z }

nonlinear part

≤ −a²ρ²−b²m²−cr^∞(aρ+bm)²

| {z }

linear part

+c²ρ²m²+a²

4 ρ²+c²ρ²m²+b² 4m²

| {z }

nonlinear part

≤ − 3a

4 −c² am²

aρ²−

3b 4 −c²

bρ²

bm²,

(2.5) where we apply Young’s inequality to the nonlinear part in the first line to obtain the first inequal- ity. Sincer^∞ defined in (2.2) is always positive for anyz∈I_z, this gives−cr^∞(aρ+bm)²≤0, so we can omit this term in the second inequality.

After we obtain the inequality as in (2.5), the exponential decay of ρ², m² follows from a smallness assumption on the initial condition. Assume the coefficients of aρ² and bm² on the RHS of (2.5) are smaller than−^a₂ and−^b₂ respectively, which is equivalent to assume

m²(0, z)≤ a²

4c², ρ²(0, z)≤ b²

4c², for allz∈Iz, (2.6) then by continuity argument, for allt >0, one has

1

2∂_t aρ²+bm²

≤ −a² 2 ρ²−b²

2m². Integrating the above equation over time, one gets,

aρ²(t) +bm²(t)≤aρ²(0) +bm²(0)− Z t

0

a²ρ(s)²ds− Z t

0

b²m(s)²ds, which implies

aρ²(t)≤aρ²(0) +bm²(0)−a² Z t

0

ρ(s)²ds, bm²(t)≤aρ²(0) +bm²(0)−b²

Z t 0

m(s)²ds.

By Grownwall’s inequality, one can get the exponential decay ofρ², m²as follows, ρ²(t, z)≤ 1

a aρ²(0, z) +bm²(0, z)

e^−at, m²(t, z)≤1

b aρ²(0, z) +bm²(0, z) e^−bt. Finally, one integrates (2.6) and the above estimates overπ(z)dz, one completes the proof of Lemma 2.1.

The difficulties of extending the results in Lemma 2.1 tokρk²_Hn

π,kmk²_Hn

π are mainly due to two reasons. First whenn= 0, the linear part in (2.5) is a negative square without any assumption onr^∞, so we can directly omit these terms in the estimates. However, if we directly do energy estimates on kρk²_Hn

π ,kmk²_Hn

π for n > 0, we have to assume Pn i=1

∂ⁱ_zr^∞

≤ O(1/n!) to make the linear part negative and this assumption is too strong. Second, the nonlinear part will be O(n(n!)²) if we directly estimate on kρk²_Hn

π,kmk²_Hn

π for n >0. This implies that one needs to

assume the initial data kρ(0)k²_Hn

π,km(0)k²_Hn

π as small as O(1/n/(n!)²) to get the exponential decay in time. We will explain it in more details in the following paragraph.

In order to simplify the notation, we set θ=aρ+bm;

ρⁿ= (ρ, ∂_zρ,· · ·, ∂_zⁿρ), similar formⁿ,θⁿ, (2.7) and letk·k²be the regular Euclidean norm,

kρⁿk²=

n

X

l=0

ρ²_l.

If we directly do energy estimates onakρⁿk²+bkmⁿk², then we will get the following inequality by taking∂_z^l (0≤l≤n) to (2.3), then multiplyingaρl, bmlrespectively and adding all equations together,

1 2∂_t

akρⁿk²+bkmⁿk²

=−a²kρⁿk²−b²kmⁿk²+

n

X

l=0

−c∂_z^l(r^∞θ) θ_l

| {z }

linear part

+

n

X

l=0

−ac∂_z^l(ρm)∂_z^lρ−bc∂_z^l(ρm)∂_z^lm

| {z }

nonlinear part

(2.8)

