A Function-Based Approach to
Systems Biology
Taesik Lee, Jeffrey D. Thomas, and Nam P. Suh
Department of Mechanical Engineering
Massachusetts Institute of Technology
tslee@mit.edu
1. Introduction
For decades engineers have studied how to best design and build complex systems such as
automobiles, skyscrapers, computers, and transportation systems. In contrast,
systems-oriented thinking is relatively new in molecular biology. We have applied the
theories and methods of Axiomatic Design (AD) [Suh 2001] to the analysis of biological
systems. Despite the fundamental dissimilarity between engineered and biological systems
(human-designed versus evolved), we find many more compelling similarities.
AD theory defines a system by the functions it performs. To understand and control
biological systems, one must establish the relationship between the system functions and
the underlying behavior of molecular interactions. For example, some of the functions of
the mammalian visual system are to record optical images, differentiate color, modulate the
amount of incident visible light, and provide depth perception. The underlying biological
entities that provide these functions are organs (e.g., eyes, visual cortex), cells (e.g., retinal
rod and cone cells), organelles (e.g., discs in outer segment), molecules (e.g., rhodopsin,
11-cis-retinal), and ions (e.g., Ca++) [Berg 1995], [Purves 1998]. The relationships between
these functions and these biological units are highly complicated. An organizing framework
is required to understand such systems.
Biological functions are hierarchical. Biological systems can be decomposed from the
system level to the organ level and ultimately down to the molecular level functions.
Similarly, biological entities can also be decomposed into a hierarchy. Once the
decomposed functions and biological entities are established and their relationships known,
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the highest-level biological functions can be related to the lowest-level biological entities.
Based on these relationships, one can predict the functions of biological systems in terms of
the molecular level behavior and interactions of biological entities.
The starting point in the AD approach is to define the functions that a system must
achieve. These are called Functional Requirements (FRs). In contrast, most molecular
biological research is focused on the interactions among physical entities. In AD, such
physical entities constitute design parameters (DPs). Interactions among biological entities
(DP-DP interactions) can be measured and quantified using analytical tools. An important
result of research on the DP-DP interactions is the pathway diagram, used ubiquitously to
represent biological systems.
In order to build a model of a biological system, the relationships between FRs and DPs
must be established. The difficulty in establishing the FR-DP relationships is due to the fact
that many biological entities (DPs) affect more than one FR. To understand the behavior of
biological systems, the complicated relationships among all the FRs and DPs of a biological
system must be determined. In Axiomatic Design, this is done using the Design Matrix.
From the perspective of AD, biological systems are probably much less complex than
they appear. Many functions that seem to be interrelated probably function independently.
The architecture of complicated regulatory networks is probably less important in overall
system function than a survey of literature would make it appear.
This paper presents a framework for understanding biological systems based on the
Axiomatic Design theory. The basic concept of Axiomatic Design theory is given in Suh
[Suh 2001]. To illustrate the approach, the process of cellular spreading onto a fibronectin
matrix has been analyzed using AD methodology.
2. Modeling of a Biological System based on Axiomatic Design
2.1. Construction of an Interaction Matrix for the Signaling Pathway that Activates
Cell Spreading
Fibroblast cells in suspension adhere to fibronectin-coated surfaces and then spread out
such that the spherical cell is converted to a flattened form. Several labs, including Sheetz et
al. [Sheetz 2001], have characterized molecular aspects of the process of cell spreading in
response to the integrin-mediated binding to fibronectin [Clark 1995]. As the cell begins to
spread it extends its membrane in the form of a lamellipodium. Extension of the
lamellipodium is mediated by a network of actin filaments. Rapid, circumferentially
-directional extension of the lamellipodium results in cell flattening in the plane of the
fibronectin layer.
Pathway diagrams have been constructed that summarize the signaling mechanism
known to link the fibronectin-integrin signal from the cell membrane to the actin-based cell
spreading behavior. Figure 1 shows an example of such signaling pathway diagram [Xiong
2004].
Using the AD framework, the pathway diagram can be represented as an Interaction
Matrix comprising known DP/DP interactions (Table 1). The symbols in the matrix indicate
3
Fibronectin
Grb2Sos
Fak
Ras
Integrin
Src
Tiam1
1chinaerin
Vav
Fgd1
Rac1
PLC
Cdc42
Pak1
LIMK
PIP5K
Ca2+/
CaM
Calcineurin
Cdc42
GAP
IP3
PIP2
WASP
ADF
VASP
Profilin
Arp2/3
CP
Figure1. The signaling network that regulates fibroblast spreading machinery [Xiong 2004].
Ellipses designate proteins and complexes. Arrows signify activation. Dots indicate inhibition.
