JUNB.in.on, MMP13.in.off, NFKB1.in.off, SMAD7.in.on.
Accepted as before.

  7. 7.

SMAD7.in.on, NFKB1.out.off $\rightarrow$ JUNB.out.off.
Accepted with confidence 6/7.

  8. 8.

SMAD7.in.on, JUNB.out.on, SMAD7.out.on $\rightarrow$ NFKB1.out.on.
Accepted with confidence 12/13.

  9. 9.

SMAD7.in.off $\rightarrow$ JUNB.in.off, MMP13.in.off.
The rule was accepted with confidence 32/32, i.e. it holds as a strict
implication also in the data. Like temporal rule 4, it underlines the
interdependency of TNF$\alpha$ and TGF$\beta$1 signaling.

  10. 10.

NFKB1.in.on, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
The rule should be rejected by reason of a small support (5/72) and confidence
(0/5). As counterexample 0010 1000 was chosen. However, as several other
counterexamples, it violates previously accepted implications, thus the rule
has been accepted in this first run of exploration.

  11. 11.

NFKB1.in.on, JUNB.out.off, MMP13.out.off, SMAD7.out.off $\rightarrow$
NFKB1.out.on.
Accepted with 7/12.

  12. 12.

NFKB1.in.off, NFKB1.out.off $\rightarrow$ JUNB.out.off.
The confidence is 5/9, but the implication is accepted as plausible: JUNB
depends on NFKB1 (BF 6); if it remains off, then also JUNB.

  13. 13.

NFKB1.in.off, JUNB.out.on, SMAD7.out.on $\rightarrow$ NFKB1.out.on.
The implication is rejected with confidence 2/6 and due to an almost inverted
premise compared to 11, but the same conclusion. Again, counterexample 3 (1001
1001) and others violate implication 12.

At this point, I restarted attribute exploration again with counterexamples
1-3, since the last two examples were judged to be important and should not be
omitted. They only had become obvious at this later point of exploration, but
the result should not depend on the order of the attributes.

  1. 1.

SMAD7.out.off $\rightarrow$ MMP13.out.off.
Subsumed by implication 2 of the first run and accepted with confidence 30/30
(remember that MMP13 was always off in the observations).

  2. 2.

MMP13.out.on $\rightarrow$ SMAD7.out.on.
Accepted with support 0 (identical to implication 3 of the first exploration
run).

  3. 3.

NFKB1.out.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.out.off.
The confidence is not high (13/20) and the implication contradicts BF 6, if a
stable expression of NFKB1 and SMAD7 is given over several time steps. Hence
counterexample 4 (0000 1010) was introduced.

  4. 4.

MMP13.out.off, NFKB1.out.off, SMAD7.out.off $\rightarrow$ MMP13.in.off.
Accepted by reason of the MMP13 measurements.

  5. 5.

JUNB.out.on, NFKB1.out.off $\rightarrow$ MMP13.in.off.
Ditto.

  6. 6.

JUNB.out.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.in.off,
MMP13.in.off, SMAD7.in.off.
Accepted with confidence 3/7, since I did not want to contradict such a
complicated, possibly artificial implication. The complexity should not be
augmented by an unreliable counterexample.

  7. 7.

SMAD7.in.on, MMP13.out.on, NFKB1.out.off, SMAD7.out.on $\rightarrow$
JUNB.out.off.
Complicated and accepted.

  8. 8.

SMAD7.in.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.out.off.
Accepted with confidence 14/18.

  9. 9.

SMAD7.in.on, JUNB.out.on $\rightarrow$ SMAD7.out.on.
The implication is a restriction of rule 4 in the first run and was accepted
with confidence 13/17. It is unanticipated by the additional reason that JUNB
and SMAD7 are regulated differently. In the adapted network (Table 8.7), the
only common influencing genes are SMAD4 and SMAD7. Possibly, the inhibition of
both by SMAD7, suggested by SMAD7.in.on, could be the relevant influence on
the transitions supporting the implication, since SMAD4 is always on in the
simulations and the data, except for a single measurement.

  10. 10.

SMAD7.in.on, JUNB.out.on, MMP13.out.on, SMAD7.out.on $\rightarrow$
NFKB1.out.on.
Complicated and accepted.

  11. 11.

SMAD7.in.off $\rightarrow$ JUNB.in.off, MMP13.in.off.
The implication is identical to rule 9 of the first run and relates to the
inhibition of TGF$\beta$1 signaling via SMAD7 and the activation of
TNF$\alpha$ signaling via JUNB.

  12. 12.

NFKB1.in.on, MMP13.out.off, NFKB1.out.on, SMAD7.out.off $\rightarrow$
JUNB.out.off.
Confidence 7/12, accepted as complicated.

  13. 13.

NFKB1.in.on, JUNB.out.on, NFKB1.out.on $\rightarrow$ SMAD7.out.on.
Accepted with confidence 23/28, similar to implication 9.

  14. 14.

NFKB1.in.on, JUNB.out.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
The restriction of implication 10 in the first run was accepted despite
confidence 0/5. There are few contradictory observations, and an antagonistic
expression is plausible by the BF 6, 12 and 14: JUNB and NFKB1 are activated
by NFKB1, SMAD7 is not. The implication constrains the coregulation of JUNB
and SMAD7 found in the previous and other implications. However, it remains
insecure and should be investigated by further experiments.

  15. 15.

3 implications related to NFKB1 are accepted as complicated.

  16. 18.

MMP13.in.on $\rightarrow$ SMAD7.in.on.
No support, accepted, similarly the next implication with MMP13.

  17. 20.

7 further implications related to MMP13 are accepted, since they are less
expressive and a data control is not possible.

  18. 27.

JUNB.in.on $\rightarrow$ SMAD7.in.on.
This again is a noteworthy, clear implication with highest confidence 38/38.
It underlines the discovered coregulation of the two genes (compare 9, 11 and
13, but also 14).

  19. 28.

JUNB.in.off, JUNB.out.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
Confidence 0/4, but the number of counterexamples in the data is not
sufficient. The biological meaning is similar to implication 14, but this time
it might be considered as a hint on a still unknown influence of JUNB on its
own expression and that of NFKB1 and SMAD7 (where BF 14, however, reflects
inhibition).

  20. 29.

SMAD7.in.on, JUNB.in.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on,
JUNB.out.off.
No support in the data, accepted.

  21. 30.

The 7 last implications concerning JUNB, together with other genes, are
accepted as too complicated or because of a small support in the set of
observated transitions.

The two rules with highest support in the resulting stem base relate MMP13
expression to SMAD7 upregulation, in the input and the output state:

$\displaystyle<104>\>$
$\displaystyle\>\text{MMP13.out.on}\rightarrow\>\text{SMAD7.out.on}$
$\displaystyle<89>\>$
$\displaystyle\>\text{MMP13.in.on}\rightarrow\>\text{SMAD7.in.on}$

In contrast, there is no implication with MMP13.on in the conclusion (already
for the simulated transitions). This reflects the weak empirical evidence for
MMP13 expression after Tgf$\beta$1 stimulation, which in turn is the reason
why there were no counterexamples to the two rules in the observed data. The
high support originates from the literature based prediction of expression
after Tgf$\beta$1 stimulation. It was no decision criterion during the
exploration aiming at qualitative, not quantitative relations. The rules can
be interpreted as follows: In the rare cases of MMP13 upregulation, it is
expected to be coregulated with SMAD7, but no positive conditions for its
expression are found.

##  Chapter 9 Conclusion and outlook

In this thesis, an FCA framework for the description and analysis of discrete
processes was developed and investigated. The knowlege bases generated by
attribute exploration suppport automatic reasoning. So it is worthwhile to
sketch connections to DL, where fast reasoning software exists. The second
section of the present chapter outlines possibilities of solving open
mathematical and logical questions. Finally, the biological applications will
be discussed, in particular the study of the ECM network related to rheumatic
diseases.