First, for the linear part, when n= 0, the linear part is automatically a negative square term, so we do not need to bound this term any more. However, whenn >0, since r^∞ in the linear term depends onz, so taking∂_z^l to the linear terms gives

linear part =−c

n

X

l=0

∂_z^l(r^∞θ)θ_l=−cr^∞kθⁿk²−c

n

X

l=1 l

X

i=1

l i

∂ⁱ_zr^∞ ∂^l−i_z θ

∂_z^lθ

. (2.9) Since−∂_zⁱr^∞(z), i≥1 are not necessarily negative, only the first term in the last equality of the above equation is negative. Therefore, we need to bound all other terms using the first negative term. By applying Young’s inequality and Cauchy-Schwatz inequality to all other terms gives,

−c

n

X

l=1 l

X

i=1

l i

∂_zⁱr^∞ ∂_z^l−iθ

∂_z^lθ

≤c 2

n

X

l=1 l

X

i=1

l i

∂_zⁱr^∞

∂_z^l−iθ²

+ ∂_z^lθ²

. n

[n/2]

ⁿ X

i=1

∂_zⁱr^∞

! kθⁿk²,

where [n/2] represents the smallest integer that is larger than or equal to n/2. The coefficient can be upper bounded by

n [n/2]

≤2ⁿ, this implies only when

n

X

i=1

|r^∞_i | ≤O r^∞

2ⁿ

, (2.10)

the RHS of (2.9) is non-positive. Obviously, the constraint (2.10) onr^∞(z) is too strong. Only a small set of steady states are included in this analysis. So we will develop another method to avoid that.

Second, for the nonlinear part in (2.8), since the two terms are similar, we only estimate the first nonlinear term. Applying Young’s inequality gives

n

X

l=0

ac∂_z^l(ρm)∂_z^lρ

≤c²

n

X

l=0

∂_z^l(ρm)² +a²

4 kρⁿk²=

n

X

l=0 l

X

i=0

l i

∂_zⁱm∂_z^l−iρ

!² +a²

4 kρⁿk²

≤

n

X

l=0 l

X

i=0

l i

∂_zⁱm²

! _l X

i=0

l i

∂_z^l−iρ²

! +a²

4 kρⁿk²≤ l

[l/2]

² m^l

2 ρ^l

2+a² 4 kρⁿk²

≤c²n n

[n/2]

2

kmⁿk²kρⁿk²+a²

4 kρⁿk²≤c²n2²ⁿkmⁿk²kρⁿk²+a² 4 kρⁿk²,

(2.11) where the first inequality comes from Cauchy-Schwatz inequality. One can get similar inequality for

Pn

l=0bc∂_z^l(ρm)∂_z^lm

. Therefore, if one ignores the linear terms in (2.8), one ends up with the following estimates,

1 2∂t

akρⁿk²+bkmⁿk²

≤ − 3a

4 −c²

a2²ⁿnkmⁿk²

akρⁿk²− 3b

4 −c²

b 2²ⁿnkρⁿk²

bkmⁿk², which implies that we have to assume

kρⁿ(0)k²,kmⁿ(0)k²≤O 1

2²ⁿn

(2.12) to get an exponential decay as follows

kρⁿ(t)k²≤O(e^−at), kmⁿ(t)k²≤O(e^−bt).

If one integrates the above two equations overπ(z)dz, then one will get the corresponding result in the Sobolev space. However, this result is too weak for large n. If the initial perturbation is smooth enough in the random space, thenkρ(0)k,km(0)k ∈H_πⁿ for any large n. However, by the above result, only for the initial random perturbationkρ(0)k²_Hn

z ,km(0)k²_Hn

z that are as small asO(1/4ⁿn), thenkρ(t)k²_Hn

z ,km(t)k²_Hn

z will decay exponentially in time.

In our analysis, we overcome the two difficulties mentioned above by adding a weight ω_i^∗ to ρi, mi. Then we will only have an assumption on the initial data that is independent ofn, furthermore, we only requirer^∞ to satisfy the following assumption.

