Mathematical Biosciences 261 (2015) 83–90

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Mathematical Biosciences

journal homepage: www.elsevier.com/locate/mbs

Toxin effect on protein biosynthesis in eukaryotic cells:

A simple kinetic model

Vladas Skakauskas a,∗, Pranas Katauskis a, Alex Skvortsov b, Peter Gray b

a Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuaniab Defence Science and Technology Organisation, 506 Lorimer st., Melbourne, VIC 3207, Australia

a r t i c l e i n f o

Article history:

Received 17 June 2014

Revised 24 December 2014

Accepted 26 December 2014

Available online 5 January 2015

Keywords:

Toxin

Ricin

Antibody

Cell receptor

Protein biosynthesis

a b s t r a c t

A model for toxin inhibition of protein synthesis inside eukaryotic cells is presented. Mitigation of this effect

by introduction of an antibody is also studied. Antibody and toxin (ricin) initially are delivered outside the cell.

The model describes toxin internalization from the extracellular into the intracellular domain, its transport to

the endoplasmic reticulum (ER) and the cleavage inside the ER into the RTA and RTB chains, the release of RTA

into the cytosol, inactivation (depurination) of ribosomes, and the effect on translation. The model consists

of a set of ODEs which are solved numerically. Numerical results are illustrated by figures and discussed.

© 2015 Elsevier Inc. All rights reserved.

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. Introduction

Ricin, a natural product of the castor bean (Ricinus communis),

s highly toxic to mammalian cells. It has been used for oncolog-

cal cancer treatment, but it also has the potential to be used as

weaponised biological agent because of its stability and ease of

roduction [1,2]. Ricin poisoning can cause severe tissue damage,

nflammatory reactions and can result in death [3]. In spite of inten-

ive biomedical research no treatment is available for ricin poison-

ng. Some promising results have been shown recently using an im-

unotherapeutic approach, i.e. application of high affinity antibodies

o neutralise the effects of ricin [4,5].

The development and production of new antibodies is an expen-

ive process that usually includes extensive experimental studies. In

his context, the retrospective evaluation of antibodies with contin-

ous experimental refinement of its binding parameters aiming at

election of the best candidate may become very time and resource

onsuming. In order to reduce this experimental burden a simple

odelling framework derived from mass-action kinetics has been

ecently proposed [6–8].

The process of toxin entry into a cell, associated with bio-chemical

eactions and their toxic effects, is well documented. This includes

eceptor–toxin binding, toxin internalization, intracellular transport

o the endoplasmic reticulum (ER) and inhibition of protein synthesis

∗ Corresponding author. Tel.: +370 5 2193084; fax: +370 5 2151585.

E-mail address: vladas.skakauskas@mif.vu.lt (V. Skakauskas).

o

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ttp://dx.doi.org/10.1016/j.mbs.2014.12.004

025-5564/© 2015 Elsevier Inc. All rights reserved.

9–12]. The modelling framework reported in [6–8] ignores some fine

etails of this complex and multi-stage process and aims to capture

hem by means of a set of coupled mass action kinetics equations.

The aim of the present study is to extend the framework reported

n [6–8] and to model the last stage of the toxicological effect of ricin:

nhibition of protein synthesis in the cell.

Protein synthesis is a complex biochemical process [13,14]. Models

f protein synthesis have been analysed in numerous publications and

t is still a topic of intensive research, see [15–32].

Protein synthesis in eukaryotic cells begins with the transcription

f a messenger RNA (mRNA) by RNA polymerase from a given gene

DNA) in the nucleus of cells. This mRNA then is transported into the

ytosol where a cellular ribosome binds to it to make a new protein

rom the specific amino acids carried by aminoacylated (‘charged’)

ransfer RNAs (tRNAs). Process of incorporation (translation) of tRNA

arried amino acids into a polypeptide chain (protein) is realised ac-

ording to the sequence of the mRNA and can be divided into three

tages: (i) initiation, in which ribosome joins the mRNA and locates

he first (initiation) codon, (ii) elongation, in which the ribosome

oves in a codon by codon stepwise manner along the mRNA, us-

ng tRNA carried molecules to incorporate the appropriate encoded

mino acid at each codon, and (iii) termination, in which the last

stop) codon is recognised by a protein release factor to direct release

f the completed polypeptide from the ribosome.