Rac1
PLC-
Cdc42
Pak1
LIMK
PIP5K
PIP2
IP3
Ca2+/CaM
Calcineurin
WASP
ADF
VASP
Profilin
Arp2/3
CP
+
+
+
+
+
+
+
+
+
+
-
+
-
+
+
+
-
CP
Arp2/3
Profilin
VASP
ADF
WASP
IP3
Ca2+/
CaM
Calcineurin
PIP2
PIP5K
LIMK
Pak1
Cdc-42
DPi
Rac1
DPj
PLC-
Table 1. The Interaction Matrix for the Fibronectin/Integrin signal in Cell Spreading. '+' ('-') at the
element (i,j) indicates that DPj activates (suppresses) DPi. This matrix represents a part of the
fibronectin-integrin signaling pathway shown in Figure 1. In this example the DPs are ordered to
correlate with the flow of information through the signaling pathway; information flow occurs in the
direction of column DPs to row DPs. The left-to-right (or top-to-bottom) sequence of DPs is not
important and can be arbitrary.
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that a signal is transduced between two specific DPs: the ‘+’ symbol designates stimulation
or activation; the ‘-’ symbol signifies inhibition or deactivation. For example, the ‘+’
symbol in the 4th row of the 1st column in Table 1 indicates that Rac1 activates Pak1. If
there were an important interaction among molecules of the same class (e.g., a
self-reinforcing or self-inhibiting mechanism), this would be indicated by a symbol in the
appropriate cell in the diagonal of the matrix (hatched cells in Table 1).
Some of the key properties of the network are readily visible by observing the structure
of the matrix: for example, 1) Since there are no symbols in the first three rows, Rac1,
PLC-, and Cdc42 must function as initiators for this part of the signaling pathway; 2) ADF,
VASP, Profilin, Arp2/3, and CP are the final products of the pathway since there is no
outgoing path from any of them (blank columns starting from ADF); 3) there are no
feedback loops. If, for example, Ca2+/calmodulin sent a retrograde activation signal to IP3,
it would be indicated in the Interaction Matrix by a ‘+’ symbol in the appropriate cell above
the diagonal; and 4) the pathway has a counteracting inputs to ADF [LIMK and PIP2 inhibit
ADF, but the PIP2-IP3-Ca2+/CaM-calcineurin chain stimulates ADF (Figure 1)].
The DP/DP interactions represented by the Interaction Matrix (Table 1) are the
comprehensive set of interactions thought to be important in transducing the
fibronectin/integrin signal. The Interaction Matrix scales easily and interaction symbols can
be replaced with mathematical descriptions of DP-DP interactions (e.g. binding constants
or activation rates).
2.2. Mapping between the Functional and Physical Domains of Biological Systems
For an engineer seeking to design a robust system using AD methodology, the starting point
is to define the functions that the system is to achieve. Once these functional requirements
(FRs) are defined, the FRs are related to DPs using the Design Matrix. As will be
demonstrated, the Design Matrix can be used to analyze FRs and their relationships to DPs
in great detail. In contrast, the Interaction Matrix exclusively describes interaction-based
molecular-level functions (e.g. activate, inhibit) and thereby does not capture other classes
of function, most notably higher-level functions that are the purpose, or raison d’etre, of a
given system.
To construct a Design Matrix for a biological system one first identifies the FRs, and the
DPs that satisfy those FRs. Once the FRs and DPs are identified, the FR/DP relationships
specified in the Design Matrix. Table 2 shows subsets of the FRs and DPs for the cell
spreading mechanism. Table 3 is a Design Matrix relating FRs to the DPs listed in Table 2b.
X’s in matrix designate a known relationship between the FR and DP. They represent
relevant functional relationships: for example, ‘actin polymerization at growing filament
tip’ (DP1.2) affects the function of ‘elongate filaments’ (FR1.2) while the ‘ratchet action’
(DP1.5) in turn affects the function of ‘provide space for growing filaments’ (FR1.5).
Complete understanding of a biological system’s functional behavior requires both the
Design Matrix (FR/DP) and the Interaction Matrix (DP/DP). The Interaction Matrix does
not address system functions whereas the Design Matrix is not exhaustive with respect to
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DPs and may thereby omit molecular components that are not relevant to performance of
functions.