Assumption 2.2. There exists a constantκsuch that, the derivative ofr^∞ in the random space can be bounded by

sup

z∈Iz

|(i+ 1)²∂_zⁱr^∞| ≤κⁱ⁺¹i!, (2.13) and it is bounded below and above by r, Rrespectively,

r≤r^∞≤R, ∀z∈I_z. (2.14)

This condition is not strict at all. Actually for any analytic functionr^∞(z) in a compact set Iz, there exists a constantC, such that

∂_zⁱr^∞

≤Cⁱ⁺¹i!, for∀i≥0, ∀z∈Iz.

Then set

κ=eC, one can always get

(i+ 1)²∂ⁱ_zr^∞

≤κⁱ⁺¹i!, for∀i≥0, ∀z∈Iz. The weight ω^∗_i we add to ρ_i, m_i inkρⁿk²,kmⁿk² is

ω^∗_i = Lⁿ⁻ⁱ κⁱ

(i+ 1)² i! ,

where κ is the constant in (2.13), L is a constant depending on κ, which we will define later.

In this weight, the term ⁽ⁱ⁺¹⁾_i! ² is used to avoid strong assumption on initial data like (2.12).

Notice that with _i!¹, the factorial in (2.11) can be absorbed into the weights, so one can get rid ofO(1/(n!)²) in the initial assumption; while the weight (i+ 1)²is used to deal withO(1/n) in the assumption. Another part of the weight ^Ln−i_κi is used to avoid strong constraint on r^∞ like (2.10). Under Assumption 2.2, the term _k¹i can be used to bound|r^∞_i |. One further notices that when iis smaller, θi will be summed for more times, so the term Lⁿ⁻ⁱ is used to balance this.

Please refer to Lemma 3.1 for details.

The ⁽ⁱ⁺¹⁾_i! ² part of the weight is first introduced in [17]; However, the assumption onr^∞and its corresponding weightLⁿ⁻ⁱ/κⁱ haven’t been developed before.

Before we present the main theorems on the sensitivity of the perturbative solution (ρ, m), we first list the frequently used notations here.

– A, Lare constants defined as,

A=

∞

X

i=1

1 i² = π²

6 , (2.15)

L=

r16Aκ²

r² + 1, (2.16)

where κis defined in (2.13).

The following Theorem is about the sensitivity of the perturbative solution (ρ, m) in the random space.

Theorem 2.3. For∀n≥0, under assumption 2.2, in addition, if initially km(0)k²_Hn

π ≤a²C₀, kρ(0)k²_Hn

π ≤b²C₀, (2.17)

then the perturbative solution to (2.3) satisfies,

kρ(t)k²_Hn

π ≤ (5νⁿn!)²

a EHⁿ_π(0)e^−at, km(t)k²_Hn

π ≤ (5νⁿn!)²

b EH_πⁿ(0)e^−bt, where EH_πⁿ(0) = akρ(0)k²_Hn

π +bkm(0)k²_Hn

π. Here C0, ν are constants independent of n, C0 = (5²2⁵Ac²)⁻¹,ν =κLandL, A, κare constants defined in (2.16), (2.15), (2.13) respectively.

Remark 2.4. The above theorem tells us that as long as the initial random perturbation around the steady state is small enough, then the perturbation will exponentially decay with a rate of e^−at, e^−bt forρ, m respectively.

3 Proof of Theorem 2.3 (The sensitivity analysis around the steady state)

In this section, we are going to analyze how EHⁿ_π =akρk²_Hn

π +bkmk²_Hn

π evolves in time by studyingEⁿ,

Eⁿ=akρⁿ_ωk²+bkmⁿ_ωk², (3.1) whereρⁿ_ω,mⁿ_ω,θⁿ_ωare similarly defined as

ρⁿ_ω= (ω^∗₀ρ, ω^∗₁∂_zρ,· · · , ω^∗_n∂ⁿ_zρ), (3.2) for weightsω_i^∗ defined as,

ω_i= (i+ 1)²

κⁱi! , ω^∗_i =Lⁿ⁻ⁱω_i. (3.3)

After taking the integration of the result for Eⁿ in the random space overπ(z)dz, we can get the results forE_πⁿ,

kρⁿ_ωk²_π= Z

kρⁿ_ωk²π(z)dz, kmⁿ_ωk²_π = Z

kmⁿ_ωk²_Hn

ππ(z)dz, E_πⁿ=akρⁿ_ωk²_π+bkmⁿ_ωk²_π. (3.4) Using the relationship between Eⁿ_π and E_Hn

π(t) = akρ(t)k²_Hn

π +bkm(t)k²_Hn

π, we can get the exponential decay forEH_πⁿ.