The effect of ricin on protein synthesis is a result of the ability

f ricin to ‘deactivate’ (damage) a ribosome and exclude it from the

ool of ribosomes available for protein synthesis (by depurinating the

-sarcin/ricin loop of the large rRNA [33,34]). This allows us to

84 V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90

Fig. 1. Ricin pathway to ribosome: 1 – spherical surface of the external compartment

(extracellular domain �e), 2 – cell membrane (external boundary of intracellular do-

main �i , 3 – spherical envelope of ER, 4 – receptor, 5 – toxin, 6 – toxin–receptor

complex, 7 – antibody, 8 – toxin-antibody complex, 9 – ribosome.

Fig. 2. Ricin chemical transformation during pathway to ribosome: values of rate

constants are given in Table 1, notations for chemical species are given in the main

text.

Fig. 3. Chemical bindings of ricin in the ribosome pool in the cell cytoplasm: values

of rate constants are given in Table 1, notations for chemical species are given in the

main text.

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propose a simple phenomenological model of the process (mass-

action kinetics) that captures the effect of ricin as a ‘supply-demand’

constraint imposed on the ribosome pool. For the sake of simplicity

we disregard the processes of aminoacylation of tRNA (i.e., activa-

tion of tRNA during which a specific amino acid is joined to tRNA),

mRNA transcription and details of its translation by a ribosome (i.e.,

reading and linking together of two neighbouring amino acids into

a polypeptide chain) and neglect post-translational modifications of

the synthesised protein.

By employing this mass-kinetic model we also demonstrate how

introduction of an antibody can mitigate inhibition of protein synthe-

sis. More specifically, we consider a model in which the toxin (ricin)

and an antibody are initially delivered outside the cell and study their

effect on cell protein synthesis in these settings.

The paper is organised as follows. In Section 2 we present the

model. In Section 3 we discuss numerical results and summary in

Section 4 concludes the paper.

2. The model

Our model aims to capture two main bio-chemical processes as-

sociated with the toxicological effect of ricin: ricin transport to the

ribosome and the effect of ricin on the ability of the ribosome to syn-

thesise protein. The geometrical settings of the system are depicted

in Figs. 1 and 3. The schemas of chemical reactions describing trans-

formations in bio-chemical processes are presented in Figs. 2 and 3.

To make predictions that may be validated in a laboratory set-

ting, the model is based on isolated, cultured cells and other factors

involved in in vivo intoxication, such as clearance rate, non-uniform

distribution throughout the tissue, etc., are not considered. The con-

centrations of reacting species used in the model are typical of those

used in laboratory studies and the number of molecules involved is

sufficiently large, so that the system is usually well described by the

deterministic equations for the mean values of fluctuating concentra-

tions. We note that this assumption may not be always valid during

the early stages of ricin transport into the cell when the number of in-

ternalised toxin molecules is small (i.e. stochastic fluctuations caused

by a finite number of molecules in the system may become significant

[35–37]

We begin with consideration of the process of ricin transport to ri-

bosome. In order to model a geometrically extended system with

a single-cell settings we assume a spatial periodicity of the sys-

tem and impose a special boundary condition at some radius of the

ingle-cell ‘compartment’ ρe � ρm (external domain), where ρm and

e are radii of the cell membrane and external sphere, respectively.

he cell is modelled as a sphere surrounded by a permeable mem-

rane (for details see [7,8]) and ER is modelled as an enclosed con-

entric sphere of a smaller radius ρn � ρm with a partially absorbing

oundary. To shorten notations we use �e for extracellular domain

ρe � ρm) and �i for intracellular space (i.e. the domain between the

R envelope and the cell membrane, ρn � ρm), see Fig. 1.

Let toxin (ricin) and antibody be initially delivered in the extracel-

ular domain (i.e. in �e). Then according to [7,8]) the concentration of

pecies in �e is governed by equations

u̇Te = −k1euTeuAe + k−1euCe

− k3er0(k2e(1 − θ)uTe − k−2eθ), uTe(0) = u0Te,

u̇Ae = −k1euTeuAe + k−1euCe, uAe(0) = u0Ae,

u̇Ce = k1euTeuAe − k−1euCe, uCe(0) = 0,

θ̇ = k2e(1 − θ)uTe − k−2eθ − keθ, θ(0) = 0

(1)

here uTe , uAe , uCe , and θ are concentration of toxin (Te), antibody (A),

oxin-antibody complex (C), and fraction of toxin-bound receptors,

espectively, u0Te

and u0Ae

are initial concentrations of the toxin and

ntibody, k1e and k−1e are association and dissociation toxin-antibody

ate constants, k2e and k−2e are association and dissociation toxin-

ntibody rate constants of toxin–receptor binding. Parameter r0 =onst is the concentration of receptors on the cell surface, so that r0θs the concentration of toxin-bound receptors [7,8].

