Computational Biology Modeling across Different Scales

Filippo Castiglione

and Francesco Pappalardo

Istituto per le Applicazioni del Calcolo, National Research Council of Italy, Viale Manzoni 30, 00185 Rome, Italy

Dipartimento di Scienze del Farmaco, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Keywords: Mathematical Biology, Computational Methods, Multi-Scale Models, Systems Biology.

Abstract: One of the most formidable challenges in modern biology is to get a unified view of the various

mechanisms governing the behavior and of the causal relationships among different parts of a living system.

It is coming clearer nowadays that to get such comprehensive picture computational models embracing

different observation levels in space and time have to be formulated to explain the enormous amount of data

deriving from -omic high throughput measurements methods. In this article we aim at giving a meaning to

the concept of multi-scale modeling in the framework of studies of biological systems with particular

interest in understanding human physiology in disease conditions.

1 INTRODUCTION

Mathematical models of natural phenomena intend

to describe reality. By means of the mathematical

formalism allowing logical reasoning over

designated variables we account for observations

made through experimentation. Defining the

variables of a mathematical model is a fundamental

step actually setting up the range of logical

deductions allowed by that model. For example, if

we use a variable to describe the changes of a

concentration of a protein in the blood we are

definitely overlooking the dynamics of the atoms

and the ions hence we cannot get any information

about the folding of the protein itself. The origin of

this oversight lays in a basic principle sometimes

referred to as the lex parsimoniae most commonly

known as the Ockam’s Razor. “Pluralitas non est

ponenda sine neccesitate” in very simple words

states that in the description of a phenomenon the

most useful model is the most parsimonious one in

terms of elements used. In this regard, following up

the example above, it makes little sense to describe

the laws governing the forces accounting for the

folding of the protein if we are interested in the half-

life of the protein and we can estimate its decay rate

by fitting a curve to a set of experimental data about

the concentration in the blood of that protein.

William of Ockham was a Franciscan monk and

logician who lived in the 14

century in a village of

the English county of Surrey. At that time the

principle of parsimony in describing and modeling a

natural phenomena was well reasoned. However

today, the situation is a “bit” different. The lex

parsimoniae is still valid and indeed very much used

when describing a phenomenon, but besides

classical mathematical models allowing for an exact

analytical approach, another modus operandi is now

commonly employed. This is what we can call the

synthetic approach consisting in constructing a

replica or toy of the studied system in terms of the

most important identified elements and the laws

governing the relationship among them. Actually

this approach is not new at all. The “engineer”

Leonardo used to construct toy models of flight

machines before attempting anything real-scale.

What is new today is that we can use digital

computers to construct toy models. We can instruct

extremely powerful CPUs to execute algorithms

representing entities and laws and we can then make

all kinds of conceptual experiments on those entities

and laws. This “digital synthetic” approach is

commonly referred to as simulation. Today, when

studying a certain natural phenomena, scientists first

identify elements and basic laws governing the

dynamics of the system, then they represent them as

data structures and algorithms and finally execute

the algorithms to observe how the system evolves.

The Ockam’s principle is still valid and used in the

first phase of this process but after that, thanks to the

fact that computers do the calculations, the

parsimony is forsaken, and the complexity of the

617

Castiglione F. and Pappalardo F..

Computational Biology Modeling across Different Scales.

DOI: 10.5220/0004405706170625

In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (BIOMED-2013), pages

617-625

ISBN: 978-989-8565-69-3

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

initial toy model is augmented by simply adding

new entities and laws. Indeed, with little difficulty

we can detail processes incorporating hypothetical

or experimentally derived knowledge. We can even

compose pre-constructed models of different parts of

the real system or arrange models describing reality

at different scales of observation. This holistic

approach is what in biology is called systems biology

(Kitano, 2002). The class of systemic models

therefore includes the one of multi-scale models.

Multi-scale modeling has been drawn a great

deal of attention in biological modeling and is

discussed in many recent articles and reviews (Qu et

al., 2011; Dada and Mendes, 2011; Southern et al.,

2008; Bassingthwaighte et al., 2005; Coveney and

Fowler, 2005; Engler et al., 2009; Grima, 2008). See

for example the interesting attempt to provide a

framework for multi-scale computational modeling

that is given in (Sloot and Hoekstra, 2010) together

with two examples showing how to bridge different

single-scale models.