The most important part in the proof is stated in the following Lemma 3.1, which will be proved later.

Lemma 3.1. Forr^∞ under Condition 2.13, and any vector function ρⁿ,mⁿ,θⁿ, the following inequalities hold

n

X

l=0

(ω_l^∗)²∂^l_z(ρm)∂_z^lρ≤ 2A

γLⁿkρⁿ_ωk²kmⁿ_ωk²+ 2γ

Lⁿ kρⁿ_ωk², ∀γ >0; (3.5)

−

n

X

l=0

(ω_l^∗)²∂_z^l(r^∞θ)∂_z^lθ≤0. (3.6) Proof. See Section 3.1.

If one multipliesω_l^∗ to the two equations in (2.8) and adds the two equations together, then sumslfrom 0 ton, one has,

1

2∂tEⁿ=−a²kρⁿ_ωk²−b²kmⁿ_ωk²−c

n

X

l=0

(ω^∗_l)²∂_z^l(r^∞θ)∂_z^lθ−c

n

X

l=0

(ω^∗_l)²∂_z^l(ρm)∂_z^lθ. (3.7) Based on (3.6) in Lemma 3.1, one can omit the third term on the RHS of (3.7). Furthermore, by settingγ=^L_8cⁿ in (3.5), one can bound the nonlinear terms by

1

2∂tEⁿ≤ −a²kρⁿ_ωk²−b²kmⁿ_ωk²+c²16A

L²ⁿ kρⁿ_ωk²kmⁿ_ωk²+a²

4 kρⁿ_ωk²+b² 4 kmⁿ_ωk²

=− 3a

4 −8c²A aL²ⁿ kmⁿ_ωk²

akρⁿ_ωk²− 3b

4 −8c²A bL²ⁿ kρⁿ_ωk²

bkmⁿ_ωk².

(3.8)

Since (3.8) is similar to (2.5) in the proof of Lemma 2.1, by the continuity arguement, one can conclude that if initially,

8c²A

aL²ⁿ kmⁿ_ω(0)k²_π≤a

4, 8c²A

bL²ⁿ kρⁿ_ω(0)k²_π ≤ b

4, (3.9)

kρⁿ_ω(t)k²_π,kmⁿ_ω(t)k²_π decay as follows, kρⁿ_ω(t)k²_π≤ E_πⁿ(0)

a e^−at, kmⁿ_ω(t)k²_π≤ E_πⁿ(0)

b e^−bt. (3.10)

Now, we need to transfer (3.9) and (3.10) to the Sobolev norm we want to estimate in the random space

kρk²_Hn

π ,kmk²_Hn π

. Since

1

n! ≤ (i+ 1)²

i! ≤5, for 0≤i≤n, so one has

1

κⁿn! ≤ω_i^∗≤5Lⁿ, for 0≤i≤n, which implies that,

1 κⁿn!

2

kρk²_Hn

π ≤ kρⁿ_ω(t)k²_π ≤(5Lⁿ)²kρk²_Hn π , and similar relationship can be obtained for kmⁿ_ω(t)k²_π and kmk²_Hn

π. Therefore, the initial re- quirement (3.9) becomes,

5²8c²A

a km(0)k²_Hn π ≤ a

4, 5²8c²A

b kρ(0)k²_Hn π ≤ b

4, then

kρk²_Hn

π ,kmk²_Hn π

will decay as follows,

kρ(t)k²_Hn

π ≤(5κⁿn!Lⁿ)²EH_πⁿ(0)

a e^−at, km(t)k²_Hn

π ≤(5κⁿn!Lⁿ)²EH_πⁿ(0) b e^−bt, where E_Hn

π(0) = akρ(0)k²_Hn

π +bkm(0)k²_Hn

π and this is obtained from (3.10). The above two equations give the final results in Theorem 2.3.