From a solution of this system we can determine the flux of the

nternalised toxin across the cell membrane into the intracellular do-

ain �i (this flux is equal to ker0θ ). Inside �i toxin moves toward

he ER driven by the mechanism of retrograde transport and even-

ually crosses the ER envelope [34]. By applying the mass-balance

V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90 85

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X

X + Rd → Cd, (12)

rguments for the toxin inside �i (i.e. by taking into account the in-

oming and outgoing fluxes across the cell membrane and the ER

nvelope) we can derive the following equation

˙ Ti = kek4er0θ − γik5euTi, uTi(0) = 0, (2)

here uTiis the toxin concentration inside �i, parameter γi is the

onstant for the outgoing toxin flux. Constants k3e = 3ρ2m/(ρ3

e − ρ3m)

nd k4e = 3ρ2m/(ρ3

m − ρ3n ), k5e = 3ρ2

n /(ρ3m − ρ3

n ) are geometrical fac-

ors that is a result of averaging of full set of equations of diffusion-

inetic model over the domain �e and �i, respectively, [7,8], see

ig. 1.

Inside the ER toxin (denote as T∗) reacts with the protein disul-

hide isomerase PDI to be cleaved into components RTA and RTB. This

rocess is governed by the reaction

DI + T∗k∗→ RTA + RTB (3)

ith some rate constant k∗.

Denote RTA, PDI and their concentrations inside the ER as Y , Z

nd uY , uZ , respectively. Taking into account the incoming toxin and

utgoing RTA fluxes (γ uTiand ηuY ) across the ER envelope and using

he mass action law we derive the system

u̇T∗ = 3

ρnγ uTi

− K∗uT∗ , uT∗(0) = 0,

u̇Y = K∗uT∗ − 3

ρnηuY , uY(0) = 0

(4)

here K∗ := k∗uZ , uT∗ is toxin T∗ concentration inside the ER, η is the

TA release to the cytosol (domain �i) rate constant, and we assume

Z = const for simplicity.

A part of RTA released to the cytosol is degraded with some rate

onstant η∗ as a misfolded ER envelope-associated protein via the

RAD (endoplasmic reticulum-associated protein degradation) path-

ay, see Fig. 3. The degradation rate also has a factor 3/ρn to account

or geometrical size of the system (result of averaging of correspond-

ng reaction–diffusion equations over the ER domain). Eq. (4) should

e considered as an approximation to the enzymatic reactions (3)

orresponding process in the eukaryotic cell.

The other part of RTA released to the cytosol binds to ribosomes

nd inactivates them (Fig. 2). The rate of this process is vT = k5e(η −∗)uY , where η∗ < η.

The solution of Eq. (4) will provide concentration of RTA, uY , that

enetrates to ribosome over the pathway depicted in Figs. 1 and 2.

his concentration will be used as an input in the model of ribosome

bility to synthesise protein under inhibiting influence of the toxin

term vT = k5e(η − η∗)uY in system (15).

Next we present a kinetic model of protein synthesis in eukaryotic

ells. Each eukaryotic mRNA contains information coding for only one

rotein and therefore in eukaryotic cells only one ribosome translates

ne mRNA. Consider the case where a protein P is synthesised in the

ytosol from n amino acids, A1, . . . , An. The model involves n elon-

ation steps joining one amino acid to a growing polypeptide chain,

ee Fig. 3. For simplicity we assume that all mRNAs after protein

ynthesis are translationally active (i.e., can bind to ribosomes and

epeat protein synthesis). For simplicity we disregard the process of

he tRNA aminoacylation and do not describe in detail the release

f amino acids from tRNA during mRNA translation, i.e., we assume

hat amino acid A1 interacts with the ribosome-mRNA complex, C0,

irectly forming a new complex C1, similarly A2 interacts with just

ormed complex, C1, forming the second new complex, C2, and so on.

e also assume that the pool of amino acids A1, A2, . . . , An is large

nough so that variation of their concentrations is insignificant.