The present article aims at giving a meaning to

the concept of multi-scale modeling in the

framework of studies of biological systems in

general with particular interest in understanding

human physiology in disease conditions. This article

provides a general introduction to the

methodological issues of multi-scale modeling. For a

more extensive reading including examples we

recommend the above-cited reviews and also

(Hunter and Nielsen, 2005; Meier-Schellersheim, et

al., 2009; Murtola et al., 2009; Schnell et al., 2007;

Bradley et al., 2011; Joshi et al., 2011).

2 LEVELS OF BIOLOGICAL

ORGANIZATION

Before we define what a multi-scale models is, it is

first necessary to make clear what it is meant to

formulate a model at a certain scale (Southern et al.,

2008). In the natural sciences, to make an

observation requires setting a temporal and a

dimensional scale. For example, freely draw from

disparate scientific fields, the phenomena of the

continental drift is better described over a time span

of million years, the evolution of a disease like

multiple sclerosis in years or decades, the immune

recognition of an infectious agent in days, the cell

cycle and circadian rhythm in twenty-four hours and

so on, to fast processes like the heart beat lasting

about a second or the fold of a protein that takes

place in microseconds and beyond. Likewise, certain

phenomena are better seen over a length or space

scale of light years, as for example the formation of

galaxies, or kilometers, like for the propagation of a

tsunami wave, or micrometers to describe the

duplication of a cell, and so on.

In general terms, while we can intuitively say if a

determined process involves cells, molecules, or

organs, it is not so simple to identify values for the

lengths at which we switch from one level to the

next (Southern et al., 2008). Major levels of

biological organization are regulated at scales of

many orders of magnitude in space and time

(Figure), with space spanning from the molecular

scale (10

-10

m) to the living organism scale (1 m), and

time from nanoseconds (10

-9

s) to years (10

s).

Figure 1: Multi-scale models of the human body targeting

complex processes span many time and length scales of

biological organization.

When combining models in a systemic way, the

challenge remains in the manner the components are

put together. Note that, in the study of complex

phenomena as for instance human pathologies, a

unified view is indeed necessary to reach a

comprehension of the various mechanisms in action

and of the causal relationships among different parts

of that complex system that is the human body (Di

Ventura et al., 2006). Complex diseases entail

phenomena ranging all scales, from observations at

the microscopic scales (from pico to micro meters)

to microscopic tissue damage and embracing

temporal events ranging from very fast processes

lasting in the order of femto seconds (for example

protein folding, protein docking, etc.) to slower

microscopic events like DNA transcription, cellular

mechanisms like meiosis or even lengthy ones like

the embryogenesis or the evolution of a disease like

diabetes or cancer (Hunter and Borg, 2003). In this

regard, there is another important aspect that should

not be left out from the whole picture. This is the

contemporary data explosion deriving from

genomic, transcriptomics, proteomics and

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metabolomics studies consisting in high dimensional

datasets produced by latest high throughput

measurements methods (Deane et al., 2002). Also,

other types of data coming from modern microscopy

and biological imaging contribute to the detailed

description of the constitutive parts and basic

structures of living organisms (Southern et al.,

2008). On that account, the current challenge

expects to relate these datasets to higher level

phenotypic characteristics and computational multi-

scale modeling approaches are set to reveal

quantitative mechanistic relationships between these

various measurements (Di Ventura et al., 2006). For

example, high throughput gene expression data can

be used to infer knowledge of the intracellular

activities that can be later ascribed to the behavior of

cells in a higher-level description; e.g., the

expression of the gene GATA3 in CD4 T

lymphocytes in a certain experimental condition

gives indication about the differentiation state of

these cells, on the pattern of cytokine secreted and

ultimately on the type of the immune response

(Santoni et al., 2008); this is an information that is

relevant to the construction of a computational

model of the immune response.

For example, we have implemented a gene

regulatory network (GRN) of the intracellular-level

gene expression dynamics to characterize the

Th1/Th2 cell differentiation, a phenomena that takes

place at the cellular (mesoscopic) level. The GRN

used represents the most extensive attempt to model

the regulatory network controlling the differentiation

of TH lymphocytes to date (Mendoza, 2006). Before

integrating the minimalistic Boolean network

dynamics in an agent-based model of the cell-cell

interaction, we identified the genes coding for

membrane receptors and those coding for soluble

molecules to be secreted by the cell, with the idea of

interpreting the former as the “input” and the latter

as the “output” of the cell (left panel of Figure 2).