3.1 Proof of Lemma 3.1

The following is the proof of Lemma 3.1.

Proof. Expanding∂_z^l(ρm) gives,

n

X

l=0

(ω^∗_l)²∂_z^l(ρm)∂_z^lρ=

n

X

l=0 l

X

i=0

(ω_l^∗)² l

i

∂ⁱ_zρ∂_z^l−im∂_z^lρ. (3.11) First notice that

(ω^∗_l)² l

i

∂_zⁱρ∂^l−i_z m∂_z^lρ

=

(l+ 1)²Lⁿ LⁱL^l−iκⁱκ^l−il!

l!

i!(l−i)!∂_zⁱρ∂_z^l−im ω^∗_l∂_z^lρ

=

(l+ 1)²

(i+ 1)²(l−i+ 1)²Lⁿ ω_i^∗∂_zⁱρ

ω^∗_l−i∂_z^l−im

ω^∗_l∂_z^lρ

≤ 2 Lⁿ

1

(i+ 1)² + 1 (l−i+ 1)²

ω^∗_i∂_zⁱρ

ω_l−i^∗ ∂_z^l−im

ω_l^∗∂_z^lρ ,

(3.12)

where the second inequality is because of

(l+ 1)²≤((i+ 1) + (l−i+ 1))²≤2(i+ 1)²+ 2(l−i+ 1)². If one sums up the first part of (3.12) overi, l, one has,

2 Lⁿ

n

X

l=0 l

X

i=0

1

(i+ 1)² ω_i^∗∂_zⁱρ

ω^∗_l−i∂^l−i_z m

ω_l^∗∂^l_zρ

≤ 1 γLⁿ

n

X

l=0 l

X

i=0

1

(i+ 1)² ω_i^∗∂_zⁱρ

ω^∗_l−i∂_z^l−im

!² + γ

Lⁿ

n

X

l=0

ω_l^∗∂_z^lρ²

≤ 1 γLⁿ

n

X

l=0 l

X

i=0

1 (i+ 1)²

! _l X

i=0

1

(i+ 1)² ω_i^∗∂_zⁱρ²

ω^∗_l−i∂_z^l−im²

! + γ

Lⁿ kρⁿ_ωk²

≤ A γLⁿ

n

X

i=0 n

X

l=i

1

(i+ 1)² ω^∗_i∂_zⁱρ²

ω^∗_l−i∂_z^l−im² + γ

Lⁿkρⁿ_ωk²

≤ A γLⁿ

n

X

i=0

1

(i+ 1)² ω^∗_i∂_zⁱρ2 n

X

l−i=0

ω^∗_l−i∂_z^l−im² + γ

Lⁿkρⁿ_ωk²

≤ A

γLⁿkρⁿ_ωk²kmⁿ_ωk²+ γ

Lⁿkρⁿ_ωk².

(3.13)

The first inequality is obtained by applying Young’s inequality, and then applying Cauchy- Schwartz inequality gives the second one. Sincel−iandiare symmetric, so the second part of (3.12) can be similarly bounded. Therefore, summing (3.12) overi, l gives an upper bound for the RHS of (3.11). This implies

n

X

l=0

(ω_l^∗)²∂_z^l(ρm)ρ_l≤ 2A

γLⁿ kρⁿ_ωk²kmⁿ_ωk²+ 2γ

Lⁿkρⁿ_ωk². For the second inequality (3.6), one first separates it into two parts,