###  9.1 Transfer to Description Logics

First I will discuss how our approach may be expressed in a standard DL and in
$\mathcal{TDL}-Lite_{Bool}$, a temporal extension by [15]. I will point at
advantages of remaining within the general framework of FCA, instead of using
the temporal expressivity of DL and attribute exploration adapted to the
construction of DL knowledge bases. Furthermore, reasons will be given to
concentrate on specific, practically relevant parts of temporal logic. I will
give hints to the embedding of the syntax and semantics given by the defined
formal contexts and their stem bases into $\mathcal{EL}$ (a DL designed for
reasoning about ontologies) and $\mathcal{TDL}-Lite_{Bool}$. It is out of the
coverage of this study to search for a DL corresponding to the expressivity of
the proposed formal contexts, not to mention its definition. However, we see a
potential of the integration of our ideas into a DL and of adapting attribute
exploration accordingly.

A transition or transitive context (analogously and easier a state context)
may be translated into a DL like $\mathcal{EL}$ as follows:

  * •

The objects are elements of the model domain $\Delta:=S$ (states).

  * •

Attributes $m\in M$ of the input states are considered as concept names
$C_{m}\in N_{C}$.

  * •

Attributes of the output states are given by concept descriptions with the
role $n\in\mathcal{N}_{r}$ (transition to the next state of a path) or
$t\in\mathcal{N}_{r}$ (transition to an arbitrary subsequent state).

Then a formal context similar to [19, p. 159] will be defined. It corresponds
to a deterministic transition context $\mathbb{K}_{t}$ (compare Table 9.1 with
Table 4.3). Nondeterminism cannot be distinguished syntactically from a
deterministic transition to a state with all alternative attributes (the first
3 lines in Table 9.1 collapse), only semantically by the formal context. As in
Table 4.3, the states are labelled by the path they belong to, e.g.
$s_{00},s_{01},s_{02}$ are the initial states of the Linux driver installation
and the Windows process with ndiswrapper or driver.windows as next steps,
respectively. (At $s_{21}$ the two Windows paths coincide.)

|  $driverLinux$ |  $ndiswrapper$ |  $driverWindows$ |  $connection$ |  $\exists\,n.driverLinux$ |  $\exists\,n.ndiswrapper$ |  $\exists\,n.driverWindows$ |  $\exists\,n.connection$
---|---|---|---|---|---|---|---|---
$s_{00}$ |  |  |  |  | $\times$ |  |  |
$s_{01}$ |  |  |  |  |  | $\times$ |  |
$s_{02}$ |  |  |  |  |  |  | $\times$ |
$s_{10}$ | $\times$ |  |  |  | $\times$ |  |  | $\times$
$s_{11}$ |  | $\times$ |  |  |  | $\times$ | $\times$ |
$s_{12}$ |  |  | $\times$ |  |  | $\times$ | $\times$ |
$s_{20}$ | $\times$ |  |  | $\times$ | $\times$ |  |  | $\times$
$s_{21}$ |  | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$
$s_{31}$ |  | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$
Table 9.1: A transition context $\mathbb{K}_{t}$ in the language of
description logics.

A transitive context $\mathbb{K}_{tt}$ might be translated by means of the
role $t$. Then the information – expressed by $n\in\mathcal{N}_{r}$ – related
to individual subsequent states is lost. Instead, $\exists t.C_{m},\,m\in M$
indicates if a reachable state is in the extent of $\\{m\\}$ in
$\mathbb{K}_{s}$ or, in the language of DL, if $C_{m}$ is interpreted by the
respective reachable state. Thus, it makes sense to combine a
DL-$\mathbb{K}_{t}$ and a DL-$\mathbb{K}_{tt}$ (Table 9.2).

|  $driverLinux$ |  $ndiswrapper$ |  $driverWindows$ |  $connection$ |  $\exists\,n.driverLinux$ |  $\exists\,n.ndiswrapper$ |  $\exists\,n.driverWindows$ |  $\exists\,n.connection$ |  $\exists\,t.driverLinux$ |  $\exists\,t.ndiswrapper$ |  $\exists\,t.driverWindows$ |  $\exists\,t.connection$
---|---|---|---|---|---|---|---|---|---|---|---|---
$s_{00}$ |  |  |  |  | x |  |  |  | x |  |  | x
$s_{01}$ |  |  |  |  |  | x |  |  |  | x | x | x
$s_{02}$ |  |  |  |  |  |  | x |  |  | x | x | x
$s_{10}$ | x |  |  |  | x |  |  | x | x |  |  | x
$s_{11}$ |  | x |  |  |  | x | x |  |  | x | x | x
$s_{12}$ |  |  | x |  |  | x | x |  |  | x | x | x
$s_{20}$ | x |  |  | x | x |  |  | x | x |  |  | x
$s_{21}$ |  | x | x |  |  | x | x | x |  | x | x | x
$s_{31}$ |  | x | x | x |  | x | x | x |  | x | x | x
Table 9.2: The apposition $\mathbb{K}_{t}\mid\mathbb{K}_{tt}^{out}$ of a
transition context and the output part of a transitive context in the language
of DL.

A transitive context is more expressive than its DL equivalent. Because the
objects of the context 9.2 are states, not transitions, only the information
regarding the next and any reachable state is kept. The following implication
of $\mathbb{K}_{tt}$ (p. 5.1) expresses: If the wrapper module and the driver
are and remain installed (at a second time point), then the connection data
has to be (is) entered:

ndiswrapper.in, driver.windows.in, ndiswrapper.out,
driver.windows.out $\rightarrow$ connection.out

Yet in the DL context of Table 9.2, already the less meaningful implication
holds $\top\rightarrow\exists t.connection$. It turns out that the role $t$
corresponds better to the operator $\operatorname{ev}$ of a temporal context.

The semantics for the $\mathbb{K}_{tmp}$ attributes from CTL is given by
$\mathbb{K}_{t}$, for $\Diamond Fm$, $\Box Gm$ and $\Box\neg Fm$ also by
$\mathbb{K}_{tt}$ (Remark 4.5.3). Two of the $\mathbb{K}_{tmp}$ attributes are
equivalent to the following concept descriptions:

$\displaystyle\Diamond Fm$ $\displaystyle\cong\exists\,t.C_{m}$
$\displaystyle\Box\neg Fm$ $\displaystyle\cong\neg\exists\,t.C_{m}$

If concept descriptions relate to states indexed by the path they belong to,
as in Table 9.2, the following temporal operator may also be expressed:

$\Diamond\neg Fm\cong\neg\exists\,t.C_{m}.$

S. Rudolph [87] and F. Baader / F. Distel [19] deal with a potentially
infinite attribute set, caused by nesting of roles like $\exists r.\exists
r.\cdots$ . In the case of cyclic concept descriptions, e.g., the potential
role depth (number of nested roles) is infinite. If such DL concepts are
interpreted temporally, they express properties of cyclic time developments.
S. Rudolph investigated attribute exploration of formal concepts with
increasing role depth (in the more expressive DL $\mathcal{FLE}$) and showed
the existence of a termination condition for finite models. However, there are
computational problems, since the number of attributes grows exponentially
with the depth. F. Distel [18] proved that a finite basis exists for the set
of all $\mathcal{EL}$-implications (or GCIs) holding in a finite model. In
[19], a single formal context with relational attributes has been defined; its
attribute exploration provides an algorithm to explicitly compute the
implicational base.

As mentioned before, the DL transition context of Table 9.1 is derived from
such a context. However, we restricted our framework to simple relational
attributes (role depth one). Since temporal attributes can also refer to
infinite paths, we do not need nested roles in this case. Roles of depth two
or three may still be meaningful to an expert, but the purpose of the present
work was not to develop a general framework for the conceptual exploration of
relations. If the number of relational attributes remains restricted and
almost all attributes occur in at least one of the output states, the set of
attributes $\exists r.C_{m},C_{m}\in\mathcal{N}_{C}$ may be fixed in advance.
Then general attribute exploration can be applied, and no algorithm handling a
growing set of attributes is needed as designed in [19].