Then we analysed the network Boolean dynamics

using classical logical methods to identify the

asymptotic regimens. In particular, three ‘attractors’

with relevant biological meaning were identified:

two leading to TH1 (P1 and P2) and one to TH2 (P3)

phenotype. For each time step of the simulation each

undifferentiated T helper cell would individually

transduce the input signals coming from the

extracellular space through the cell receptors (right

panel of Figure 2) into a micro-dynamics of the gene

regulatory network eventually falling (or not) in one

of the attractors. In the case one of the possible

attractors is reached, then rule is fired and the cell

becomes a Th1 or Th2, otherwise the cell remains in

the undifferentiated state. More formally, we

obtained a partition of the space of all possible

configurations Ω0,1



(17 are the genes of the

GRN) considering hyper spheres of radius two

centred in P1, P2 and P3, that is, 



∈

Ω: ,   2,   1,2,3, where ∀,  ∈

Ω, 



,





∑















and 



Ω



∪





∪



) is the remaining space. Note that





∩



∅ while 



∪



∩



∅. The

rule states that, at time 1, undifferentiated Th

cells at time , whose internal network state belongs

to 



∪



, are marked as Th1; those with internal

state in 



are marked Th2 and all the rest do not

differentiate.

Figure 1: Left panel: The GRN used to control the

differentiation of Th cells. Nodes correspond to

genes/molecules involved in the Th1/2 switch. Connectors

ending with an arrow indicate activation while those

ending with a dot indicate inhibition. Right panel: The

differentiation of each uncommitted Th cell depends on

the concentration of input cytokines surrounding it. These

cytokines determine the activation level of the

corresponding input nodes, i.e., if cx denotes the

concentration of an input cytokine, then the activation

level of the corresponding input node is given by [m ·

/(θ

+ c

)] where [x] denotes the smallest integer greater

than x, m are the activation levels and θ is a constant.

The resulting automaton was able to reproduce a

dynamics that was consistent with macroscopic

observable phenomena at the cell population level

still remaining compatible with a realistic gene

expression profile at the microscopic level (Santoni

et al., 2008). This example shows that the two levels

of description (intracellular and extracellular) can

realistically be integrated supposed that (i) the

intracellular gene regulatory network is biologically

sound and allows for relevant asymptotic regimens

and (ii) the stable dynamics at the lower level can be

rationally translated into an action (the rule) at the

upper level.

As already mentioned, mathematical models that

try to describe such mechanisms, usually fix a

spatial and the temporal scale and describe the

system with a mathematical or computational (i.e.,

algorithmic) formalism (Dada and Mendes, 2011;

Engler et al., 2009; Qu et al., 2011). Computers do

ComputationalBiologyModelingacrossDifferentScales

619

the rest as they provide the dynamics by executing

(resolving) the rules just described in mathematical

formalism. The dynamics is dependent on

parameters and initial conditions so that one

generally tries hypothetical scenarios modifying

those initial conditions to get a feeling of the

systems behavior (Meier-Schellersheim et al., 2009;

Schenell, 2007). This process leads itself in

discovering new knowledge. However, the problem

is that the real system is in general not isolated

hence a local description is not sufficient to disclose

crucial mechanisms. It comes quite clear that one of

the reasons why biological phenomena are

intrinsically complex is because they are influenced

by variables that are outside a single level of

space/temporal description.

If we take into consideration the space, a good

way to define a scale is to selectively assign

processes to their position within a biological

hierarchy i.e., whether they represent interactions

between organs, within a tissue, between cells, and

so on. We can refer to these hierarchical positions as

levels of a biological organization. A relevant note

to this question is expressed in (Southern et al.,

2008), namely, when comparing different organisms

with each other, the specific spatial-temporal scales

in standard international units may be quite

different, even when looking at the same level of

biological organization and it would therefore be

beneficial for multi-scale modeling in bio-medicine

to refer to these levels of organization.

Biological systems can be thought as hierarchical

structures, i.e., genes that encode proteins; proteins

that are building blocks of organelles and cells; cells

that form tissue and organ; organs that form

organisms; and organisms that give origin to

individuals and populations. Different levels

communicate each other in the sense that lower

levels affect the higher ones and vice versa. For

example proteins regulate gene expression.

Therefore, in a biological system, interactions can

occur both at the same scale (such as interactions

between different cells) as well as between scales.

This originates a very complex system in which one

has to deal with multiple spatial and temporal scales

and feedback loops.