Protein synthesis can be described by the schematic reactions oc-

urring in cytoplasm (i.e. in �i), Figs. 1 and 3,

+ Rk2

�k−2

C0, (5)

j−1 + Aj

k3j

�k−3j

Cj, j = 1, . . . , n, (6)

nk4→ P + X + R. (7)

ere k2, k3j, k4 and k−2, k−3j are the forward and reverse reaction rate

onstants and X := mRNA. Denoting by uR, uX , uP , uC0, uCj

, uAjand αR,

X , αP , αC0, αCj

for j = 1, . . . , n concentrations and degradation rate

onstants of respective species and applying the mass action law we

erive the following system of ODEs

u̇R = vstR − k2uXuR + k−2uC0

+ k4uCn− αRuR,

u̇X = vstX − k2uXuR + k−2uC0

+ k4uCn− αXuX,

u̇C0= k2uXuR − k−2uC0

− k31uC0uA1

+ k−31uC1− αC0

uC0,

u̇Cj= k3juCj−1

uAj− k−3juCj

− k3j+1uCjuAj+1

+ k−3j+1uCj+1− αCj

uCj, j = 1, . . . , n − 1,

u̇Cn= k3nuCn−1

uAn− k−3nuCn

− k4uCn− αCn

uCn,

u̇P = k4uCn− αPuP

(8)

here vstR and vst

X are permanent supply rates of ribosomes and mRNAs

n healthy functioning cells. An important property of system (8) is

he existence of two steady-state points (u±R , u±

X , u±P , u±

C0, . . . , u±

Cn),

iz.,

u±X = vst

X − ξu±C0

αX,

u±R = vst

R − ξu±C0

αR,

u±C0

=q ±

√q2 − 4ξ 2vst

X vstR

2ξ 2,

u±Cj

= aju±C0

, aj =j∏

s=1

gs, j = 1, . . . , n,

u±P = k4u±

Cn

αP,

q = ξ(vstX + vst

R )+ βαXαR

k2,

ξ = k31uA1+ αC0

− k4an − k−31g1,

β = k−2 + k31uA1− k−31g1 + αC0

,

gj = k3juAj

k−3j + k3j+1uAj+1+ αCj

− k−3j+1gj+1

, j = 1, . . . , n − 1,

gn = k3nuAn

k−3n + k4 + αCn

(9)

here upper index ± corresponds to a different sign in

he third expression. It can be proved that only the point

u−R , u−

X , u−P , u−

C0, . . . , u−

Cn) is positive and therefore it corresponds

o a feasible solution.

In the presence of toxin the reactions for protein synthesis (5)–(7)

hould include the new processes of the toxin–ribosome binding and

nactivation of the toxin-ribosome-mRNA complex (see Fig. 1):

+ Rk1→ Rd, (10)

+ Rk2

�k−2

C0, (11)

k3

86 V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90

Table 1

Parameters of simulations (j = 1, 2, . . . , n; mv – model value).

Parameter Values Ref.

ke 3.3 × 10−5 s−1 [41,42]

k1e 2.6 × 10−16 cm3 s−1 [40]

k2e 2.1 × 10−16 cm3 s−1 [4]

k−1e 5.2 × 10−2 s−1 [40]

k−2e 1.4 × 10−4 s−1 [4]

k1 2.15 × 10−16 cm3 s−1 mv

k2 2.15 × 10−16 cm3 s−1 mv

k−2 10−4 s−1 mv

k3 1.66 × 10−15 s−1 mv

k3j 1.5 × 10−14 cm3 s−1 mv

k−3j 10−4 s−1 mv

k4 0.5 s−1 mv

K∗ 0.8 s−1 mv

η, η∗ 2 × 10−2, 10−2 cm s−1 mv

γi 3 × 10−4 cm s−1 mv

ρn 2 × 10−4 cm [46]

ρm 10−3 cm [46]