Thus, standard FCA algorithms could be used. Nevertheless, our approach is
generalisable to more complicated relational attributes; also Relational
Concept Analysis (RCA) [45] may be applied. Inversely, attribute exploration
software using fast DL reasoners could be used for our special formal
contexts.

In the approach of [15] (see Section 3.2.3), a flow of time is represented by
$\mathbb{N}_{0}$, and the interpretation consists in a family of succeeding
situations $(\mathcal{I}_{n})_{n\in\mathbb{N}_{0}}$ defined by the
interpretation function $\mathcal{I}$. According to [18, p. 48 f.], the family
can be seen as a set of temporally changing models with a constant domain
$\Delta$ for the concept description language $(\mathcal{L},\mathcal{I})$
given by $\mathcal{TDL}-Lite_{Bool}$, and each model correponds to a formal
context. This is a more expressive framework than ours and may not easily be
translated into the defined formal contexts. On the other hand,
$\mathcal{TDL}-Lite_{Bool}$ is based on the linear time logic LTL, whereas
attributes of the temporal context are definable by the nondeterministic,
branching time logic CTL.

A state context $\mathbb{K}_{s}$ can be translated into
$\mathcal{TDL}-Lite_{Bool}$ as follows:

  * •

The domain $\Delta$ does not correspond to the object set, but to the
attribute set $M$ of $\mathbb{K}_{s}$.

  * •

The object names $a_{i}$ are in one-to-one correspondence to the attributes by
$a_{i}^{\mathcal{I}(n)},\>i\in\\{0,...,|M|-1\\},n\in\mathbb{N}_{0}$.

  * •

The object intent of a state $s\in S$ could be identified with a single
concept name $A_{0}$. Then $A_{0}^{\mathcal{I}(n)}:=s^{\prime}$, but the whole
syntactic information concerning attributes holding in a state would be lost.
A better alternative is to define $|M|$ DL concepts $A_{i}$ by
$A_{i}^{\mathcal{I}(n)}:=a_{i}^{\mathcal{I}(n)}$, if
$a_{i}^{\mathcal{I}(n)}\in s^{\prime}$, else
$A_{i}^{\mathcal{I}(n)}:=\emptyset$. Then the assertion $A_{i}(a_{j})$ holds,
if $i=j$ and $a_{j}^{\mathcal{I}(n)}$ is contained in the intent of $s$.

  * •

Then every line of a state context corresponds to an interpretation
$\mathcal{I}(n)_{C}$ without role names:

$\mathcal{I}(n)_{C}:=(\Delta,a_{0}^{\mathcal{I}(n)},...,A_{0}^{\mathcal{I}(n)},...),$

where the object intent is given by the current interpretation of the DL
concepts $A_{i}$. This is a bijection only if the state context is not
clarified.

This translation defines a concept description language
$(\mathcal{L},\mathcal{I})$ of which $\mathbb{K}_{s}$ is a model. GCIs
$C_{1}\sqsubseteq C_{2}$ hold iff $C_{1}^{\mathcal{I}(n)}\rightarrow
C_{2}^{\mathcal{I}(n)}$ holds in $\mathbb{K}_{s}$ for all
$n\in\mathbb{N}_{0}$, with $C_{1}:=A_{i}\sqcap...\sqcap
A_{j},\>C_{2}:=A_{i}\sqcap...\sqcap A_{j}\sqcap...\sqcap A_{k}$, for
$i<j<k\leq|M-1|$ (compare [18, p. 48-50]).

It is only mentioned here that the interpretation
$P_{i}^{\mathcal{I}(n)}\in\Delta\times\Delta$ of local roles $P_{i}$ may be
given by a transition context.

[15] investigate also a weak temporalisation of $\mathcal{EL}$. In
$\mathcal{TL}_{\mathcal{EL}}$ existential role restrictions are admitted, but
no local roles nor negation, and temporal operators are restricted to $F$
(eventually) for concept constructors and $G$ (always) for formula.

$C:=\top\mid A_{i}\mid C_{1}\sqcap C_{2}\mid\exists T_{i}.C\mid F\,C$

In this logic GCIs are satisfiable in the most general model (where all
concepts and roles are interpreted by the whole domain at every time point),
but it is undecidable whether a GCI is a consequence of a finite set of GCIs
[15, Theorem 11]. In our “pure” FCA framework however, we are safe that such
problems do not occur: For each formal context the specific implications are
decidable, even in linear time relative to the size of the stem base.

A detailed investigation of decidability and complexity issues of the
presented approach in comparison to different DLs is out of reach of this
thesis.

###  9.2 Open mathematical and logical questions

We have established the foundation to exploit manifold mathematical results of
FCA for the analysis of gene expression dynamics and of discrete temporal
transitions in general. In Chapter 6 mathematical questions related to
background knowledge and to the mutual dependency of the defined formal
contexts and their stem bases were solved. While it was not feasible to
develop a complete rule calculus for a large class of temporal implications,
the method was applied to special cases and possibilities of tackling the
general problem were discussed.

A related question should be further analysed: How can attribute exploration
be split into partial problems for these special contexts? For instance, one
could focus on a specific set of genes (Remark 5.1.1) or temporal attributes
first, which is understandable as a scaling (p. 4.3). Then, the decomposition
theory of concept lattices will be useful, which permits an elegant
description by means of the corresponding formal contexts [41, Chapter 4].

The price of the logical completeness of attribute exploration is its
computational complexity. Computation time strongly depends on the logical
structure of the context, and there exist cases where the size of the stem
base is exponential in the size of the input [67]. It was proved recently that
the basic step – recognising whether a subset of attributes is a pseudo-intent
– is coNP-complete [21]. However, deriving an implication from the stem base
is possible in linear time, related to the size of the base, and the Prolog
queries in Section 7.4 were very fast.

In addition to the integration of background knowledge, attribute exploration
can be simplified and shortened, if implications are decided without the
necessity to generate all possible transitions. For that purpose, model
checking could be a promising approach [33], [29], or the structural and
functional analysis of Boolean networks by an adaptation of metabolic network
methods in [66]. In this study, determining activators or inhibitors
corresponds to the kind of rules found by our method, and logical steady state
analysis indicates which species can be produced from the input set and which
not. Another direction of research would be to conclude dynamical properties
of Boolean networks from their structure and the transition functions, for
instance by regarding them as polynomial dynamical systems over finite fields
and by exploiting theoretical work in the context of cellular automata [70,
Section 4 and 6].

###  9.3 Assessment of the biological applications

The analyses in the ECM study (Chapter 8) were based on literature data
refering to healthy human SFB. These findings were fine-tuned and adapted to
gene expression time course data triggered by TGF$\beta$1 and TNF$\alpha$ in
SFB from RA and OA patients. Both the assembly of previous knowledge and the
adaptation of the Boolean functions gave detailed insights into disease-
related regulatory processes. To the best of our knowledge, this is the first
dynamical model of ECM formation and degradation by human SFB.

Manual adaptation of the network may be superseded by algorithms of network
inference [50] in order to achieve a scalability to larger networks. Then,
however, knowledge and data are integrated by criteria fixed in advance and
hardly controllable in their effects, not by flexible and open expert
decisions. Moreover, the relation of the validation by network inference and
by computing the stem base should be clarified, or an expert centered
attribute exploration should be applied.

The fidelity of the obtained temporal rules was reinforced by the comparison
of simulated and observed time series data, first manually, then automatically
by computing the stem base of the combined transitive context. One of the
strengths of the FCA method applied here is its ability to give a complete,
but minimal representation of a data set. This complete overview of temporal
rules enabled us to find new relationships. The most unexpected result is the
expression of TNF at some time points following TNF$\alpha$ stimulation,
whereas it is commonly assumed that SFB do not express TNF [27], [83].
Similarly, our experimental data as well as our simulation results support
MMP9 expression in SFB thus corroborating the majority of the literature
regarding expression of this protease [92], [114]. Here, it is important to
note that a contamination of the SFB population with macrophages (potentially
contributing to MMP9 production) can be excluded due to the SFB isolation
protocol, resulting in a pure SFB population [116]. We also found that MMP1
was induced in the absence of FOS after TNF$\alpha$ stimulation, whereas MMP13
was not expressed despite reports about its induction by NF-$\kappa$B, JUN or
FOS. These facts indicate that the regulation of MMP expression may be more
diverse than presently known and that it still represents a relevant research
target to elucidate the role of SFB in the pathophysiology of rheumatic
diseases.