In theory, one can develop a model of a

biological system (such as a cardiac cell or the heart)

consisting of the genes and proteins, or even the

atoms. In practice however, existing computational

tools are yet insufficient for this task. It should be

noted that experiments are done at many scales,

ranging from single molecules or proteins to whole

organs and organisms, and therefore, experimental

information exists at different scales. Therefore,

relying on different experimental data, a model can

be formulated using two main approaches, i.e., top-

down or bottom-up (Alberghina and Westerhoof,

2008).

If one chooses to take into account the individual

elements and their interactions, studying the

resulting biological system as a consequence of the

emergent behavior of its single components, then the

bottom-up approach takes place. The advantage of

this type of approach is that it is adaptive and robust,

in the sense that if the available biological

knowledge varies, one can adapt the new knowledge

to the specific components of the model, in a very

selective way. Moreover this kind of approach is

suitable for studying the emergent properties of

systems consisting of a large number of interacting

elements. The intensive computer power required is

the main disadvantage for the bottom-up approach

and can be sometimes even prohibitive. Moreover,

the model itself can become too complicated to

control.

Instead, one can decide not to look straight into

the details of the individual elements, but to consider

the system at the macroscopic level, using

experimental observations as guidelines during the

formulation of the model. The clear advantage of

this approach is that it is relatively simple. On the

other hand, the adaptability and the robustness of the

model are less evident compared with the bottom-up

approach. Moreover, it should be highlighted that

the variables and parameters in these models are

largely phenomenological without direct connection

with detailed physiological parameters. Due to this

reason, it may sometimes happen that the top-down

approach does not correctly reveal the actual

responsible mechanism, e.g., when there are

multiple mechanisms for the same behavior or a

single mechanism resulting in multiple phenomena.

When existing components have to be integrated

with some new part a third design principle, named

“middle-out”, is used (Hunter and Viceconti, 2009).

3 MODELING ACROSS

DIFFERENT

SCALES – FILLING THE GAP

Going from the lowest scale to higher levels one can

choose among different modeling choices. Intra-

cellular modeling approaches aim at a detailed,

mechanistic description of molecular processes

occurring inside single cells. These models usually

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adopt the differential equation description to predict

the molecular kinetics of specific cellular pathways

starting from experimentally determined parameters.

These models consist of mass action or Michaelis-

Menten kinetic rate-law equations describing the

changes of molecular concentrations. An example of

a bi-domain model describing a phenomenon at a

level that originates from the microscopic dynamics

at a smaller space scale is the wave propagation in

reactive media belonging to the class of the so-called

Belousov– Zhabotinsky reaction. In a simple form

(called the “oregonator” model) it may be

understood in terms of the following schema (Tyson,

1994) including an autocatalytic reaction

A+Y→X+P, X + Y → 2P, A + X → 2X + 2Z, 2X

→ A + P,

󳋡

B + Z → hY + Q, where the variables

represents concentrations of specific molecules (e.g.,

bromomalonicacid, carbon dioxide, etc.) and h is a

constant. Translated to ordinary differential

equations the system is dX/dt =AY −XY +AX−2X

dY/dt = AY − XY + hBZ, dZ/dt = 2AX – BZ, where

A, B and P are held constant. The solution of this

system has an oscillatory dynamics that, transposed

to two spatial dimensions, describes a propagating

wave. The bi-domain “nature” of the model in this

example lays in the emergence of the wave at a level

that is above the one chosen to describe the

phenomena, that is the molecular level of the

reactants (Murray, 2003).

An alternative to differential equations for intra-

cellular models is the microsimulation of reactions

within cells where the number of reagents is a small

number (due to current computational limitations).

The method developed many decades ago and

known as the Gillespie algorithm (Gillespie, 1976;

Gillespie, 1977) allows to accurately simulating

chemical or biochemical systems of reactions

generating a statistically correct trajectories as

possible solutions of a stochastic equation as for

example the differential equations corresponding to

the time-evolution of stochastic processes that

proceed by jumps (e.g., Markov jump process

(Bailey, 1990)). A simplified version of this

equation is the master equation describing the time

evolution of the probability

of a system to be in a

set of states with regard to a continuous time

variable t. The most familiar form of a master

equation is a matrix form



(1)

where M is the matrix specifying the connections. At

a higher scale level of description, tissues or whole

organs are modeled in two different ways: either as

functional compartments or system units or as a

collection of microscopic components (e.g., cells).