Table 2. Decomposition of cell spreading.
a. FR0: Cell Spreading
Number
1
2
Functional Requirements
Generate force to extend lamellipodia
Orient force to extend lamellipodia
Design Parameters
A specific ultrastructure of cytoskeleton
Fibronectin-Integrin signal
b. FR1: Generate force to extend lamellipodia
Number
1.1
Functional Requirements
Provide actin monomer substrate
1.2
Elongate Filaments
1.3
Maintain appropriate network structure
1.4
1.5
Rigidify Filament Network
Provide Space for Growing Filaments
Design Parameters
Filament fragmentation
Actin polymerization at growing filament
tip
Cross-linked network of short actin
filaments
Alpha actinin [Svitkina 1999]
Ratchet action [Mogilner 2003]
c. FR1.1: Provide actin monomer substrate
Number
1.1.1
1.1.2
1.1.3
1.1.4
Functional Requirements
Prepare
actin
monomer
for
depolymerization
Depolymerize actin monomer from the
filament
Transform
ADP-bound
actin
to
ATP-bound actin
Deliver ATP-bound actin to the growing
end of the filament
Design parameters
ADP-bound actin [Mullins 1998]
ADF [Didry 1998]
Profilin [Mullins 1998]
Concentration gradient between
growing end and the other end
the
d. FR1.3: Maintain appropriate network structure
Number
Functional Requirements
1.3.1
Branch actin filaments
1.3.2
Stop growing actin filaments
Design parameters
Active Arp2/3 at end of growing filament
[Mullins 1998]
Capping Protein (CP) [Schafer 1996]
Table 3. Design matrix, [A], captures the functional relationships between FRs and DPs.
FR1.1
FR1.2
FR1.3
FR1.4
FR1.5
DP1.1
X
DP1.2
DP1.3
DP1.4
DP1.5
X
X
X
X
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2.3. Coupling, Complexity and the Design Matrix
In engineered systems, the Independence Axiom states that FRs must be independent of one
another. Thus in an ideal design there are only one-to-one FR-DP relationships (these are
called “uncoupled” designs, as in Table 3). If it is not possible to achieve complete
independence of FRs, one can still achieve an acceptable design by ensuring that none of
the FR-DP pairs (or chains) creates a circular-loop relationship (which is called a
“decoupled” design). In this case, if FRs need to be modified, DPs can be changed in a
specific sequence so that the impact on FRs due to the off-diagonal element is propagated in
a sequence given in the matrix. Since uncoupled and decoupled systems satisfy the
Independence Axiom, they are capable of satisfying the target values of FRs.
When off-diagonal elements exist and form a circular-loop(s), the design is “coupled”.
Coupled design means that changes in any single DP in the loop will require that all DPs in
the loop must be re-set to, if any, the exact values at the same time to satisfy the FRs. As a
consequence, it is difficult to satisfy the FRs and the system is not adaptable.
Advances in molecular biological research methods, especially genomic and proteomic
technologies, have made cataloguing DP-DP interactions far easier than characterizing
FR-DP relationships in biological systems. While a comprehensive list of possible DP-DP
interactions will have undisputed value, such data may also impede the characterization of
biological systems by making it difficult to differentiate interactions that are important to
system function from those interactions that are not; in the absence of FR-DP relationship,
studies of DP-DP interactions may make biological systems appear overly complicated.
For example, two FRs important in cell spreading, branching and stopping growth
(Table 2d), are uncoupled (Table 4) and thereby independent. However, the Interaction
Matrix (Table 5) that describes the two DPs, Arp2/3 and Capping Protein (CP), shows a
complicated interaction between them. As Table 5 suggests, PIP2 has the opposite impact
on Arp2/3 and CP: PIP2 inhibits CP whereas it stimulates Arp2/3 through the
WASP-mediated pathway. The apparent coupling between branching (FR1.3.1) and
capping (FR1.3.2) is due to DP-DP interactions in the pathway rather than a coupled FR/DP
relationship.
Table 4. The diagonal Design Matrix indicates FR/DP relationship is uncoupled.
FR1.3.1
FR1.3.2
DP1.3.1
X
DP1.3.2
X
Table 5. Interaction matrix relevant to the targets, Arp2/3 and CP.
PIP2
PIP2
WASP
Arp2/3
CP
WASP
+
+
-
Arp2/3
CP
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2.4. The Design Matrix as a Foundation for Mathematical Modeling of Biological
Systems: Cross-scale Modeling
The Design Matrices above (Tables 3 and 4) have been simplified by the use of the symbol
“X” to indicate FR-DP relationships. FR-DP relationships can be specifically characterized
by mathematical equations (Table 6). In the case that all FR-DP relationships for a system
can be represented mathematically, the Design Matrix serves as a complete mathematical
model of that system. As discussed earlier, not all the DPs in the signaling pathway appear
in Design Matrix. The change in a given DP, say DPi, in the Design Matrix may be the
outcome of a signal cascade involving dozens of DPs (DP chains) in the Interaction Matrix.