ρe 1.8 × 10−3 cm [46]

u0Ae

6.02 × 1013 cm−3 [46]

u0Te

3.01 × 1013 cm−3 [46]

r0 1.27 × 109 cm−2 [46]

u∗ 6.02 × 1013 cm−3 [46]

uZ , uAj6.02 × 1013 cm−3 mv

vstR 6.02 × 1010 cm−3 s−1 mv

vstX 6.02 × 1010 cm−3 s−1 mv

b

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w

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Cj−1 + Ai

k3j

�k−3j

Cj, j = 1, . . . , n, (13)

Cnk4→ P + X + R. (14)

Here T is the RTA component of ricin in �i, Rd := RT is an inac-

tive complex of RTA bound to ribosome (R), Cd := XRd is an inactive

complex of Rd bound to mRNA, k1, k2, k3, k3j, k4 and k−2, k−3j are

forward and reverse reaction rate constants, see Figs. 1, 3. Applying

the mass-action law to reactions (10)–(14) of the new processes we

arrive at the following set of equations:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u̇T = vT − k1uT uR, uT(0) = 0,

u̇R = vR − k1uT uR − k2uXuR + k−2uC0

+ k4uCn− αRuR, uR(0) = u−

R ,

u̇Rd= k1uT uR − k3uXuRd

− αRduRd

, uRd(0) = 0,

u̇X = vX − k2uXuR + k−2uC0− k3uXuRd

+ k4uCn− αXuX,

uX(0) = u−X ,

u̇C0= k2uXuR − k−2uC0

− k31uC0uA1

+ k−31uC1− αC0

uC0,

uC0(0) = u−

C0,

u̇Cj= k3juCj−1

uAj− k−3juCj

− k3j+1uCjuAj+1

+ k−3j+1uCj+1− αCj

uCj, uCj

(0) = u−Cj

, j = 1, . . . , n − 1,

u̇Cn= k3nuCn−1

uAn− k−3nuCn

− k4uCn− αCn

uCn, uCn

(0) = u−Cn

,

u̇P = k4uCn− αPuP, uP(0) = u−

P ,

vR = vstR uP/u−

P ,

vX = vstX uP/u−

P ,

vT = k5e(η − η∗)uY .

(15)

Here vX , vT , vR are supply rates of mRNA (X), RTA (T), and ribosome

(R), uT is RTA concentration in �i. As in Eq. (4), system (15) is used

to approximate enzymatic reactions (10)–(14) corresponding pro-

cesses. Initial values uR(0) = u−R , uX(0) = u−

X , uP(0) = u−P , uCj

(0) = u−Cj

,

j = 0, 1, . . . , n, are determined by Eq. (9) for the positive steady-state

of system (8).

Since the production (supply) rates of ribosomes and mRNAs (pa-

rameters vR and vX in Eq. (15)) are explicitly dependent on protein

concentrations uP , a decrease of uP can eventually make the inhibition

of protein synthesis irreversible (when vR and vX become zero). The

selection of vR = vstR , vX = vst

X (i.e. independent of protein concentra-

tions) corresponds to asymptotic (recovery) state of the cell after all

toxin in the system has been consumed. Both selections have been

used in our simulations.

From pharmacological perspective the most important question

is how introduction of an antibody affects the life functions of the

cell. This effect can be characterised by comparing the cell protein

production with and without antibody. More precisely, we use con-

centrations uP|(u0Ae

=0, u0Te

>0), uP|(u0Ae

>0, u0Te

>0), and define the antibody

protection measure (potential) as a relative reduction of the inter-

nalised toxin concentration

δ(t) =uP|(u0

Ae=0, u0

Te>0)

uP|(u0Ae

>0, u0Te

>0)

. (16)

By definition, 0 ≤ δ(t) ≤ 1 with the lower values of δ(t) corre-

sponding to the more profound therapeutic effect of antibody treat-

ment.

