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Gene Regulation and Systems Biology 2011:5 89–104

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Gene Regulation and Systems Biology

P e R S P e c T i v e

Gene Regulation and Systems Biology 2011:5

89

Biomolecular Self-Defense and Futility of High-Specificity

Therapeutic Targeting

Simon Rosenfeld

National cancer institute, ePN 3108, 6130 executive Blvd., Rockville, Maryland 20892, USA.

corresponding author email: sr212a@nih.gov

Abstract: Robustness has been long recognized to be a distinctive property of living entities. While a reasonably wide consensus has

been achieved regarding the conceptual meaning of robustness, the biomolecular mechanisms underlying this systemic property are

still open to many unresolved questions. The goal of this paper is to provide an overview of existing approaches to characterization of

robustness in mathematically sound terms. The concept of robustness is discussed in various contexts including network vulnerability,

nonlinear dynamic stability, and self-organization. The second goal is to discuss the implications of biological robustness for individual-

target therapeutics and possible strategies for outsmarting drug resistance arising from it. Special attention is paid to the concept of

swarm intelligence, a well studied mechanism of self-organization in natural, societal and artificial systems. It is hypothesized that

swarm intelligence is the key to understanding the emergent property of chemoresistance.

Keywords: biological robustness, swarm intelligence, biological networks, chemoresistance, cancer therapeutics, dynamic stability,

adaptivity

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Gene Regulation and Systems Biology 2011:5

Introduction

Robustness has been long recognized to be a distinctive

property of living entities. According to,1 robustness

is the ability of biological systems “to maintain

phenotypic stability in the face of diverse perturbations

arising from environmental changes, stochastic

events and genetic variations.” A detailed account of

the concept of biological robustness has been given

in.2 In this paper, robustness is defined as “a property

that enables the system to maintain its functionalities

against external and internal perturbations.” Being

a universal mechanism of maintaining integrity of

life, robustness also contributes to drug resistance

and imposes stringent requirements in drug design.3

In particular, single molecular targeting has been

shown to have low efficacy in many complex

diseases like cancer or diabetes.4 On the other hand,

notable success of non-steroidal anti-inflammatory

drugs (NSAID) in treating or alleviating wide range

of medical conditions suggests that low-specificity

multiple targeting may be more efficient in therapeutic

modification of complex biomolecular networks.5

While there exists a reasonably wide consensus

regarding the conceptual meaning of robustness and

its all-pervading importance in cellular functioning

and medical applications, the biomolecular mecha-

nisms underlying this systemic property are still open

to many unresolved questions. In the literature, the

attempts of theoretical explanations rarely go beyond

the analogies of automatic control theory with

strong emphasis on the concept of feedback loops.2,6

Import of engineering analogies into biology often

comes with a heavy price of tacitly adopted, but poorly

substantiated, assumptions such as linearization, sta-

tionarity, stability, and others.7 Due to mathematical

difficulties, one of the most salient features of biolog-

ical systems, that is, multiple interactions between the

system’s components, often remains beyond the scope

of existing theories. Theoretically sound handling of

these interactions inevitably leads to strongly nonlin-

ear dynamical systems of very high dimensions. The

mathematical construct of network with links to non-

linear dynamics and graph theory provides a natural

description of such systems.8–10 The property of robust-

ness is inherent in many natural and societal systems.

Notable examples include Internet, social networks,

insect colonies, and ecological systems, to name just

a few. However, it should be noted that robustness

is not an inalienable self- evident property of all net-

works. In order to formulate more precisely which

networks are indeed robust and which are not, several

prerequisites are required. First, a mathematically

definitive and self-consistent description of the net-

works should exist. Second, the concept of robustness

should be formulated in an unambiguous quantitative

manner. Third, the methods should exist for estimat-

ing the quantitative measures of robustness from

observational data. Based on the literature currently

available, it may be rightfully stated that the math-

ematical science of robustness is still in its infant

stage, and relatively few examples of reasonably well

founded methodologies have been proposed so far. As

indicated in Ref:2 “Given the importance of robust-

ness for the understanding of the principles of life and

its medical implications, it is an intriguing challenge

to formulate a mathematically solid, and possibly

unified theory of biological robustness that might

serve as a basic organizational principle of biologi-

cal systems. Such a unified theory could be a bridge

between the fundamental principles of life, medical

practice, engineering, physics and chemistry. This is

a difficult challenge in which a number of issues have

to be solved, particularly to establish mathematically

well-founded theories. However, the impact would be

enormous.”