Concerning the formation of collagen type I fibres by COL1A1 and COL1A2
proteins following the stimulation with TGF$\beta$1, a constitutive expression
of COL1A2 was calculated. Based on these data, COL1A1 has to be regarded as
the critical switch for the formation of collagen I. In contrast, the
corresponding literature generally postulates a co-regulation of both genes,
due to similarities in their promoters [58], [28]. This difference suggests
that the regulation of COL1A1 and COL1A2 may not have been fully elucidated so
far possibly pointing at COL1A1 as a more promising target for the exploration
of fibrosis.

Our analyses also show that TNF$\alpha$-induced signalling predominantly
results in the activation of ETS1 and NFKB1, whereas TGF$\beta$1-related
signal transduction is ultimately regulated via proteins of the SMAD family.
Defined intervention addressing these signalling modules, alone or in
combination with established therapies targeting TNF$\alpha$ (e.g.
etanercept), may therefore improve the efficiency and outcome of current anti-
rheumatic treatment [12]. Alternatively, the present results may be employed
to define subpopulations of RA patients in characteristic phases of RA (active
inflammatory early versus burnt-out/fibrotic late) and tailor anti-rheumatic
treatment to the particular needs of the respective phase [30].

The semantics of a transitive context implies rules to be valid for the
complete simulated and observed time interval. Thus, besides on co-regulation
and contrary regulation, the focus is on (positive or negative) attributes of
gene expression processes which will always hold after a given class of
states. Since in the ECM study always was restricted to the observation time
of 12 h, meaningful results were obtaned. This type of rules is even more
appropriate in cases of a dramatic, permanent switch of the cellular
behaviour, like for sporulation of B. subtilis (Chapter 7). An exploration of
a temporal context was sketched in Section 5.1 for the computer example. Also
in biological applications it will give supplementary insight, in particular
by the eventually attributes.

Exact logical rules (implications) are a precondition for the uniqueness and
minimality of the stem base, in contrast to association rule mining. Data
discretisation is one possibility of handling biological imprecion and
flexibility and to filter out noise resulting from measurements [71]. The
presented method could be developed further by integration of “blurred”
methods like fuzzy FCA [10] or rough sets [39]. Clustering methods could be
applied for data preprocessing.

Data discretisation inevitably causes loss of information. Carefully
evaluating the method, we tried to keep as much important information as
possible and set low thresholds for differential gene expression. A recently
developed FCA-based method avoids predefined discretisation but computes an
ordered set of interval pattern structures depending on the observed values
[62]. Thus, a data set may be described without loss of information or by
means of any desired granularity.

Because the mathematical framework of FCA is very general and open, many
adaptations of the presented methods are possible. According to the mainstream
of applied FCA research, they should aim at the reduction of complexity as
indicated. Then FCA methods can demonstrate their strength best: a clear
account and visualisation of a conceptual logic. Beside this, multifarious
refinements are possible, according to current approaches of modelling
dynamics within systems biology. For instance, the introduction of more fine-
grained expression levels is possible, e.g. in the sense of qualitative
reasoning [64].

However, despite the rough discretisation to two levels off and on, the
complexity of even relatively small networks such as our ECM network and the
completeness of the attribute exploration algorithm led to a large number of
temporal rules. High support of a rule (often correlated to its simplicity)
was used as an indicator for the most meaningful hypotheses about co-
regulation, mutual exclusion, and/or temporal dependencies not only between
single genes, but also between small sets of (functionally related) genes. The
inspection of the rule sets by experts should be aided by automatic
preselection. Iceberg concept lattices lead to a set of “important” rules with
high support, by taking advantage of the duality between the stem base and the
concept lattice of a formal contexts [93].

We applied also complete and in-depth investigation of a small set of
interesting genes by interactive attribute exploration. Using this procedure
for the knowledge base construction, single rules can be validated manually or
by a supporting computer program, or even new experiments can be suggested.
While for the automatic stem base computation we applied a strong validation
criterion requiring rules to hold for all simulated and observed transitions,
the expert can evaluate thresholds of support and confidence. This may reduce
noise or eliminate measurement errors. Importantly, these decision criteria
can be freely combined with relevant knowledge, and the method does not depend
on mathematical problems of association rule mining, since the expert decides
about strict implications. As a main result, several temporal rules were found
and confirmed, which express a coregulation of JUNB and SMAD7, i.e. an
activation of TNF$\alpha$ signalling together with an inhibition of
TGF$\beta$1 signalling (partly also for the inverse case). This is
biologically plausible in general, but was surprising regarding the Boolean
functions. Thus, by simulation, data and expert validation we discovered a
temporal invariant of the network.

Expert decisions are not always obvious. During the first run of the
exploration it turned out that too restrictive implications had been accepted
and therefore different plausible counterexamples could not be introduced. In
the small example, restarting the exploration with the counterexamples was no
major problem, but it would be helpful to keep correct implications. This is
possible if the implications accepted after the first error are checked again
and used as background knowledge for the further exploration [20, p. 14].

Corresponding to the stem base, but with the supplementary information of
extents (states or transitions), Hasse diagrams of the investigated formal
contexts like in Figure 7.2 give detailed (albeit less compact) insight into
gene regulatory processes. They should be generated for small subsets of
genes. Nested line diagrams as implemented in ToscanaJ [7] offer the
possibility to investigate achievable subsets of genes and to combine two of
them at a time according to a chain of questions.

Combining two well-developed algebraic, discrete and logical methods – Boolean
network construction and FCA – it was possible to include human expert
knowledge in all different phases (assembly of the network, adjustment to the
data, choice of relevant temporal rules and interactive attribute
exploration), with the exception of the challenging data discretisation step.
“Digital” and “analog” thinking are combined by our approach. Bioinformatic
algorithms are often complicated and – even if carefully evaluated – can
produce hardly interpretable results. Against this tendency, we wanted to
avoid uncritical trust in the objectivity of a method and let the main task of
interpretation with human experts. Hence, the main biological results are not
only single findings, but within the interdisciplinary collaboration the
medical, biological and bioinformatic scientists as well as myself were
inspired to complex reflections about regulatory processes. We got new
insights regarding existing knowledge and data, which are verifiable by
further experiments.

## Bibliography

  * [1] Bibliosphere.  http://www.genomatix.de/products/BiblioSphere/
BiblioSpherePE1.html.