For consistent comparison of the effect of different parameters we

transform Eqs. (1), (2), (4), (8), (9), (15) to the non-dimensional form

by using scales of τ∗ (time), l (length), and u∗ (concentration) that will

e specified in the next section. By substituting variables,

ρ = lρ̄, t = τ∗ t̄, r0 = lu∗r̄0,

f = ¯f u∗, f = uTe, uAe

, uCe, uT∗ , uZ, uY , uT , uR, uX

uRd, uP, uC0

, uCj, uAj

,

f = ¯f u∗/τ∗, f = vX, vR, vstX , vst

R ,

f = ¯f/(τ∗u∗), f = k1e, k2e, k∗, k1, k2, k3, k3j,

f = ¯f l/τ∗, f = γi, η, η∗,f = ¯f/τ∗, f = k−1e, k−2e, ke, k4, k−2, αRd

,

αX, αR, αP, αC0, αCj0, k−3j,

f = ¯f/ l, f = k3e, k4e, k5e

(17)

ith j = 1, . . . , n into Eqs. (1), (2), (4), (8), (9), (15) we can deduce

he same systems, but now in the non-dimensional form expressed

y using the dimensionless (overscored) quantities. Therefore, for

implicity in what follows, we omit the bar and treat system (1), (2),

4), (8), (9), (15) as the non-dimensional.

. Numerical results

To solve Eqs. (1), (2), (4), (15) we initially evaluated a number of

tandard MATLAB ODE solvers, viz. ode45 and ode113, ode15s [38,39].

he solver ode15s found to be most efficient (probably due to the stiff-

ess of underlying problem) and it is mainly used in our numerical

alculations. Our selection of the values of parameters for Eqs. (1),

2), (4), (8), (9), (15) was motivated by the values available in the

literature (see Table 1) with an extended range to allow exploration

and illustration of the various pharmacokinetic regimes possible in-

side the eukaryotic cell. Where experimental data were not available

we assume the value of unknown parameters typical for intracellu-

lar transport processes. Constants k1e and k−1e are from the study by

cGuinness and Mantis [40] on the kinetics of binding of a protective

onoclonal antibody (2B11) to ricin. The rate constants k2e and k−2e

re from paper of Prigent et al. [4]. Constant ke is from study by Lord

nd Spooner [41] and van Deurs et al. [42].

Values of the other kinetic parameters of the model is difficult to

stimate. To the best of our knowledge they have not been measured

irectly and reported literature. The selected values of all parameters

V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90 87

(a)

(b)

Fig. 4. Influence of forward toxin-antibody reaction rate constant k1e and parameter

γi on protein concentration uP (a) and antibody protection factor δ (b) in case vR =vst

R uP/u−P , vX = vst

X uP/u−P . k1e = 0.013, γi = 0.03 (1); k1e = 0.0042, γi = 0.03 (2); k1e =

0.0013, γi = 0.03 (3); k1e = 0.013, γi = 0.001 (4). Solid and dashed lines correspond to

value of uP determined for u0Ae

= 1 and u0Ae

= 0.

a

f

2

f

p

b

t

a

fl

δn

e

c

1

t

p

s

t

e

a

v

e

s

r

T

t

m

m

v

(a)

(b)

Fig. 5. Effect of variation of parameter η∗: 1.75 (1), 1.5 (2), 1.25 (3), 1 (4) on protein

concentration uP (a) and antibody protection factor δ (b) in case vR = vstR uP/u−

P , vX =vst

X uP/u−P . Solid and dashed lines correspond to value of uP determined for u0

Ae= 1 and

u0Ae

= 0.

t

s

i

p

f

αl

a

(

t

a

p

t

n

v

t

M

a

(

t

u

i

a

a

k

s

v

u

re presented in Table 1. Some rationale behind this selection can be

ound in [41–45].

Proteins in living organisms are formed by the bonding of only

0 amino acids to make up long polymer chains. Proteins are made

rom as few as 50 to as many as 3000 amino acids. To study such

olypeptide chain synthesis we can numerate each joined amino acid

y different index, e.g., Ai, i = 1, 2, ..., 50. We perform most calcula-

ions for the case where protein is synthesised from twenty (n = 20)

mino acids. To conduct some sensitivity analysis we study the in-

uence of the protein size on the antibody protection parameter,

(t). To this end we solved system (1), (2), (4), (8), (9), (15) for

= 50, 100, 200, and 300.

The results of the numerical solutions are presented in Figs. 4– 8. In

ach figure (excluding Fig. 8) we present the plots of protein con-

entration uP(t) (normalised with a reference value u∗ = 6.02 ×013 cm−3 typical for intercellular processes [46]) and antibody pro-

ection factor δ(t) (16) as functions of some model parameters. These

arameters include rate constants and the steady supply rates of ribo-

omes and mRNAs (vstR and vst

X ), see Table 1. For consistent comparison

he results are presented in a non-dimensional form using the refer-

nce values u∗ = 6.02 × 1013 cm−3, l = 10−2 cm, and τ∗ = 1 s−1 as

scale of concentration, length and time, respectively. These scaling

alues are typical for intercellular transport processes [46]. In all leg-

nds we use values of non-dimensional parameters (bar omitted for

implicity). Values of parameters that are not shown in legends cor-

espond to non-dimensional ones determined by using values from

able 1.