These two modeling paradigm use a completely

different point of view in describing a functional

unit as a tissue or organ. In the former case the organ

is seen as a black box with known input-output

relationship. This relation is typically derived from

known facts and ultimately realized by differential

equations linking stimulus with response or input to

output or causes to effects. These kind of

phenomenological models do not attempt to give an

explanation of the observed behavior whereas they

aim merely at reproducing it. They are quite useful

when combined together to offer a bigger picture.

The latter modeling paradigm proposes to represent

a tissue as an array of individual units (i.e., cells)

exchanging signals with the environment. A

noticeable example of these multicellular systems

has been originally developed to study the growth of

solid tumors (Drasdo et al., 1995; Drasdo, 2000),

and has later on been applied to simulate the

function (the regeneration) of complex organs like

the liver (Hoehme et al., 2010).

There are a number of ongoing projects whose

aim is to simulate a whole cell (e.g., virtual cell

(Schaff et al., 1977), e-cell (Normile, 1999;

Takahashi et al., 2004)), whereas efforts aiming at

simulating whole systems or organs are, for

example, models of the heart (Hunter and Nielsen,

2005), of the liver (Holzhütter et al., 2012), and of

the skeletal system (Viceconti, 2012). Other efforts

aim at creating computational platforms suite to

integrating various physiological processes (Eissing

et al., 2011). These are integrative systems biology

challenges that target the simulation of complex

biological systems through multi scale integration of

different mathematical and computational models.

The approach is the so-called middle-out strategy

proposed by Brenner, (1998) and Noble (2002;

2006), based on the principle that, in biology, there

is no privileged level for the description of a certain

phenomenon and that the inter-level causal

relationships are driven by interactions between

multiple levels. An application of the same modeling

principle to nutritional sciences can be found in de

Graaf et al., (2009) where the authors describe how

multi-scale models integrating processes from the

cellular up to the physiological levels are indeed

necessary in answering important nutritional

questions.

The use of different modeling paradigms

however, introduces gaps between scales. Multi-

scale modeling, besides modeling the individual

system components, needs to address the issue of

how to bridge the gaps between different

ComputationalBiologyModelingacrossDifferentScales

621

methodologies and between models at different

scales. Unfortunately, there is not a specific or

simple way to achieve this goal, but there are quite a

number of empirical principles and methods that can

provide some hint. For instance, adaptive mesh

refinement in lattice models (Plewa et al., 2005) is

used to scale down the details of a certain process,

the Hidden Markov Models (Baum and Petrie, 1966)

are used to deduce higher scale logics from the

observation of lower scale patterns, equation free

methods (Kevrekidis et al., 2003) based on the

execution of microscopic simulation models

allowing for computing the evolution equation of a

system at a higher (e.g., coarse) level, etc.

Systems biology is the main area in which one

can find this help. The goal of systems biology is to

consider a biological system from a holistic

perspective, and use both experiments and modeling

and the interactions between experiments and

modeling to reveal how the system behaves (Kitano,

2002; Kohl et al., 2010).

Specific modeling choices at a lower length scale

favor the integration of information at higher scales

and vice versa. For example, the individual- or

agent-based modeling approach at the mesoscopic

level (Castiglione et al., 2007) can be integrated to

the microscopic intracellular description for which

we can adopt either the continuous approach (as in

Ribba et al., 2006, that integrates cell cycle

regulation and macroscopic tumor dynamics with

the aim with the aim of mathematically investigating

this therapeutic failure the anti-metastatic agents

called inhibitors of metalloproteinases), or Boolean

networks to model intracellular events (like the

regulation of gene activation as in the differentiation

of T lymphocytes (Santoni et al., 2008)). In other

words, taken out the necessary approximation, a

fruitful approach in constructing large-scale

mechanistic models is given by combining

mechanistically detailed kinetic models (either

continuous - equations based - or discrete - boolean

networks) and coarse-grained (i.e., individual- or

agent-based) models (Smallbone et al., 2007).

Interestingly, it has been shown recently that

complex system behavior is often largely defined by

the interaction topology among the various model

components (Brown et al., 2004; Gutenkunst et al.,

2007). This finding further supports the expectation

that in order to obtain meaningful predictions most

likely only a few molecular processes need to be

described in great detail with precise parameters

estimates, while the rest of the system can be

described using the coarse-grained interaction

topology (de Graaf et al., 2009).