This paper is intended to fulfill, at least in part,

the overall goal formulated above. In particular,

it provides an overview of existing approaches to

characterization of robustness in mathematically

sound terms. Among many aspects of robustness

and many ways of conceptualizing this systemic

property, special attention has been paid in this paper

to the concept of swarm intelligence, a well studied

mechanism of self-organization in many natural,

societal and artificial systems. The second goal is to

discuss the implications of biological robustness for

individual-target therapeutics and possible strategies

for outsmarting drug resistance arising from it.

Quantitative Measures

of Robustness

Intuitively, it seems natural to consider robustness

as some sort of stability. This qualitative analogy,

however, is a shaky basis for introducing the

quantitative measures of robustness. The concept

of robustness is wider than the concept of stability.

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Gene Regulation and Systems Biology 2011:5

91

As discussed in,2,11,12 robustness is a property of

maintaining the functional stability but not necessarily

the structural invariance and phenotypic stability.

In contrast, stability per se is the characteristic of

dynamic behavior of a system with a pre-specified

and invariant configuration; hence, dynamic stability

may be seen as a simple form of robustness. We

now consider several approaches to quantifying the

property of robustness.

Robustness as tolerance to attack

and resistance to damage

In this approach, robustness is seen as a characteristic

of the network in its ability to perform certain

functions under adverse conditions. It is postulated

that efficiency in performing these functions strongly

depends on the existence and density of alternative

pathways between the network’s nodes. If some links

between the nodes are broken then average lengths of

the pathways between any two nodes selected at random

may increase and the network may even become

fragmented. This increase in the average length of the

path is interpreted as degradation in performance.13,14

Such a notion of robustness is introduced on a purely

intuitive level; telecommunication networks, traffic

infrastructures, social networks, power grids, citation

networks and many others provide fertile ground for

supporting such intuition. Adopting this notion as

a starting point for further logical and mathematical

constructs, one may move on towards quantification

of robustness. To this end, the concept of efficiency

should be defined. Among many possibilities of

the kind, perhaps the simplest and intuitively most

appealing one can be introduced as follows.14 Suppose

that dij is the shortest number of steps which are

necessary to travel, or transmit information, from the

node i to the node j within the network, G; this number

is often called the network distance. Efficiency, εij, of

the {i, j} link is defined as εij

efficiency as the average of the pair-wise efficiencies

over the network

ij d

=

−1, and the global

E

N N

(

dij

i j G

( )G =

)

−

≠ ∈∑

1

1

1

(1.1)

Vulnerability to damage caused by the deactivation

of the set of nodes {i}, V{i}, may now be defined as

decrease in global efficiency: V{i} = E(G) − E{i}(G),

and the global vulnerability to damage as the average

of individual vulnerabilities over all the subsets {i} of

the same size

V G

( )

N

VG

( ),

i

i

{ }

{ }

=

∑

1

(1.2)

where N is the total number of possible combinations

{i}. Given these definitions, robustness may be

quantified as the inverse or the opposite of the V (G):

the less vulnerable is a network the more robust it is.

Two notes are in order regarding this approach to

robustness. Firstly, not all the networks of interest,

especially those of biological nature, are readily

amenable to such a definition. As an ad hoc example, in

population dynamics, the predator-prey relationships

within the food webs can hardly be characterized in

terms of transmission of information of some sort or

travel between the nodes. In molecular biology, genetic

regulatory networks can not be always reduced to the

pair-wise interactions. The list of examples for which

the above formulated quantification of robustness

is not well suited may be continued. Secondly, the

concept of robustness introduced above does not have

any direct links to the notion of dynamic stability.

In part, this is because in the above outlined graph-

theoretical approach to robustness, neither the links

nor the nodes are assumed to possess any dynamic

time-dependent properties; essentially such networks

are static.