For consistent comparison, most plots correspond to the same

oxin and antibody initial concentrations and the same ribosome and

RNA supply rates.

Plots in Figs. 4, 5 and 6, 7, 8 correspond to the ribosome and

RNA supply rates normalised by the steady state concentrations,

R = vstuP/u−, vX = vstuP/u− and vR = vst , vX = vst , as described in

R P X P R X

he previous section (see Eq. (15)). The explicit dependence of the

upply rates on the protein concentration, uP , forms a feedback loop

n our system and eliminates ribosome and mRNA supply without

rotein production (an essential consistency condition for a normally

unctioning cell). For plots in Figs. 4–8 degradation rate constants

P , αR, αX , αRd, αC0

, . . . , αCn are equal to 10−3 s−1. Solid and dashed

ines in Figs. 4–8 correspond to the modelling cases with and without

ntibody, respectively.

Fig. 4 illustrates the evolution of protein concentration, uP(t),Fig. 4a) and antibody protection parameter, δ(t), (Fig. 4b) for

hree values of the rate constant, k1e = 0.013, 0.0042, 0.0013 (toxin-

ntibody forward rate), and two values of the cytosol-ER ‘interface’

arameter, γi = 0.001, 0.03 (cytosol-ER absorbtion rate). One can see

he decrease of uP as k1e decreases or γi increases. Apparently, for the

o-antibody case, function uP does not depend on k1e. Fig. 4b pro-

ides an example of the monotonic decrease of the antibody protec-

ion factor δ(t)with time that was observed in this set of simulations.

ore specifically, for a given time the parameter δ(t) increases with

decrease of k1e (forward toxin-antibody rate) and/or increase of γi

cytosol-ER absorption rate).

Fig. 5 shows the effect of the RTA destruction rate constant η∗ on

he time evolution of uP(t) (Fig. 5a) and δ(t) (Fig. 5b). The value of η∗sed in these simulations are η∗ = 1, 1.25, 1.5, 1.75 and illustrate the

ncrease of uP and δ as η∗ (RTA destruction rate parameter) grows. We

lso studied the influence of parameters k1, k3, and k4 and observed

slight decrease of uP as k1 and k3 grow and/or k4 decreases.

As mentioned above, Fig. 6 illustrates the effect of rate constant

1e on the behaviour of functions uP(t) and δ(t) in the case where the

upply rates of ribosome and mRNA are given by the steady values,

R = vstR and vX = vst

X . In the presence of antibody, the evolution of

P is always non-monotonic: for different values of parameters uP

88 V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90

(a)

(b)

Fig. 6. Influence of forward toxin-antibody reaction rate constant k1e and parame-

ter γi on protein concentration uP (a) and antibody protection factor δ (b) in case

vR = vX = 0.001. k1e = 0.013, γi = 0.03 (1); k1e = 0.0042, γi = 0.03 (2); k1e = 0.0013,

γi = 0.03 (3); k1e = 0.013, γi = 0.001 (4). Solid and dashed lines correspond to value of

uP determined for u0Ae

= 1 and u0Ae

= 0.

(a)

(b)

Fig. 7. Effect of variation of parameter η∗: 1.75 (1), 1.5 (2), 1.25 (3), 1 (4) on protein

concentration uP (a) and antibody protection factor δ (b) in case vR = vX = 0.001. Solid

and dashed lines correspond to value of uP determined for u0Ae

= 1 and u0Ae

= 0.

