THRESHOLD LOGIC GENE REGULATORY MODEL

Prediction of Dorsal-ventral Patterning and Hardware-based Simulation of

Drosophila

Tejaswi Gowda, Samuel Leshner, Sarma Vrudhula

School of Computing and Informatics, Arizona State University, Tempe, AZ.

Seungchan Kim

Division of Computational Biology, Translational Genomics Research Institute, Phoenix, AZ.

Keywords:

Gene Regulation, Threshold Logic, Drosophila embryo patterning, Modeling and Simulation of Gene Systems.

Abstract:

Precise characterization of gene regulatory mechanisms is a fundamental problem in developmental biology.

In this paper we present a new gene regulatory network (GRN) model which is based on threshold logic (TL).

Two different set of genes are responsible for the cell patterning of the Drosophila embryo. By using the

proposed threshold logic gene regulatory model (TLGRM), we derive the different gene regulatory rules for

the gene products involved. We use these rules to model and explain the interaction between the genes. Very

large or complex gene regulatory networks are difﬁcult to simulate using a general purpose CPU. Specialized

programmable hardware provides additional concurrency and is an alternative to a large and expensive cluster

of machines. The steady state gene expression predicted by the model clearly mimics the actual wild-type

gene expression along the dorsal-ventral axis in the Drosophila embryo. We thus demonstrate that for a

well characterized gene regulatory system, the nature and topology of interaction is enough to model gene

regulation. We also demonstrate through proof of concept that using hardware-based simulation, it is possible

to achieve orders of magnitude of performance improvement over conventional CPU-based simulation.

1 INTRODUCTION

Understanding how different genes interact with each

other to give rise to the different cell types in an organ-

ism is an active area of research in development biol-

ogy (Albert and Othmer, 2003). The human genome

project was the ﬁrst step in this direction (Collins

et al., 2003). Now that the methods for gene sequenc-

ing have greatly improved (Heller, 2002) it is possi-

ble to obtain huge amounts of gene expression data.

This data can be analyzed and interpreted to obtain

the different regulatory interactions (activation and

inhibition) between different gene products (mRNA,

protein and protein complexes). Models from math-

ematics and computer science have are been used

to model the interaction of genes (Dougherty et al.,

2007) There are two types of gene regulatory mod-

els. One assumes the expression of the gene products

to be continuous functions which interact with each

other in continuous time (e.g: (Gursky et al., 2001),

(Reinitz and Sharp, 1995)). Another group of mod-

els assume that gene expression takes place in dis-

crete levels and gene interaction takes place in dis-

crete time (e.g: (Shmulevich et al., 2002), (Albert and

Othmer, 2003), (Kauffman, 1993)). The challenge

is to come up with a model that captures the tempo-

ral and spatial characteristics of gene regulation, and

which is easy to construct and simulate. It is generally

agreed that merely specifying the type of interaction

between genes is not enough to characterize gene reg-

ulation (Albert and Othmer, 2003). In this work we

try to show that for at least two gene regulatory sys-

tems knowing the kind of interaction between genes

will sufﬁce to construct an expressive model which

can mimic the actual biological process. For the em-

bryo patterning problem that is considered in this pa-

per, we assume a ﬁxed number of discrete levels of

expression for gene products. This approach is well

suited for this purpose as once the gene is either ex-

pressed or not expressed in a cell, it remains that way

to enable the cell to develop towards its determined

fate.

Although the Drosophila embryo patterning problem

presented in this work uses only a general purpose

212

Gowda T., Leshner S., Vrudhula S. and Kim S. (2008).

THRESHOLD LOGIC GENE REGULATORY MODEL - Prediction of Dorsal-ventral Patterning and Hardware-based Simulation of Drosophila.

In Proceedings of the First International Conference on Biomedical Electronics and Devices, pages 212-219

DOI: 10.5220/0001054302120219

 SciTePress

CPU for simulation, simulation of large GRNs cannot

be done without a large cluster of CPUs. We propose

an inexpensive hardware-based simulation technique

for GRNs using specialized ﬁeld-programmable gate

arrays.