−

n

X

l=0

(ω^∗_l)²∂_z^l(r^∞θ)∂^l_zθ=−

n

X

l=0

(ω_l^∗)²r^∞∂_z^lθ∂^l_zθ−

n

X

l=1 l

X

i=1

(ω_l^∗)² l

i

∂_zⁱr^∞∂^l−i_z θ∂_z^lθ

=−r^∞kθⁿ_ωk²− 2 Lⁿ

n

X

l=1 l

X

i=1

1

(i+ 1)²+ 1 (l−i+ 1)²

ω^∗_i∂_zⁱr^∞

ω_l−i^∗ ∂_z^l−iθ

ω_l^∗∂^l_zθ

≤ −r^∞kθⁿ_ωk²+ 1 γLⁿ

n

X

l=1 l

X

i=1

1

(i+ 1)²(ω_i^∗r_i^∞) ω^∗_l−iθ_l−i

!²

+ 1

γLⁿ

n

X

l=1 l

X

i=1

1

(l−i+ 1)²(ω^∗_ir^∞_i ) ω_l−i^∗ θ_l−i

!² + 2γ

Lⁿ

n

X

l=0

kθⁿ_ωk²

≤ −r^∞

2 kθ_ωⁿk²+ 4 L²ⁿr^∞

n

X

l=1 l

X

i=1

1

(i+ 1)²(ω^∗_ir^∞_i ) ω_l−i^∗ θ_l−i

!²

+ 4

L²ⁿr^∞

n

X

l=1 l

X

i=1

1

(l−i+ 1)²(ω_i^∗r_i^∞) ω^∗_l−iθl−i

!² ,

(3.14)

where the second equality is obtained by applying (3.12), then applying Young’s inequality gives the first inequality, and settingγ=^Ln₄^r∞ gives the last inequality. The second and third terms

in the last inequality are similar to the first term in the second line of (3.13), so according to the fourth line in (3.13), (3.14) can be further simplified to

−

n

X

l=0

(ω_l^∗)²∂^l_z(r^∞θ)θl

≤ −r^∞

2 kθⁿ_ωk²+ 4A L²ⁿr^∞

n

X

l=1 l

X

i=1

1

(i+ 1)²+ 1 (l−i+ 1)²

(ω^∗_ir^∞_i )² ω_l−i^∗ θ_l−i2

=−r^∞

2 kθⁿ_ωk²+4A r^∞

n

X

l=1 l

X

i=1

L⁻²ⁱ

(i+ 1)² + L⁻²ⁱ (l−i+ 1)²

(ωir_i^∞)² ω_l−i^∗ θl−i2

≤ −r^∞

2 kθⁿ_ωk²+4Aκ² r^∞

n

X

l=1 l

X

i=1

L⁻²ⁱ

(i+ 1)² + L⁻²ⁱ (l−i+ 1)²

ω^∗_l−iθl−i2

=−r^∞

2 kθⁿ_ωk²+4Aκ² r^∞

n

X

i=1

L⁻²ⁱ (i+ 1)²

n

X

l=i

ω^∗_l−iθ_l−i²

+4Aκ² r^∞

n

X

i=1

L⁻²ⁱ

n

X

l=i

ω^∗_l−iθ_l−i2

(l−i+ 1)²

(3.15)

where the first equality comes from the definition ofω_i^∗ in (3.3), and the second inequality is by Assumption 2.2,

sup

z∈Iz

ω_i∂ⁱ_zr^∞2

≤κ². Furthermore, since

n

X

i=1

L⁻²ⁱ (i+ 1)² ≤

n

X

i=1

L⁻²ⁱ≤ 1

(L²−1) ≤ r² 16Aκ², by the definition ofLin (2.16). Inserting it back to (3.15) gives

−

n

X

l=0

(ω^∗_l)²∂_z^l(r^∞θ)θ_l≤ −r^∞

2 kθⁿ_ωk²+r

4kθ_ωⁿk²+r

4kθⁿ_ωk²≤0, which completes the proof for the second inequality (3.6).

4 The gPC-SG method

4.1 The numerical method

In this section, we will introduce a numerical method for model (2.1), which enjoys spectral accuracy in the random space.

