Figure 1.
and (B) components of a single LUT entry.
Figure 2.
Connectivity graph of the SPN model.
The fifteen genes and proteins considered in the SPN model are represented (white nodes). The incoming edges to a node originate in the nodes that are used by
to determine its transition. Shaded nodes represent the spatial signals (states of WG, HH and
in neighbouring cells). Note that SLP – derived from an upstream intra-cellular signal – is an input node to this network. The self-connection it has represents the steady-state assumption:
. Notice also that this graph represents the fully synchronous version of the model, where modifications concerning PH and SMO have been made (see main text for details).
Figure 3.
A parasegment in the SPN model.
Cells are represented horizontally, where the top (bottom) row is the most anterior (posterior) cell. Each column is a gene, protein or complex – a node in the context of the BN model. The specific pattern shown corresponds to the wild-type initial expression pattern observed at the onset of the segment polarity genes regulatory dynamics (); Black/on (white/off) squares represent expressed (not expressed) genes or proteins.
Table 1.
Boolean logic formulae representing the state-transition functions for each node in the SPN (four-cell parasegment) model.
Figure 4.
The ten attractors reached by the SPN.
These attractors are divided in four groups: wild-type, broad-stripe, no segmentation and ectopic. More specifically: (a) wild-type, (b) variant of (a), (c) wild-type with two stripes, (d) variant of (c), (e) broad-stripe, (f) no segmentation, (g) ectopic, (h) variant of (g), (i) ectopic with two
stripes, and (j) variant of (i). The wild-type attractor (a) is referred to as
in the results and discussion sections.
Figure 5.
(A) Subset of LUT entries of an example automaton that prescribe state transitions to on (1); white (black) states are 0 (1). (B) Wildcard schema redescription; wildcards denoted also by grey states. Schema
is highlighted to identify the subset of LUT entries
it redescribes. (C) Two-symbol schema redescription, using the additional position-free symbol; the entire set
is compressed into a single two-symbol schema:
. Any permutation of the inputs marked with the position-free symbol in
results in a schema in
. Note that
redescribes the entire set of entries with transition to on and thus
. Since there is only one set of marked inputs, the position-free symbol does not require an index. Although this figure depicts only the schemata obtained for the subset of LUT entries of
that transition to on, entries that do not match any of these schemata transition to off (since
is a Boolean automaton).
Figure 6.
McCulloch & Pitts representation of Expression (1).
The conjunction clauses in Expression (1) for the example automaton are directly mapped onto a standard McCulloch & Pitts network with two layers. On one layer the two literal enputs are accounted for by a threshold unit (at the top) with threshold
. There is also a group-invariant enput with permutation subconstraints on both Boolean states. Two threshold units on the same layer are used to capture these. The threshold unit on the left accounts for the permutation subconstraint
. It thus has as incoming s-units the inputs
and threshold
. In a similar way, the threshold unit on the right accounts for the subconstraint
. When all the constraints (literal and group-invariant) are satisfied, the last threshold unit (second layer) ‘fires’ causing the transition to on.
Figure 7.
Every s-unit is a circle, labelled according the automaton's input it represents and coloured according to its state: black is on and white is off (here we use light-blue for a generic state). The t-unit (schema) is represented using a larger circle. One of its halves is coloured, and the other labelled with the t-unit's threshold . Fibres can be single, or branched. In this example there are branching fibres only, where fibre fusions represent all possible combinations of two out of the three s-units.
Table 2.
Connectivity rules in canalizing maps.
Figure 8.
Canalizing map of example automaton characterized by a single schema.
(A) Since (shown on top) has
, the corresponding s-units for literal enputs
are directly linked to the t-unit for
with single fibres;
. (B) Adding the subconstraint
of the group-invariant enput
. In this case,
, so there is only one subset
and thus a single branch from each s-unit in the group-invariant, fused into a single ending. The threshold becomes
. (C) Finally, we add the second subconstraint
of the group-invariant enput
, which has the same properties as the subconstraint integrated in (B). The final threshold of the t-unit is given by (9), therefore
. Notice that only the input-combinations that satisfy the constraints of Expression (1) for
can lead to the firing of the t-unit; in other words, the canalizing map is equivalent to schema
.
Figure 9.
Procedure for obtaining the canalizing map of a group-invariant subconstraint.
(A) subconstraint defined by , where
. The first step is to consider the s-units (in state 0) for the four input variables in the group invariant enput
. Next we identify all the subsets
of these s-units containing
s-units:
(shown with dotted arrows). From every s-unit in each such subset
, we take an outgoing fibre to be joined together into a single fibre ending as input to the t-unit. Finally, we increase the threshold of the t-unit by the total number of subsets, that is
. (B) An example of the same procedure but for
and
:
.
Figure 10.
x. (A) complete set of schemata for
, including the transitions to on shown in Figure 5 and the transitions of off (the negation of the first).(B) canalizing map; t-units for schemata
and
were merged into a single t-unit with threshold
(see main text). (C) effective connectivity, input symmetry and input redundancy of
.