The magnitude of DPi may be a function of many variables, e.g., the signal level and the
kinetics of biochemical reactions that are triggered by the signal. Whenever quantitative
data are available, a model can be developed that incorporates both the Design Matrix and
the Interaction Matrix. Precise determination of a complete {FR}-{DP chains} relationship
requires extensive knowledge of biomolecular interactions in their finest granularity: a
formidable challenge.
Table 6. Symbols in the Design Matrix are replaced by mathematical relationships between FRs and
DPs. FR1.3.1 “Branch actin filament” is represented by the reaction rate of branching, abranching, and
FR1.3.2 “stop growing actin filament” is represented by the reaction rate of capping, acapping. The
concentrations of each molecule, uArp2/3 and uCP, serve as metrics for the DPs. For individual actin
filament the reaction rates can be represented by the rate constant of reaction and the concentrations of
the reactants [Xiong 2004]. where kbranching is the branching reaction rate constant and kcapping is the
rate constant for the capping reaction. f, , kBT, uG-ATP represent the total resistance force, the filament
length increment per monomer, the thermal energy, and the concentration of G-ATP, respectively. f
is therefore equivalent to the activation energy barrier. f and uG-ATP can be estimated to be 50-500
pN/m and ~8.5M [Xiong 2004], while  and kBT are known to be 2.2nm and 4.2 pNnm,
respectively [Mogilner 2002].
DP1.3.1, uArp2/3
FR1.3.1
abranching
FR1.3.2
acapping
DP1.3.2, uCP
 f  2
  uG-ATP
k branching  exp  
 k BT 
k capping
The hierarchical nature of FRs and DPs can be leveraged to establish a framework for
generating models that cross multiple scales of organization (tissue level, cellular,
molecular, etc.). High-level FRs and the DPs are first decomposed into lower and lower
levels until as many details as necessary are included. The mapping the FRs to DPs in the
biological systems was illustrated previously [Thomas 2004].
After decomposing a system into FRs and DPs, functions (denoted f ) are selected that
relate the FRs and DPs at each level. f is equivalent to the elements of the Design Matrix.
Owing to the decomposition process, DP domains are also related to the FR domain one
level below in the hierarchy (“sublevel”). Thus, sub-level FRs are enumerations of the
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parent-FR, and they are specific to their parent-level DP. The relationship between the
parent DP and its sub-level FRs is denoted here by g.
With the notion of f and g, one can now express the cross-scale modeling as
  g  f
FR 1  f 1 g 1, 2 f
2
2, 3
LEAF
DP
LEAF

(1)
FR1 is a highest-level (level 1) FR. f i is a mapping function between the FRs and the DPs at
level i. gi,i+1 is a function relating the parent DP at level i to the sub-level FRs at level i+1.
The inner-most function of equation (1) is f LEAF(DPLEAF), which takes the lowest-level DPs
(“leaf” level) as an input. Thus, equation (1) represents a model encompassing multiple
scales, and describes the highest-level FRs in terms of the lowest-level DPs.
4. Conclusion
The Design Matrix of AD, which was developed to create the science base for design of
human-engineered systems, offers an important new perspective in the field of systems
biology. Most significantly, the Design Matrix maps the physical elements of a system to
the system’s functions. In contrast, most current studies of biological systems are focused
on interactions between physical elements.
The application of AD theory to biological systems results in several important
conclusions with implications for biology and medicine. The Independence Axiom states
that in robust systems, functions are maintained independently. When accurate models of
biological systems are developed, the key FR-DP relationships (i.e. diagonal elements) may
be used in regulating a system or in targeting therapeutic molecules with minimal side
effects. AD theory predicts that disease states are the result of off-diagonal elements
becoming large non-zero terms and thereby creating coupled systems with reduced
robustness. They may also be the result of the degradation of DPs and the decrease in the
values of the diagonal elements. For example, aging may be the consequence of dominant
off-diagonal elements or of cumulative coupling.
The Design Matrix provides a foundation for the establishment of mathematical models
of biological systems. When FR-DP relationships are sufficiently characterized,
mathematical representations of the FR-DP relationships can be used as Design Matrix
elements. Although current models of biological systems are limited by knowledge of FRs,
the Design Matrix highlights these unknowns and therefore the questions important for
further studies. Models based upon the Design Matrix can be scaled to accommodate
hundreds of FR-DP interactions. The hierarchical nature of the Design Matrix facilitates the
building of models that traverse molecular, subcellular, cellular, tissue and physiological
scales.
The application of the Design Matrix and the concepts of AD theory provide a useful
framework for advancing biological knowledge, especially in systems biology.
9
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