  * [2] Concept Explorer.  http://sourceforge.net/projects/conexp/.
  * [3] NCBI Gene Expression Omnibus.  http://www.ncbi.nlm.nih.gov/geo/.
  * [4] Pathway studio.  http://www.ariadnegenomics.com.
  * [5] Pubmed.  http://www.ncbi.nlm.nih.gov/pubmed/.
  * [6] R. A free software environment for statistical computing and graphics.  http://www.r-project.org/.
  * [7] ToscanaJ. An FCA suite.  http://toscanaj.sourceforge.net/.
  * [8] AM Abeles and MH Pillinger.  The role of the synovial fibroblast in rheumatoid arthritis: cartilage destruction and the regulation of matrix metalloproteinases.  Bull NYU Hosp Jt Dis, 64:20–24, 2006.
  * [9] B. Alberts et al.  Molecular Biology of the Cell, with CD-ROM. Taylor & Francis, London, 5th rev. ed. edition, 2008.
  * [10] Christina Alcalde, Ana Burusco, and Ramón Fuentes-González.  L-fuzzy concepts and linguistic variables in knowledge acquisition processes.  In Marzena Kryszkiewicz and Sergei Obiedkov, editors, CLA-2010. Proceedings of the 7th International Conference on Concept Lattices and Their Applications, Sevilla, Spain, October 19-21, 2010., pages 50–59. University of Sevilla, 2010.
  * [11] R Altman, E Asch, D Bloch, G Bole, D Borenstein, K Brandt, W Christy, TD Cooke, R Greenwald, and M Hochberg.  Development of criteria for the classification and reporting of osteoarthritis. classification of osteoarthritis of the knee. diagnostic and therapeutic criteria committee of the american rheumatism association.  Arthritis Rheum, 29:1039–1049, 1986.
  * [12] ET Andreakos, BM Foxwell, FM Brennan, RN Maini, and M Feldmann.  Cytokines and anti-cytokine biologicals in autoimmunity: present and future.  Cytokine Growth Factor Rev, 13:299–313, 2002.
  * [13] P Angel and M Karin.  The role of Jun, Fos and the AP-1 complex in cell-proliferation and transformation.  Biochim Biophys Acta, 1072:129–157, 1991.
  * [14] FC Arnett, SM Edworthy, DA Bloch, DJ McShane, JF Fries, NS Cooper, LA Healey, SR Kaplan, MH Liang, and HS Luthra.  The American Rheumatism Association 1987 revised criteria for the classification of rheumatoid arthritis.  Arthritis Rheum, 31:315–324, 1988.
  * [15] A. Artale, R. Kontchakov, C. Lutz, F. Wolter, and M. Zakharyaschev.  Temporalising tractable description logics.  In Proceedings of the Fourteenth International Symposium on Temporal Representation and Reasoning. IEEE Computer Society Press, 2007.
  * [16] H Asahara, K Fujisawa, T Kobata, T Hasunuma, T Maeda, M Asanuma, N Ogawa, H Inoue, T Sumida, and K Nishioka.  Direct evidence of high DNA binding activity of transcription factor AP-1 in rheumatoid arthritis synovium.  Arthritis Rheum, 40:912–918, 1997.
  * [17] F. Baader, D. McGuinness, D. Nardi, and P.F. Patel-Schneider.  The Description Logic Handbook: Theory, Implementation and Applications.  Cambridge University Press, 2003.
  * [18] Franz Baader and Felix Distel.  A finite basis for the set of $\mathcal{EL}$-implications holding in a finite model.  In Raoul Medina and Sergei A. Obiedkov, editors, ICFCA, volume 4933 of Lecture Notes in Computer Science, pages 46–61. Springer, 2008\.
  * [19] Franz Baader and Felix Distel.  Exploring finite models in the description logic $\mathcal{EL}_{gfp}$.  In Sébastien Ferré and Sebastian Rudolph, editors, ICFCA, volume 5548 of Lecture Notes in Computer Science, pages 146–161. Springer, 2009.
  * [20] Franz Baader and Baris Sertkaya.  Usability issues in description logic knowledge base completion.  In Sébastien Ferré and Sebastian Rudolph, editors, ICFCA, volume 5548 of Lecture Notes in Computer Science, pages 1–21. Springer, 2009.
  * [21] Mikhail A. Babin and Sergei O. Kuznetsov.  Recognizing pseudo-intents is coNP-complete.  In Marzena Kryszkiewicz and Sergei Obiedkov, editors, CLA-2010. Proceedings of the 7th International Conference on Concept Lattices and Their Applications, Sevilla, Spain, October 19-21, 2010., pages 294–301. University of Sevilla, 2010.
  * [22] A Barchowsky, D Frleta, and MP Vincenti.  Integration of the NF-kappaB and mitogen-activated protein kinase/AP-1 pathways at the collagenase-1 promoter: divergence of IL-1 and TNF-dependent signal transduction in rabbit primary synovial fibroblasts.  Cytokine, 12:1469–1479, 2000.
  * [23] I Berger and Y Shaul.  c-Fos antagonizes the junD gene positive autoregulatory loop; a novel c-Fos role in promoter switching.  Gene, 211:375–382, 1998.
  * [24] BM Bolstad, RA Irizarry, M Astrand, and TP Speed.  A comparison of normalization methods for high density oligonucleotide array data based on variance and bias.  Bioinformatics, 19:185–193, 2003.
  * [25] V Bours, J Villalobos, PR Burd, K Kelly, and U Siebenlist.  Cloning of a mitogen-inducible gene encoding a $\kappa$B DNA-binding protein with homology to the rel oncogene and to cell-cycle motifs.  Nature, 348:76–80, 1990.
  * [26] Peter Burmeister.  Formal Concept Analysis with ConImp: Introduction to the Basic Features.  University of Technology, Darmstadt. http://www.mathematik.tu-darmstadt.de/burmeister/ConImpIntro.pdf, 2003.
  * [27] GR Burmester, B Stuhlmuller, G Keyszer, and RW Kinne.  Mononuclear phagocytes and rheumatoid synovitis. Mastermind or workhorse in arthritis?  Arthritis Rheum, 40:5–18, 1997.
  * [28] C Buttner, A Skupin, and EP Rieber.  Transcriptional activation of the type I collagen genes COL1A1 and COL1A2 in fibroblasts by interleukin-4: analysis of the functional collagen promoter sequences.  J Cell Physiol, 198:248–258, 2004.
  * [29] Nathalie Chabrier-Rivier et al.  Modeling and querying biomolecular interaction networks.  Theoretical Computer Science, 325(1):25–44, 2004.
  * [30] PG Conaghan, MJ Green, and P Emery.  Established rheumatoid arthritis.  Baillieres Best Pract Res Clin Rheumatol, 13:561–575, 1999.
  * [31] H. de Jong et al.  Qualitative simulation of the initiation of sporulation in Bacillus subtilis.  Bulletin of Mathematical Biology, 66:261–299, 2004.
  * [32] E. Dimitrova, J. McGee, and R. Laubenbacher.  Discretization of time course data.  http://polymath.vbi.vt.edu/discretization/DimitrovaMcGeeLaubenbacher.pdf, 2005\.
  * [33] J. Esparza.  Model checking using net unfoldings.  Sci. Comput. Programm., 23:151–195, 1994.
  * [34] GS Firestein and AM Manning.  Signal transduction and transcription factors in rheumatic disease.  Arthritis Rheum, 42:609–621, 1999.
  * [35] GS Firestein, MM Paine, and BH Littman.  Gene expression (collagenase, tissue inhibitor of metalloproteinases, complement, and HLA-DR) in rheumatoid arthritis and osteoarthritis synovium. Quantitative analysis and effect of intraarticular corticosteroids.  Arthritis Rheum, 34:1094–1105, 1991.
  * [36] B. Ganter, G. Stumme, and R. Wille.  Formal Concept Analysis – Foundations and Applications, volume 3626 of LNAI.  Springer, Heidelberg, 2005.
  * [37] Bernhard Ganter.  Attribute exploration with background knowledge.  Theor Comput Sci, 217(2):215–233, 1999.
  * [38] Bernhard Ganter.  Contextual attribute logic of many-valued attributes.  In B. Ganter, S. Stumme, and R. Wille, editors, Formal Concept Analysis – Foundations and Applications, number 3626 in LNAI, pages 101–113, Heidelberg, 2005. Springer.
  * [39] Bernhard Ganter and Christian Meschke.  A formal concept analysis approach to rough data tables.  In RSFDGrC, pages 117–126, 2009.