Figure 11.
Micro-level canalization for the Automata in the SPN model.
Schemata for inhibition (transitions to off) and expression (transitions to on) are shown for each node (genes or proteins) in model. In-degree (), input redundancy (
), effective connectivity (
), and input symmetry (
) are also shown.
Figure 12.
Quantification of canalization in the SPN automata.
Relative input redundancy is measured in the horizontal axis () and relative input symmetry is measured in the vertical axis (
). Most automata in the SPN fall in the class II quadrant, showing that most canalization is of the input redundancy kind, though nodes such as CIR and
display strong input symmetry too.
Figure 13.
Canalizing Maps of individual nodes in the SPN model (part 1).
The set of schemata for each automaton is converted into two CMs: one representing the minimal control logic for its inhibition, and another for its expression. Note that denotes the state of node
in both neighbour cells:
and
, where
is one of the spatial-signals
, HH, or WG (see text).
Figure 14.
Canalizing Maps of individual nodes in the SPN model (cont).
The set of schemata for each automaton is converted into two CMs: one representing the minimal control logic for its inhibition, and another for its expression. Note that denotes the state of node
in both neighbour cells:
and
, where
is one of the spatial-signals
, HH, or WG (see text).
Figure 15.
Dynamics Canalization Map for a single cell of the SPN Model.
Also depicted are pathway modules (pink) and
(blue), whose initial conditions depend exclusively on the expression and inhibition of input node SLP and of WG in neighbouring cells (the nWG spatial-signals).
,
(see details in text).
Figure 16.
Dynamical unfolding of the (single-cell) SPN with partial input configurations.
The eight initial partial configurations tested correspond to the combinations of the steady-states of intra- and inter-cellular inputs SLP, WG and
(and where
HH and
are merged into a single node,
). The specific state-combinations of these three variables is depicted on the middle (white) tab of each dynamical unfolding plot. Initial patterns that reach the same target pattern are grouped together in five groups
to
(identified in the top tab of each plot). The six input configurations in groups G1, G2, and G3 depict the dynamics where pathway module
is involved (nodes involved in this module are highlighted in pink.) The two input configurations in G4 and G5 depict the dynamics where pathway module
is involved (nodes involved in this module are highlighted in blue.) Three of the eight combinations are in
because they reach the same final configuration which, although partial, can only match the attractor
. There are five possible attractor patterns of the SPN model for a single cell, shown in bottom right inset:
to
(see
background). Attractors reached by each group are identified in the bottom tab of each plot. Groups
and
both unfold to an ambiguous target pattern that can end in
or
for
, and
or
for
. Finally, the initial partial configurations in groups
and
are sufficient to resolve the states of every node in the network.
Table 3.
Macro-level canalization in the wildcard MC sets converging to wild-type in the SPN.
Figure 17.
Two-Symbol schemata with largest number of position-free symbols, obtained from redescription of
Xwt. The pair were the two-symbol schemata obtained in our stochastic search; both include 4 pairs of symmetric node-pairs, each denoted by a circle and a numerical index.
Table 4.
Macro-level canalization in the wildcard MC sets converging to wild-type in the SPN.
Figure 18.
Enput power in the wild-type basin of attraction of the spatial SPN model.
Enput power is shown for each of the four sets of MCs considered in our analysis: (A) , (B)
, (C)
and (D)
. A parasegment is represented by four rounded rectangles, one for each cell, where the anterior cell is at the top, and posterior at the bottom. Since enput power is computed for every node in each of its two possible states, every cell rectangle has two rows of circles. The bottom row (marked on the sides with a white circle on the outside) corresponds to enput power of the nodes when off, while the top row is the enput power when the same nodes are on (marked on the sides with a dark circle). Each circle inside a cell's rectangle corresponds to the power of a given enput in the corresponding subset of MCs identified by the letters A to D. Full power is highlighted in red, other values in blue and scaled, while null power is depicted using small grey circles. Full power occurs only for enputs that are present in every MC (and configurations) of the respective set, whereas null power identifies nodes that are never enputs in any MC – always irrelevant for the respective dynamical behaviour.
Figure 19.
A MC not requiring in wild-type attractor basin.
When proteins are allowed to be expressed initially, the second necessary condition, reported in [38], ceases to be a necessary condition, as discussed in the main text; in the MC shown, and
can be in any state and the network still converges to the wild-type attractor.
Figure 20.
‘Extreme’ configurations converging to wild-type in the SPN model.
(A) A configuration with the minimal number of nodes expressed that converges to wild-type, and its corresponding MC: 32 nodes are irrelevant, 24 must be unexpressed (off), and only 4 must be expressed (on). (B) The opposite extreme condition where 16 genes and proteins are unexpressed and all other 44 are expressed.
Figure 21.
Wild-type enput disruption in the SPN model.
Each coordinate in a given diagram (each corresponding to a tested enput) contains a circle, depicting the proportion of trials that converged to attractor
when noise level
was used. Red circles mean that all trajectories tested converged to
.