Robustness as manifestation

of dynamic stability

In this approach, the system of interest is considered

as a nonlinear dynamical system whose behavior may

be described, at least in principle, through the laws

of interaction between the system’s components. Let

S be a dynamical system whose governing equation

is written in the form

dx/dt = F (x|θ), (1.3)

where x(t) is the time-dependent vector characterizing

the state of the system, and θ is the vector of structural

parameters of the system. Let also {xp} be a set of

fixed points, that is, the points in which F(xp|θ) = 0.

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92

Gene Regulation and Systems Biology 2011:5

In applications, such points are often referred to as

the points of equilibrium or steady states. It should be

noted, however, that in general existence of a point

of equilibrium does not mean that this equilibrium is

necessarily stable, and therefore does not automati-

cally imply that xp are the steady states. In order to

infer stability of a system at a fixed point, x0 ∈{xp},

one needs to linearize (1.3) thus transforming it to the

form

F(x) = J (x0) (x − x0), (1.4)

where J(x0) = ||∂Fi/∂xj|| is the Jacobian matrix.

According to general theory, the system is stable if all

the eigenvalues of the J(x0) have negative real parts.

This condition guarantees that any initial perturbation

will exponentially decrease with time.15 If at least one

of eigenvalues has a positive real part then the system

is unstable (a more detailed discussion may be found

in the works12,16 by this author). It should be noted

that in multidimensional systems, the conditions

of Jacobian stability impose a set very stringent

constraints of high algebraic order (such as Routh-

Hurwitz and Lyapunov criteria)17 and have very little

chance to materialize naturally.18 For example, it has

been shown numerically in19 that multidimensional

equations of chemical kinetics almost certainly are

unstable in the Jacobian sense. This conclusion has

far reaching implications.11 It suggests that observed

robustness must have much deeper roots than those

associated with the Jacobian stability. Furthermore,

the Jacobian analysis of stability provides little guid-

ance regarding the patterns of long term behavior of

the system. A key concept in studying such behavior

is the quantity called phase space compressibility,

χ(t). It is defined as the trace of the time-dependent

Jacobian matrix

χ( )( )

tt J t

ii

i

==∑

Tr ( )

J

=1

N

(1.5)

This quantity is the measure of the rate of relative

decrease or increase of the phase space volume

moving with the flow along the system’s trajectories

in its phase space. If χ(t) . 0 then the trajectories

initially lying within some small domain, Ω(t|t0), will

diverge with time, and the distance between them will

grow to infinity when t → ∞. Such behavior signifies

high sensitivity to initial conditions, and is equivalent

to asymptotic dynamic instability. In the opposite

case, when χ(t) , 0, the phase volume, Ω(t|t0), is

contracting with time. This means that initially distant

trajectories become closer to each other and ultimately

will enter a certain compact set in the phase space to

stay there forever. This situation is usually referred to

as asymptotic dynamic stability. If the volume Ω(t|t0),

is deforming with time uniformly in all directions then

initially distant trajectories asymptotically approach

the same limiting trajectory thus forming a limit cycle.

The concept of asymptotic dynamic stability provides

an avenue for quantification of robustness. If the system

is asymptotically stable then knocking the system out

of its repertoire (trajectory) will have no lasting effect

because the system is supposed to return back to the

same asymptotic domain. Note that the concept of

asymptotic dynamic stability is a concise expression of

existence of multiple negative feedback loops. On top

of that, since the limit cycles in asymptotically stable

systems may be multidimensional and inseparable,

the concept of asymptotic dynamic stability is also

a concise, mathematically self-consistent, expression

of interaction between the feedback loops belonging

to different dimensions of the system.