In (Albert and Othmer, 2003) it is shown that

by using two discrete states of gene expression and

Boolean logic for gene regulation rules it is possible

to explain the action of segment polarity genes in the

creation of segments along the anterior-posterior axis

of a Drosophila embryo. While it has been demon-

strated that Boolean logic rules are enough to describe

gene interaction, there is little understanding on how

to construct Boolean rules for a generic gene regula-

tory system. We propose that the Boolean rules used

in gene regulation belong to a special class of func-

tions called threshold functions. These functions are

a subset of Boolean functions. Using this small subset

of functions we derive regulatory relationships from

the gene interaction graph. We then demonstrate that

it is possible to model the interaction of two differ-

ent sets of genes with these rules. These genes are

responsible for the dorsal-ventral patterning and seg-

ment formation along the anterior posterior axis of the

Drosophila embryo.

A threshold element (TE) has n binary inputs, x

..., x

, and a single binary output, y. Its internal pa-

rameters are a threshold T and weights w

, ..., w

where each weight w

is associated with x

. The val-

ues of T and w

(i = 1, ..., n) may be any real num-

bers (Dertouzos, 1965; Winder, 1965). The input-

output relation of a TE is deﬁned as follows:

y =



1 if

∑

i=0

≥ T

0 otherwise

(1)

The weighted sum in Equation (1), denotes arithmetic

summation. Example: Figure 1 shows a threshold

element that implements the Boolean function y =

(b + c). Input a, b, c are assigned the weights −2, 1

and 1 respectively and the threshold (T ) is assigned

the value of 1. It can be seen that the logic function

realized by this threshold element is y = a

(b + c).

Figure 1: A threshold element. Figure 2: A perceptron.

Boolean AND and OR functions are threshold (Der-

touzos, 1965). Threshold elements are also ca-

pable of implementing complex Boolean functions.

e.g: ab(c + d) + cd(a + b). However, it should be

noted that a single TE cannot represent all Boolean

functions, as not all Boolean functions are thresh-

old (Muroga, 1971; Kohavi, 1970). The TE is sim-

ilar to a single perceptron (commonly encountered

in neural network literature) with the step function as

the activating function (Russell and Norvig, 2003), as

shown in Figure 2.

Simulation on a large gene regulatory network can

be a slow, computation-intensive process, largely due

to the fact that a general purpose CPU can only per-

form so many operations at one time (DeHon, 1996).

Consider the sequence of events the CPU must per-

form to compute the next state for each gene product

in the network. The CPU must ﬁrst obtain a copy of

the current state of every gene product in the simula-

tion. Next the updating rules are applied to each gene

product in the simulation sequentially, using the cur-

rent states of the gene products as inputs. Each next

state computation requires one or more instructions,

depending on the rule being applied. Each next state

that is computed must also be saved. Once all next

state computations are complete, the current states are

replaced with the next states and the process repeats.

The total time t required by the CPU for each time

step in the simulation is given in Equation (2), where

instruction

is the time per instruction, n

instruction

is the

average number of instructions per next state compu-

tation, and n

gene

is the number of gene products in the

simulation.

t = (t

instruction

)(n

instruction

)(n

gene

) (2)

This is a highly inefﬁcient method of simulation.

Since the next state computation of each gene prod-

uct depends only on the current states of the other

gene products in the simulation, it should be possible,

given the proper resources, to update all gene prod-

ucts in parallel. This can be achieved using a special-

ized ﬁeld-programmable gate array.

The ﬁeld-programmable gate array, or FPGA, is

composed of a matrix-like arrangement of conﬁg-

urable logic blocks, or CLBs, which are joined to-

gether by interconnect resources (Brown, 1992). The

CLBs and interconnect can be programmed via on-

chip memory to realize any design that it possesses

the resources to implement. Figure 3 shows the con-

ceptual FPGA architecture.

If the number of input parameters in the next state

computation of a single gene product is reasonably

low, the next state computation for a single gene prod-

uct can be implemented and synchronized using a sin-

gle CLB. Conﬁgurable logic blocks synchronized us-

ing the same clock signal operate in parallel, thus us-

ing a specialized FPGA the next states of all gene

THRESHOLD LOGIC GENE REGULATORY MODEL - Prediction of Dorsal-ventral Patterning and Hardware-based

Simulation of Drosophila

213

Figure 3: Conceptual FPGA architecture.

products can be computed simultaneously and then

synchronized for use in the next time step.