(a)

(b)

Fig. 8. Dependence of relative protein concentration uP(t)/u−P (a) and antibody pro-

tection factor δ (b) on amino acids number n: 20 (1), 50 (2), 100 (3), 200 (4), 300 (5) in

case vR = vX = 0.001, k1e = 0.013, γi = 0.001.

monotonically drops to local minima and then monotonically grows

to various saturation limits. But in the case where no antibody is

present uP monotonically decreases to different saturation limits. This

difference is driven by the shape of the rate function vT(t)(rate of toxin

release into the cytosol). In the presence of antibody it has a sharp

maximum at finite times that correspond to a drop of protein produc-

tion uP . The plots in Fig. 6b illustrate the dependence of antibody pro-

tection factor, δ(t), on the same parameters (dependency on the pa-

rameterγi, cytosol-ER absorption rate, is in agreement with the trends

reported in Fig. 4). We remark that contrary to the results presented

in Fig. 4b all curves in Fig. 6b tend to the non-zero stationary values.

Fig. 7 illustrates effect of the RTA destruction rate constant η∗on the time behaviour of protein production uP(t) (Fig. 7a) and

antibody protection factor δ(t) (Fig. 7b). The values of the other

parameters are the same as in Fig. 6. From Figs. 6 and 7 we ob-

serve the similar behaviour of functions uP(t) and δ(t) for cases with

and without antibody (including local minima and various satura-

tion limits). It is worth noting that according to our model when

toxin Te is completely consumed and degraded the protein con-

centration uP will grow and the cell will eventually recover. The

recovery time depends on the initial toxin and antibody concen-

tration and other parameters of the model and for some cases it

may become so large, so it would be impractical to reach the re-

covery stage. In order to explicitly model the event of cell death one

can impose a threshold condition either on recovery time or protein

production.

As a part of the sensitivity analysis of the model we also estimate

the effect of the number amino acids (protein size) on the output

of the model. In Fig. 8 we present functions uP/u−P , δ(t) calculated

for different values of n with all other parameters being equal. Cal-

culations show that the function uP/u− increases with n and tends

P

V. Skakauskas et al. / Mathematical Biosciences 261 (2015) 83–90 89

t

t

a

p

t

c

t

o

4

e

b

c

a

n

a

t

p

a

o

m

p

p

o

e

p

r

e

s

c

l

i

t

m

d

e

t

r

A

(

m

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

3463–3470.

o a saturation limit which is determined by n. Fig. 8b demonstrates

he increase of δ(t) as the protein size n grows. Importantly, the δ(t)lways increases with the protein size n.

The plots depicted in Figs. 4–8 demonstrate the dependence of

rotein production uP(t) and antibody protection parameter δ(t) on

he main parameters of the model. Although the presented modelling

ases do not comprehensively cover the parameter space of the model

hey can be used as illustrative (and sometimes revealing) examples

f the main drivers for the model outputs.

. Conclusions

Summarizing, we have presented a simple kinetic model of the

ffect of ricin on protein synthesis in the eukaryotic cell. The model is

ased on the mass-action law and consists of a set of coupled ODEs. It

an be readily solved numerically with any conventional ODE solver

nd applied to a variety of experimental and operational settings. A

umber of specific cases that include protective antibody have been

nalysed in detail. We found that the effect of ricin on protein produc-

ion can be a convoluted function of time. In contrast, the protection

otential of antibody calculated for the same modeling scenarios is

lways a decaying function of time (at least within the assumptions

f the model). This supports the observations that an antibody is the

ost effective at earlier stage of ricin poisoning. The proposed model

rovides a predictive tool and quantitative metrics for such a com-

arison.

The presented study is the first step in building an analytical model

f a variety of complex effects of ricin on the protein synthesis in the

ukaryotic cell. More work is needed to improve the fidelity of the

roposed model and this would include refining some biochemical

eactions (to cater for enzymatic rates), reducing uncertainty in the

stimation of rate constants (either by reprocessing available data

ets or by setting up a targeted experimental program), as well as in-

orporating stochastic fluctuations (that may become important for

ow toxin concentration) and including some relevant features of the

n vivo settings. This model may then be used as an accurate predictive

ool in toxicological studies. Nevertheless, even in its current form the

odel adequately captures the main trends in available experimental

ata and can be useful in ‘what-if’ analysis to define optimal param-

ters of protecting antibodies for a given test case. It may also help

o reduce the experimental burden on a pharmacological program by

educing the parameter space for antibody evaluation studies.

cknowledgements

V.S. and P.K. were supported by the Research Council of Lithuania

project number MIP-052/2012).

We wish to thank two anonymous referees for their valuable com-

ents to the manuscript and their constructive suggestions.

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