The very multi-scale nature of novel models in

computational biology makes their development

particularly challenging, not just from a biological

point of view but also from a mathematical and

computational perspective. Moreover, given the

availability of already published models targeting a

single scale, the sharing and reusing of such models

has become an issue. A prominent attempt at solving

this problem is provided by the Physiome Project

(Bradley et al., 2011; Hunter and Borg, 2003), which

aims at developing a framework for the modeling of

the “whole” human body. As part of that initiative,

the mark-up language CellML was introduced with

the aim of establishing a world-wide-adopted

standard in the development of cellular level that are

modeled as sets of ODEs (Garny et al., 2008).

Similarly, FieldML has been defined to model

processes on the tissue and organ level that are

represented as sets of PDEs (Christie et al., 2009).

Along with CellML, another standard called

Systems Biology Markup Language (SBML) (Hucka

et al., 2003) has been proposed and is now beginning

to make a significant impact on the modeling

community as a means to exchange models.

However, neither CellML nor SBML include

explicit directives to deal with the problem of

implementing a multi-scale computational model,

although there are attempts to address this issue

(Baylei, 1990).

Regardless the integration framework one

decides to use there are few aspects that need to be

taken into account when developing a multi-scale

model. In general, the time scales on which the

lower-level processes occur are much faster than

those on which the higher-level processes occur.

Usually this means that the lower-level processes

can be assumed to occur instantaneously and can

therefore be included as a representation of some

kind of field at the higher level. The switch to a

model at a higher level of organization is usually

determined by the need to ensure that the

calculations can be performed in reasonable time

(Southern et al, 2008). When coupling together

independent models of processes that occur on

different scales or as part of different physical

systems (as is in multi-organ systems) it is enticing

to simply couple existing components (i.e., software)

for the separate models to one another. However,

this does not take into account how inaccuracies in

the values of the variables that are passed between

the two models may affect the combined model -

one variable may be accurate enough in one model

but when these models are coupled may first

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introduce errors into the solution of the other model,

and in turn the solution of the combined model. In

order to prevent these inaccuracies from occurring

one should consider the whole as a single model

rather than the combination of two simpler ones. For

instance, we can consider a microscopic simulator at

the cellular level can be coupled with the description

of the intracellular signaling activating a specific

cellular pathway. In this example the differentiation

of T lymphocytes into the phenotypes Th1, Th2,

Treg and Th17 is described at a cellular level by

means of individual entities (e.g., agent-based)

whereas the gene regulation is described (at variance

with the example above which use a Boolean

network) by a system of differential equations

describing activation level of each gene of the gene

network represented with the following equation

()

(1 )(1 )













(2)

where 



is the activation level of the i

gene, 



and





are parameters relative to the network topology

and C is a constant (Mendoza and Pardo, 2010).

Here the lower level description of gene activation is

determined at each upper-level time step by solving

the system of ODEs, and the cell differentiation is

executed at the upper level on the basis of the

information coming from the gene expression levels.

This procedure is iteratively executed at each time

step and for each lymphocyte.

4 CONCLUSIONS

In the study of complex biological phenomena it is

necessary to develop a unified view of the various

mechanisms in action and of the causal relationships

among different parts of that complex system (Di

Ventura et al., 2006; Kitano, 2002). In this article we

have briefly described the problems faced when one

wants to link mathematical or computational models

across different time and length scales.

In many areas of biology and physiology, multi-

scale and multi-physics models are very much

acclaimed, Although there exist an abundant

literature for multi-scale models in science and

engineering domains (Fish, 2010; Weinan, 2011), a

lot remains to be done in terms of translating these

mathematical theories and methodologies to the

domains of biology and physiology (Evans et al.,

2008; Caiazzo et al., 2011; Tahir et al., 2011).

A key unsolved issue is how to represent

appropriately the dynamical behaviors of a high-

dimensional model of a lower scale by a low-

dimensional model of a higher scale, so that it can be

used to investigate complex dynamical behaviors at

even higher scales of integration (Qu et al., 2011).

The use of different modeling techniques,

introduces gaps between scales. Multi-scale

modeling, besides modeling the system, needs to

address the issue of how to bridge the gaps between

different methodologies and between models at

different scales. Unfortunately, there is no specific

or simple way to tell how to achieve this objective,

but there are empirical principles and methods that

can be of help. The goal of computational systems

biology to consider a biological system from a

holistic perspective, and use both experiments and

modeling to reveal how the system behaves (Kitano,

2002; Kohl et al., 2010). It is certainly one of the

main research fields that can benefit from the use of

multi-scale models and, at the same time, provide

methodologies for their development.

ACKNOWLEDGEMENTS

This work was partially supported under the EC

contract FP7-ICT-2011-9, No.600803 (MISSION-

T2D)".

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