For random variable z with probability density function π(z), there exists a corresponding orthogonal polynomial basis{Φi}^∞_i=0 with respect to the measureπ(z)dz, which is orthonormal to each other in the weightedL²_π inner product,

Z

I_z

ΦiΦjπ(z)dz=δij, (4.1)

where δ_ij is the Kronecker delta function. The K-th order subspace is therefore spanned by {Φi}^K_i=0. As a popular numerical method, the generalized Polynomial Chaos stochastic Galarkin (gPC-SG) method is to find the approximate solution in the truncated K-th order subspace.

That is, define the approximation solution of the perturbativeρ, m in the form of, ˆ

ρ^K(t, x, z) =

K

X

i=0

ˆ

ρ_i(t, x)Φ_i(z), mˆ^K(t, x, z) =

K

X

i=0

ˆ

m_i(t, x)Φ_i(z), (4.2) then insert ˆρ^K, ˆm^K into (2.2) and do Galerkin projection, so the approximation solution ˆρ^K,mˆ^K satisfies,







∂tρˆ^K,Φj

π=

−(a+acr^∞) ˆρ^K−bcr^∞mˆ^K−cρˆ^Kmˆ^K,Φj

π, 0≤j≤K, ∂tmˆ^K,Φj

π =

−(b+bcr^∞) ˆm^K−acr^∞ρˆ^K−cρˆ^Kmˆ^K,Φj

π, 0≤j ≤K.

(4.3) Equivalently, (4.3) can be written as a system of the deterministic coefficients of ˆρ^K, ˆm^K, i.e.

the vector functions ˆρ^K(t, x) = ( ˆρ₀(t, x),· · ·,ρˆ_K(t, x))^>, ˆm^K(t, x) = ( ˆm₀(t, x),· · ·,mˆ_K(t, x))^>

satisfiy,



















∂_tρˆ^K =−aρˆ^K−acΥ ˆρ^K−bcΥ ˆm^K−c



 X

i,j

ˆ m_iS_ij^l ρˆ_j





K

l=0

,

∂_tmˆ^K =−bmˆ^K−bcΥ ˆm^K−acΥ ˆρ^K−c



 X

i,j

ˆ m_iS_ij^l ρˆ_j





K

l=0

,

(4.4)

with initial data, ˆ

ρj(0) =hρ(0, z),Φji_π, mˆj(0) =hm(0, z),Φji_π, 0≤j≤K.

HereS^l,Υ are symmetric matrices defined as S^l_ij=

Z

Iz

ΦiΦjΦlπ(z)dz, Υij= Z

Iz

r^∞ΦiΦjπ(z)dz. (4.5)

4.2 Main results and proof sketch

We will prove that the approximate solution obtained by the gPC-SG from solving the de- terministic system (4.4) has spectral accuracy. We will decompose the error of the approximate solution into two parts, one is the projection error, another is the Galerkin error. The first part is determined by the regularity of the solution (ρ, m) in the random space, while the second part is determined by the stability of the Galerkin system (4.4).

Define the projection of the analytic perturbative solution (ρ, m) onto the subspace {Φ_i}^K_i=0 as,

¯ ρ^K :=

Z

ρΦ^Kdπ(z)

·Φ^K, m¯^K :=

Z

mΦ^Kdπ(z)

·Φ^K, (4.6)

where Φ^K(z) = (Φi)^K_i=0 is the vector function that contains all basis functions up to the K-th order. Then we can decompose the error of the approximation solution ˆρ^K,mˆ^K

into two parts, ρ−ρˆ^K= (ρ−ρ¯^K)

| {z }

%^K_ρ

+ ( ¯ρ^K−ρˆ^K)

| {z }

ε^K_ρ

, (4.7)

m−mˆ^K = (m−m¯^K)

| {z }

%^K_m

+ ( ¯m^K−mˆ^K)

| {z }

ε^K_m

, (4.8)