  * [40] Bernhard Ganter and Sebastian Rudolph.  Formal concept analysis methods for dynamic conceptual graphs.  In ICCS 2001, LNAI 2120, pages 143–156, Heidelberg, 2001. Springer.
  * [41] Bernhard Ganter and Rudolf Wille.  Formal Concept Analysis – Mathematical Foundations.  Springer, Heidelberg, 1999.
  * [42] F Gaultier, A Foucault-Bertaud, E Lamy, AL Ejeil, SM Dridi, N Piccardi, A Piccirilli, P Msika, G Godeau, and B Gogly.  Effects of a vegetable extract from lupinus albus (LU105) on the production of matrix metalloproteinases (MMP1, MMP2, MMP9) and tissue inhibitor of metalloproteinases (TIMP1, TIMP2) by human gingival fibroblasts in culture.  Clin Oral Investig, 7:198–205, 2003.
  * [43] EM Gravallese, JM Darling, AL Ladd, JN Katz, and LH Glimcher.  In situ hybridization studies of stromelysin and collagenase messenger RNA expression in rheumatoid synovium.  Arthritis Rheum, 34:1076–1084, 1991.
  * [44] H. P. Gumm.  Universelle Coalgebra.  In Thomas Ihringer, Allgemeine Algebra, pages 155–208. Heldermann, 2003.
  * [45] Mohamed Rouane Hacene, Marianne Huchard, Amedeo Napoli, and Petko Valtchev.  A proposal for combining formal concept analysis and description logics for mining relational data.  In Sergei O. Kuznetsov and Stefan Schmidt, editors, ICFCA, volume 4390 of Lecture Notes in Computer Science, pages 51–65. Springer, 2007.
  * [46] H Hacker and M Karin.  Regulation and function of IKK and IKK-related kinases.  Sci STKE, 2006:re13, 2006.
  * [47] Z Han, DL Boyle, AM Manning, and GS Firestein.  AP-1 and NF-kappaB regulation in rheumatoid arthritis and murine collagen-induced arthritis.  Autoimmunity, 28:197–208, 1998.
  * [48] ML Handel and L Girgis.  Transcription factors.  Best Pract Res Clin Rheumatol, 15:657–675, 2001.
  * [49] JA Hartigan and MA Wong.  A k-means clustering algorithm.  Applied Statistics, 28:100–108, 1979.
  * [50] Michael Hecker, Sandro Lambeck, Susanne Toepfer, Eugene van Someren, and Reinhard Guthke.  Gene regulatory network inference: Data integration in dynamic models – a review.  BioSystems, 96:86–103, 2009.
  * [51] JM Hernandez, DH Floyd, KN Weilbaecher, PL Green, and K Boris-Lawrie.  Multiple facets of junD gene expression are atypical among AP-1 family members.  Oncogene, 27:4757–4767, 2008.
  * [52] SI Hirai, RP Ryseck, F Mechta, R Bravo, and M Yaniv.  Characterization of junD: a new member of the jun proto-oncogene family.  Embo J, 8:1433–1439, 1989.
  * [53] Jens Hollunder, Maik Friedel, Andreas Beyer, Christopher T. Workman, and Thomas Wilhelm.  DASS: efficient discovery and p-value calculation of substructures in unordered data.  Bioinformatics, 23(1):77–83, 2007.
  * [54] John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman.  Introduction to Automata Theory, Languages, and Computation.  Addison Wesley, Upper Saddle River, N.J., 2nd edition, 2001.
  * [55] LC Huber, O Distler, I Tarner, RE Gay, S Gay, and T Pap.  Synovial fibroblasts: key players in rheumatoid arthritis.  Rheumatology (Oxford), 45:669–675, 2006.
  * [56] Thomas Ihringer.  Allgemeine Algebra.  Heldermann, Lemgo, 2003.
  * [57] E Karouzakis, M Neidhart, RE Gay, and S Gay.  Molecular and cellular basis of rheumatoid joint destruction.  Immunol Lett, 106:8–13, 2006.
  * [58] G Karsenty and RW Park.  Regulation of type I collagen genes expression.  Int Rev Immunol, 12:177–185, 1995.
  * [59] S.A. Kauffman.  The Origins of Order: Self-Organization and Selection in Evolution.  Oxford University Press, New York, 1993.
  * [60] Stuart Kauffman, Carsten Peterson, Björn Samuelsson, and Carl Troein.  Random Boolean network models and the yeast transcriptional network.  PNAS, 100(25):14796–14799, 2003.
  * [61] Stuart A. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets.  J Theor Biol, 22:437–467, 1969.
  * [62] Mehdi Kaytoue, Sergei O. Kuznetsov, Amedeo Napoli, and Sébastien Duplessis.  Mining gene expression data with pattern structures in formal concept analysis.  Information Sciences, 181(10):1989 – 2001, 2011.
  * [63] G Kervizic and L Corcos.  Dynamical modeling of the cholesterol regulatory pathway with Boolean networks.  BMC Syst Biol, 2:99, 2008.
  * [64] Ross D. King, Simon M. Garrett, and George M. Coghill.  On the use of qualitative reasoning to simulate and identify metabolic pathways.  Bioinformatics, 21(9):2017–2026, 2005.
  * [65] RW Kinne, E Palombo-Kinne, and F Emmrich.  Activation of synovial fibroblasts in rheumatoid arthritis.  Ann Rheum Dis, 54:501–504, 1995.
  * [66] S. Klamt.  A methodology for the structural and functional analysis of signaling and regulatory networks.  BMC Bioinformatics, 7(56), 2006.
  * [67] S.O. Kuznetsov and S.A. Obiedkov.  Counting pseudo-intents and #P-completeness.  In J. Schmid, editor, ICFCA 2006, volume 3874 of LNCS, pages 306–308, Heidelberg, 2006. Springer.
  * [68] PF Lambert, MJ Ludford-Menting, NJ Deacon, I Kola, and RR Doherty.  The Nfkb1 promoter is controlled by proteins of the Ets family.  Mol Biol Cell, 8:313–323, 1997.
  * [69] Serge Lang.  Algebra.  Springer, Heidelberg, 3rd edition, 2002.
  * [70] R. Laubenbacher.  Algebraic models in systems biology.  In H. Anai and K. Horimoto, editors, Algebraic Biology 2005, Proceedings of the First International Conference on Algebraic Biology – Computer Algebra in Biology, pages 33–35, Tokyo, 2005. Universal Academy Press.
  * [71] R Laubenbacher and B Stigler.  A computational algebra approach to the reverse engineering of gene regulatory networks.  J Theor Biol, 229:523–537, 2004.
  * [72] S Martin, Z Zhang, A Martino, and JL Faulon.  Boolean dynamics of genetic regulatory networks inferred from microarray time series data.  Bioinformatics, 23:866–874, 2007.
  * [73] SS McCachren.  Expression of metalloproteinases and metalloproteinase inhibitor in human arthritic synovium.  Arthritis Rheum, 34:1085–1093, 1991.
  * [74] M Mizui, Y Isaka, Y Takabatake, Y Sato, H Kawachi, F Shimizu, S Takahara, T Ito, and E Imai.  Transcription factor Ets-1 is essential for mesangial matrix remodeling.  Kidney Int, 70:298–305, 2006.
  * [75] Susanne Motameny, Beatrix Versmold, and Rita Schmutzler.  Formal concept analysis for the identification of combinatorial biomarkers in breast cancer.  In Raoul Medina and Sergei A. Obiedkov, editors, ICFCA, volume 4933 of Lecture Notes in Computer Science, pages 229–240. Springer, 2008\.
  * [76] EP Newberry, D Willis, T Latifi, JM Boudreaux, and DA Towler.  Fibroblast growth factor receptor signaling activates the human interstitial collagenase promoter via the bipartite Ets-AP1 element.  Mol Endocrinol, 11:1129–1144, 1997.
  * [77] Peter Øhrstrøm.  Time and logic: A.N. Prior’s formal analysis of temporal concepts.  In Sébastien Ferré and Sebastian Rudolph, editors, Formal Concept Analysis, 7th International Conference, ICFCA 2009, Darmstadt, Germany, May 21-24, 2009, Proceedings, volume 5548 of Lecture Notes in Computer Science, pages 66–81, Berlin Heidelberg, 2009. Springer.
  * [78] Charles Sanders Peirce.  How to make our ideas clear.  In C. Hartshorne and P. Weiss, editors, Collected papers, Cambridge / Mass., 1931 - 1935. Harvard University Press.
  * [79] SG Pereira and F Oakley.  Nuclear factor-kappaB1: regulation and function.  Int J Biochem Cell Biol, 40:1425–1430, 2008.
  * [80] John L. Pfaltz and Christopher M. Taylor.  Closed set mining of biological data.  In BIOKDD02: Workshop on Data Mining in Bioinformatics, 2002.