Robustness as manifestation

of multistability

Existence of multiple attracting domains in complex

dynamical systems (for brevity, often called

multistability) provides a mechanistic basis for a

switch-like behavior in which a system can make

a sudden jump from one attractor to a drastically

different attractor under seemingly gradual change

in stimulus, environmental factors or small random

perturbations.20 In molecular biology, multistability

is considered to be an important mechanism of cell

differentiation.21 As mentioned above, in multidimen-

sional systems, local fixed points almost certainly are

unstable in the Jacobian sense. However, existence

of multiple fixed points may drastically change the

scenario of the system’s behavior: it can travel from

one fixed point to another thus creating very complex

but dynamically stable patterns. Such patterns of

behavior have been experimentally observed in a

number of biological phenomena; the circadian

clock is a prominent example.22 Multiplicity of attrac-

tors creates complexity of the behavior and may serve

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Gene Regulation and Systems Biology 2011:5

93

as an indirect measure of the number of alternative

repertoires available for the system. The key question

arising in this context is how many attractors may exist

in the system of interest? Evaluation of the number of

attractors in multidimensional systems is a difficult

mathematical task. A notable example amenable to

direct analytical exploration has been considered in.23

In this example, dynamics of the system is governed

by the Lotka-Volterra equation, the mathematical

model widely used in population dynamics of inter-

acting species24,25

dx

dt

xx x

i

iN

i

i i ikk

k

N

=+=

=∑

εα

;, ..,1,

1

(1.6)

where {xi} are the population abundances and {εi}

are the corresponding rates of production. It has been

demonstrated in this work that existence of multi-

ple attractors is associated with the existence of the

autocatalytic cycles and can be found through the

eigenvalues of the interaction matrix, αik. Another

promising approach has been developed in26 for the

dynamical systems presented as random Boolean

(Kauffman) networks. It has been shown that in

such systems the number of attractors grows with the

system’s size. Generally, the question of number of

attractors in a large dynamical system is wide open

for further inquiry.

Robustness as tolerance

to variations of structural parameters

Tolerance to perturbation of structural parameters is

yet another property of dynamical systems that may

be interpreted as a form of robustness. As an ad hoc

example of such perturbations, let us recall that in

complex biochemical systems the kinetic rates are

temperature-dependent through the Arrhenius factor

exp(Q/RT), where Q is the caloric effect of reaction,

R is the universal gas constant, and T is the absolute

temperature.27 Hence, even moderate variations in

ambient temperature may cause drastic changes in

kinetic rates and overall dynamics of the system.

To formalize this concept, let us suppose that in the

equation (1.3) governing the system, the structural

parameters, θ, are subject to some perturbation, δθ.

Obviously, trajectories of the perturbed system will

also change, and the question arises how sensitive

are the solutions to this modification. The core

quantitative characteristic to reflect this sensitivity is

the matrix

∂∂

Fik

/ θ

(often called sensitivity matrix).

Given identical initial conditions, the evolution of

differences between the perturbed and unperturbed

solutions is described by the equation

0

−

( |) ( | )

x t

{ [x( |

i

F

)|]

[x( | )| ]} ,

t

θ θ

t

ii

i

x tt

F dt

δδδ+−=++

∫

θθθθθ θθ

(1.7)

which for sufficiently small perturbations reduces to

0

( | )

x t

(/),

δθ δθ

k

=∂∂

∑

k

∫

t

iik

F dt

θ

(1.8)

Sensitivity analysis proved to be a highly efficient

instrument in the design of robust engineering systems.

With the advent of high throughput data gathering

techniques and computational systems biology,

the concepts of sensitivity analysis began to gain

popularity in the analysis of complex biomolecular

phenomena.28 It should be noted, however, that in

highly nonlinear systems, even a small change of

parameters may throw the system into an entirely

different dynamic regime (the phenomenon known

as bifurcation).15 High degree of nonlinearity is quite

typical in the biomolecular world; hence, despite

obvious usefulness, applicability of the essentially

linear sensitivity analysis to complex biological

phenomena may be limited.

Robustness as tolerance

to random perturbations

Any biological system is functioning in the presence

of uncontrolled, and mostly unknown, disturbances

covered by the blanket term noise. There are numerous

ways of including noise in the system’s dynamics29

among which the additive model is the simplest and

intuitively most appealing

dx/dt = F(x|θ) + ξt, (1.9)

where ξt is a stochastic process. Mathematically

tractable results may be obtained by transforming

the stochastic differential equation (1.9) into the