The total time required by the FPGA for each

time step is thus given solely by the inverse of the

maximum operating frequency, given by the worst-

case next state computation delay for the slowest

gene product in the simulation. The dependence on

the number of gene products in the simulation is no

longer a factor in the delay per time step. Despite the

fact that FPGAs typically operate at frequencies sev-

eral times lower than that of a general purpose CPU,

the elimination of this dependence can improve the

speed of the simulation by orders of magnitude for

large gene regulatory networks.

FPGAs are thus capable of reducing the amount

of real time required to complete a single simulation

by computing the next state of all gene products con-

currently, however the FPGA’s advantage of parallel

computation does not stop at the system level. Given

enough resources, multiple copies of a complete sys-

tem model can be mapped to the FPGA and simu-

lated concurrently, increasing performance yet again

by a factor equivalent to the number of simulations

running in parallel.

2 APPROACH

The motivation for using threshold logic comes form

the “French ﬂag” model of cell pattern forma-

tion (Wolpert et al., 2002). A chemical whose con-

centration varies over a region, and is responsible for

cell patterning is called a morphogen (Wolpert et al.,

2002). A morphogen can create different cells be-

cause its concentration varies from cell to cell. The

morphogen diffuses from the source (region of high

concentration) to the sink (region of lower concen-

tration). The concentration of the source and sink is

constant and hence the gradient is maintained.

The cells derive positional information from one

or more morphogens before differentiating into a par-

ticular type of cell. Threshold logic naturally sup-

ports this kind of multiple logic levels. The weighted

sum

∑

i=0

in Equation 1 can be compared against

(n − 1) thresholds to generate an output of n levels.

e.g: If the weighted sum is compared against one

threshold we get binary output and if it’s compared

against two thresholds we get ternary output, and so

on. Threshold logic works in the same way irrespec-

tive of the number of levels. This is not true for

Boolean logic.

Another reason to use threshold rules is because

there are only two types of actions that one gene prod-

uct can have on another – activation or inhibition. Ac-

tivation can be thought of as a positive inﬂuence and

inhibition as a negative inﬂuence. In a threshold func-

tion, these two kinds of inﬂuences can be modeled

by the polarity of the input weight. We use a pos-

itive weight for activators and a negative weight for

inhibitors. It is also known that for some genes the

action of inhibitors is stronger than that of the activa-

tor. This can be easily represented in the model by

assigning different magnitudes of positive and nega-

tive weights.

The impact of different activators and inhibitors

can be incorporated into the model by using different

weights. Example: If g

has greater impact on on g

than g

, then in the TE for g

, w

> w

As shown later, these observations lead us to for-

mulate simple guidelines that can be used to derive

the TL rules for gene regulation. This rule generation

is simpler than generating Boolean rules. In the rest

of the paper we introduce the new model and demon-

strate its usefulness by using it to model pattern for-

mation in the Drosophila embryo.

The conﬁgurable logic block is a programmable

cell designed to implement all or part of the next state

computation for a single gene product, and then syn-

chronize the output for use in the next time step.

A standard CMOS implementation of the conﬁg-

urable logic block is shown in Figure 4. This is a

simple modiﬁed version of the CLB used in most

general purpose FPGAs, tailored speciﬁcally for per-

forming and synchronizing next state computations

for individual gene products. The unnecessary ﬂex-

ibility added by more general purpose designs has

been removed, yielding a faster, more area efﬁcient

functional unit. Inputs in the ﬁgure represent the cur-

rent states of the gene products that promote or in-

hibit the expression of the gene product the CLB rep-

resents. The next state of the gene product the CLB

represents is observed at the output.

The CLB is composed of two primary compo-

nents: the lookup table, which provides the next state

computation, and the ﬂip-ﬂop, which provides syn-

chronization. The lookup table observes the unique

combination of logic states on all of its inputs (1 =

expressed, 0 = unexpressed) and accesses a single bit

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

214

Figure 4: Component-level CMOS implementation of n-

input conﬁgurable logic block.

Figure 5: Connection block and switch matrix interconnect

components.

entry in an array stored in memory, containing the de-

sired output (Francis, 1992). This value is passed as

input to the ﬂip-ﬂop which updates its output with the

same value and maintains it until the next time step.