  * [81] AE Postlethwaite, MA Holness, H Katai, and R Raghow.  Human fibroblasts synthesize elevated levels of extracellular matrix proteins in response to interleukin 4.  J Clin Invest, 90:1479–1485, 1992.
  * [82] Aviv Regev and Ehud Shapiro.  The pi-calculus as an abstraction for biomolecular systems.  In G. Ciobanu and G. Rozenberg, editors, Modelling in Molecular Biology, Natural Computing Series. Springer, 2004.
  * [83] C Ritchlin.  Fibroblast biology. Effector signals released by the synovial fibroblast in arthritis.  Arthritis Res, 2:356–360, 2000.
  * [84] G Rogler and T Andus.  Cytokines in inflammatory bowel disease.  World J Surg, 22:382–389, 1998.
  * [85] S Ross and CS Hill.  How the Smads regulate transcription.  Int J Biochem Cell Biol, 40:383–408, 2008.
  * [86] R Rossignol, B Faustin, C Rocher, M Malgat, JP Mazat, and T Letellier.  Mitochondrial threshold effects.  Biochem J, 370:751–762, 2003.
  * [87] Sebastian Rudolph.  Relational Exploration – Combining Description Logics and Formal Concept Analysis for Knowledge Specification.  Universitätsverlag Karlsruhe, December 2006.  Dissertation.
  * [88] T Schlitt and A Brazma.  Current approaches to gene regulatory network modelling.  BMC Bioinformatics, 8(Suppl 6):S9, 2007.
  * [89] P Shannon, A Markiel, O Ozier, NS Baliga, JT Wang, D Ramage, N Amin, B Schwikowski, and T Ideker.  Cytoscape: a software environment for integrated models of biomolecular interaction networks.  Genome Res, 13:2498–2504, 2003.
  * [90] JS Smolen and G Steiner.  Therapeutic strategies for rheumatoid arthritis.  Nat Rev Drug Discov, 2:473–488, 2003.
  * [91] L. Jason Steggles, Richard Banks, Oliver Shaw, and Anil Wipat.  Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach.  Bioinformatics, 23(3):336–343, 2007.
  * [92] KM Stuhlmeier.  Mepacrine inhibits matrix metalloproteinases-1 (MMP-1) and MMP-9 activation in human fibroblast-like synoviocytes.  J Rheumatol, 30:2330–2337, 2003.
  * [93] G. Stumme, R. Taouil, Y. Bastide, N. Pasquier, and L. Lakhal.  Computing iceberg concept lattices with TITANIC.  Data and Knowledge Engineering, 42:189–222, 2002.
  * [94] Y Sun, L Wenger, CE Brinckerhoff, RR Misra, and HS Cheung.  Basic calcium phosphate crystals induce matrix metalloproteinase-1 through the Ras/mitogen-activated protein kinase/c-Fos/AP-1/metalloproteinase 1 pathway. Involvement of transcription factor binding sites AP-1 and PEA-3.  J Biol Chem, 277:1544–1552, 2002.
  * [95] Michael Thielscher.  FLUX: A logic programming method for reasoning agents.  Theory and Practice of Logic Programming, 5(4-5):533–565, 2005\.
  * [96] C.A. Tomas et al.  DNA array-based transcriptional analysis of asporogenous, nonsolventogenic clostridium acetobutylicum strains SKO1 and M5.  J. Bacteriol., pages 4539–4547, 2003.
  * [97] F Verrecchia and A Mauviel.  Transforming growth factor-beta signaling through the Smad pathway: role in extracellular matrix gene expression and regulation.  J Invest Dermatol, 118:211–215, 2002.
  * [98] F Verrecchia, M Pessah, A Atfi, and A Mauviel.  Tumor necrosis factor-alpha inhibits transforming growth factor-beta/Smad signaling in human dermal fibroblasts via AP-1 activation.  J Biol Chem, 275:30226–30231, 2000.
  * [99] Hartmut von Hentig.  Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozeß.  Klett, Stuttgart, 1972.
  * [100] JH Ward.  Hierarchical grouping to optimize an objective function.  J Am Stat Assoc, 58:236–244, 1963.
  * [101] Natalia Wegner et al.  Peptidylarginine deiminase from Porphyromonas gingivalis citrullinates human fibrinogen and $\alpha$-Enolase. Implications for autoimmunity in rheumatoid arthritis.  Arthritis and Rheumatism, 62(9):2662–2672, 2010.
  * [102] HG Welgus, JJ Jeffrey, and AZ Eisen.  The collagen substrate specificity of human skin fibroblast collagenase.  J Biol Chem, 256:9511–9515, 1981.
  * [103] Z Werb.  Proteinases and matrix degradation.  Textbook of Rheumatology, pages 300–321, 1989.
  * [104] LA White and CE Brinckerhoff.  Two activator protein-1 elements in the matrix metalloproteinase-1 promoter have different effects on transcription and bind Jun D, c-Fos, and Fra-2.  Matrix Biol, 14:715–725, 1995.
  * [105] LA White, TI Mitchell, and CE Brinckerhoff.  Transforming growth factor beta inhibitory element in the rabbit matrix metalloproteinase-1 (collagenase-1) gene functions as a repressor of constitutive transcription.  Biochim Biophys Acta, 1490:259–268, 2000.
  * [106] Karl Erich Wolff.  Interpretation of automata in temporal concept analysis.  In Uta Priss, Dan Corbett, and Galia Angelova, editors, Proceedings of the 10th International Conference on Conceptual Structures (ICCS 2002), volume 2393 of Lecture Notes in Computer Science, pages 341–353. Springer, 2002.
  * [107] K.E. Wolff.  States of distributed objects in conceptual semantic systems.  In F. Dau, M.-L. Mugnier, and G. Stumme, editors, Conceptual Structures: Common Semantics for Sharing Knowledge, number 3596 in LNAI, pages 250–266, Heidelberg, 2005. Springer.
  * [108] K.E. Wolff.  States, transitions, and life tracks in temporal concept analysis.  In B. Ganter, S. Stumme, and R. Wille, editors, Formal Concept Analysis – Foundations and Applications, number 3626 in LNAI, pages 127–148, Heidelberg, 2005. Springer.
  * [109] J. Wollbold, R. Huber, D. Pohlers, D. Koczan, R. Guthke, R. Kinne, and U. Gausmann.  Adapted Boolean network models for extracellular matrix formation.  BMC Systems Biology, 3(77):21 July 2009, 2009.
  * [110] Johannes Wollbold.  Attribute exploration of discrete temporal transitions.  In Alain Gély et al., editor, Contributions to ICFCA 2007 – 5th International Conference on Formal Concept Analysis, pages 121–130, Clermont-Ferrand, 2007.
  * [111] Johannes Wollbold, Reinhard Guthke, and Bernhard Ganter.  Constructing a knowledge base for gene regulatory dynamics by formal concept analysis methods.  In K. Horimoto et al., editor, Algebraic Biology 2008, LNCS 5147, pages 230–244, Heidelberg, 2008. Springer.
  * [112] Johannes Wollbold, René Huber, and Karl Erich Wolff.  Conceptual representation of gene expression processes.  In Karl Erich Wolff, Dmitry E. Palchunov, Nikolay G. Zagoruiko, and Urs Andelfinger, editors, Knowledge Processing and Data Analysis. Proceedings of KONT/KPP 2007, LNAI 6581, pages 79–100, Heidelberg, 2011. Springer.
  * [113] JJ Wu, MW Lark, LE Chun, and DR Eyre.  Sites of stromelysin cleavage in collagen types II, IX, X, and XI of cartilage.  J Biol Chem, 266:5625–5628, 1991.
  * [114] M Xue, L March, PN Sambrook, and CJ Jackson.  Differential regulation of matrix metalloproteinase 2 and matrix metalloproteinase 9 by activated protein C: relevance to inflammation in rheumatoid arthritis.  Arthritis Rheum, 56:2864–2874, 2007.
  * [115] Y Yamanishi and GS Firestein.  Pathogenesis of rheumatoid arthritis: the role of synoviocytes.  Rheum Dis Clin North Am, 27:355–371, 2001.
  * [116] T Zimmermann, E Kunisch, R Pfeiffer, A Hirth, HD Stahl, U Sack, A Laube, E Liesaus, A Roth, and E Palombo-Kinne.  Isolation and characterization of rheumatoid arthritis synovial fibroblasts from primary culture – primary culture cells markedly differ from fourth-passage cells.  Arthritis Res, 3:72–76, 2001.

## Index

  * A- and B-concepts of time §3.2.1
  * ABox §3.2.3
  * Always §4.5
  * Armstrong rules §2.2, §6.3, Proposition 6.3.1
  * Association rule §5.2
  * Attribute Definition 2.1.1
  * Attribute exploration Chapter 1, §2.2
    * complexity §9.2
    * Next Closure algorithm Chapter 5
  * Automaton §3.1.1
    * Acceptor §3.1.1
  * Boolean network Chapter 1, §3.3.2, Definition 4.3.2, item 2, Table 7.1, §8.2.2, §9.2
  * Branching time §3.2.1, §3.2.2
  * Category §3.1
  * Chronicle §3.2.1
  * Closure operator §2.1, §6.1.1
  * Closure system §6.1.1
  * Closure system, closed set §2.1
  * Coalgebra §3.1, Definition 3.1.2
  * Completeness (logics) §2.2
  * Computation Tree Logic (CTL) §3.2.2
  * Concept Explorer §2.2, §6.5, Figure 7.2, Figure 7.2
  * Concept lattice (hierarchy) Figure 2.1, §2.1
  * Confidence §5.2
  * ConImp §2.2, §6.4, §6.5
  * Counterexample §2.2
  * Description logics (DL) §3.2.3, §9.1
  * Discretisation §2.1, §8.2.3, §8.3.4, §9.3
  * Endofunctor §3.1
  * Entity §3.3.2, 1st item
  * Eventually §4.5
  * Extent Figure 2.1
  * Filter (lattice) Figure 7.2
  * Fluent 2nd item
  * Formal concept Definition 2.1.7
  * Formal Concept Analysis (FCA) Chapter 2
  * Formal context Definition 2.1.1
    * apposition Definition 2.1.5, §2.3, §4.5, §6.2.2, Table 9.2
    * clarified §2.1
    * derived §2.1
    * many-valued Definition 2.1.2, Table 4.2
    * reduced §2.1
  * Functor §3.1
  * Galois connection §6.1.1
  * Gene regulatory network §7.1, Figure 8.4, §8.2.2, §8.3.1, §9.3
  * Hasse diagram §2.1, §9.3
  * History §3.2.1
  * Ideal (lattice) Figure 7.2
  * Implication §2.2
    * holds §2.2
    * respected §2.2
  * Intent Figure 2.1
  * Kripke structure §3.1.2, §4.3
  * Labelled Transition System with Attributes (LTSA) §3.1.3
  * Lattice Figure 2.1
    * Boolean §2.2
    * complete Chapter 1, §2.1
  * Lectic order Chapter 5
  * Liveness §3.2.2, §4.5
  * Model §3.2.1, §5.3, §6.1.1, §9.1
    * checking 1st item, §9.2
  * Morphism §3.1
  * Nested line diagram Remark 5.1.1, §9.3
  * Never §4.5
  * Next §4.5
  * Non-redundancy (logics) §2.2
  * Object Definition 2.1.1
  * Output function §3.1.1, §3.1.2, §4.2, §4.3
  * Petri net Chapter 1, §7.1
  * Process
    * deterministic Definition 3.1.3, §4.3, §4.5, §6.3, §8.2.5
    * gene regulatory §7.1
    * nondeterministic §3.1.2, §3.2.1, §3.2.2, §6.3, §7.2, §9.1
  * Prolog §5.2, §7.2, §7.4, §8.2.8
  * Propositional tense logic §3.2.1
  * Pseudo-intent §2.2, §5.1
  * Relational Concept Analysis (RCA) §9.1
  * Role (DL) §3.2.3, 3rd item, §9.1
    * depth §9.1
    * local §3.2.3, §9.1, §9.1
  * Safety §3.2.2, §4.5
  * Scaling Definition 2.1.4, §4.2, §9.2
    * biordinal Table 4.2
    * dichotomic Table 2.2, §4.3, §4.5, Remark 6.2.3, §8.3.9
    * nominal §2.1, Table 3.1, §4.2
  * Semantic inference §6.1.1
  * Semantics Chapter 1, §2.2, Chapter 3, Table 3.2, §4.5, Chapter 5
  * Semiautomaton §3.1.3
  * Semiproduct of formal contexts Definition 2.1.6, §4.3
  * Soundness (logics) §2.2
  * State §2.3, item 1, §3.1, Example 3.1.4, Example 3.1.6
    * final §3.1.1, §3.1.2, §3.1.3, §3.1.3, 4th item, §4.5, §5.1
    * input Example 3.1.6, §4.3, Definition 4.5.2, 2nd item
    * output Example 3.1.6, §4.3, §4.3, §7.2, §8.2.5, 3rd item
    * steady 2nd item, §5.1, §8.2.4
  * State context §4.2
    * $\mathcal{TDL}-Lite_{Bool}$ §9.1
  * Stem base Chapter 1, §2.2, §2.2, §2.2
  * Support §5.1, §5.2, §9.3
  * Syntax Chapter 1, Chapter 3, Table 3.2
  * TBox §3.2.3
  * Temporal Concept Analysis (TCA) §2.3
  * Temporal context Definition 4.5.2
  * Test context §6.2.2
  * Theorem
    * Map Reconstruction §3.1.4
    * of Duquenne-Guigues Theorem 2.2.2, §6.3
  * Transition Example 3.1.6
  * Transition context §4.3
    * DL Table 9.1
  * Transition function §3.1.1, §3.1.2, §4.3, Definition 4.3.2
  * Transitive context §4.4
    * DL Table 9.2
  * Until §4.5
  * $\pi$-calculus §3.3.1

## Attachments

  1. 1.

Ehrenwörtliche Erklärung

  2. 2.

CD with supplementary files and the developed R scripts

## Ehrenwörtliche Erklärung

Hiermit erkläre ich ehrenwörtlich, dass mir die geltende Promotionsordnung der
Fakultät für Mathematik und Informatik der Friedrich-Schiller-Universität Jena
bekannt ist. Ich versichere, dass ich die vorliegende Arbeit selbständig, ohne
unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen
Hilfsmittel angefertigt habe. Die aus fremden Quellen direkt oder indirekt
übernommenen Gedanken sowie persönliche Mitteilungen sind als solche kenntlich
gemacht. Eigene Prüfungsarbeiten wurden nicht verwendet. Bei der Auswahl und
Auswertung des Materials sowie bei der Herstellung des Manuskripts haben mich
die in der Danksagung genannten Personen unterstützt. Ich habe nicht die Hilfe
von Vermittlungs- bzw. Beratungsdiensten in Anspruch genommen. Niemand hat von
mir unmittelbar oder mittelbar geldwerte Leistungen für Arbeiten erhalten, die
im Zusammenhang mit dem Inhalt der vorgelegten Arbeit stehen.

Weiterhin erkläre ich, dass ich diese Dissertation noch nicht als
Prüfungsarbeit für eine staatliche oder andere wissenschaftliche Prüfung
eingereicht habe. Auch habe ich noch keine Dissertation bei einer anderen
Hochschule eingereicht.

Erfurt, 9. Mai 2011 Johannes Wollbold