Interconnect is used to join conﬁgurable logic

blocks together into one cohesive, functionally cor-

rect system. There are two types of interconnect com-

ponents that are of major importance, the connection

block and the switch matrix (Brown, 1992). Both of

these components are featured in Figure 5.

A connection block is a collection of switches that

directly connect the CLBs to the segments of the in-

terconnect. As the ﬁgure indicates, connection blocks

are located adjacent to CLBs on the FPGA.

A switch matrix is a collection of switches that

connect segments of the interconnect to each other.

As the ﬁgure indicates, switch matrices are located

where groups of perpendicular wire segments inter-

sect on the FPGA. Each wiring segment that enters a

switch matrix can connect to up to three others.

Preparing a ﬁeld-programmable gate array for

simulation consists of three separate steps. First, each

of the gene products in the model is assigned to one or

more of the available CLBs. Second, the components

of the interconnect are conﬁgured to determine the in-

teraction between the CLBs. Finally, the complete

placement and routing scheme is replicated to max-

imize the usage of the FPGA’s available resources.

Gene products are assigned to conﬁgurable logic

blocks with the modular structure of the system in

mind. This is especially important when the GRN

models multicellular interactions. The following rout-

ing assumptions are made when assigning gene prod-

ucts to CLBs:

• The majority of interactions take place between

gene products within the same cell.

• Some interactions are expected to take place be-

tween gene products of adjacent cells.

• Few (if any) interactions occur between gene

products of distant cells.

Using these assumptions, a system can be decom-

posed in modules of cells, and modules of cells into

individual gene products. Cells are organized into a

uniform arrangement of CLBs so that they will be

able to ﬁt together easily with adjacent cells. Like-

wise, complete systems are organized into a uniform

arrangement of cells so as to maximize the number of

simulations that will be able to run concurrently on

the FPGA.

Since nearly all interactions take place within a

single cell or between adjacent cells, interconnect

programming relies on direct connections and short

wire segments to assemble disconnected CLBs into a

fully functioning system.

3 METHODS

Drosophila is one of the best understood of all devel-

opmental systems at the genetic level (Wolpert et al.,

2002). Drosophila, like many other insects starts its

life as an egg, develops into a larva and then un-

dergoes metamorphosis to become an adult. We are

mostly interested in the embryonic gene regulation

that determines the biological fate of the different re-

gions of the embryo. Like all animals with bilateral

symmetry, Drosophila is patterned along two distinct,

independent axes: the anterior-posterior axis and the

dorsal-ventral axis (Wolpert et al., 2002).

Early Drosophila embryo patterning occurs within

a multi-nucleate syncytial blastoderm. This is formed

because of repeated rounds of nuclear division with-

out any cytoplasmic division (Wolpert et al., 2002).

The absence of cells in the early embryo is really

important, because the mRNA and proteins can now

diffuse easily throughout the embryo. This diffu-

sion creates a concentration gradient of different tran-

scription factors and morphogens. The interaction

between different gene products creates a sharp and

stable concentration gradient of proteins. The pres-

ence or absence of certain proteins determine the fate

THRESHOLD LOGIC GENE REGULATORY MODEL - Prediction of Dorsal-ventral Patterning and Hardware-based

Simulation of Drosophila

215

of the cell (when cellularization occurs). The con-

centration gradient of maternal mRNA and proteins

is known (Wolpert et al., 2002; Roth et al., 1989).

The interaction between the different gene products

that determine the dorsal-ventral patterning is also

known (Wolpert et al., 2002). We make use of this

data to build a threshold logic model of gene interac-

tion. We simulate the rules we generate making use

of the concentration gradient of the maternal proteins

as the initial state. The maternal proteins are the pro-

teins generated by maternal genes. Maternal genes

are the genes expressed in the mother fruit ﬂy and not

the fertilized egg (Wolpert et al., 2002).

Patterning along the axes is similar to the French

ﬂag patterning problem described earlier. The expres-

sion of zygotic genes in localized regions differenti-

ate the cells in the embryo. These different cells de-

termine the germ layers that will eventually develop

out of them. This patterning starts when the embryo

is still a syncytial blastoderm. So the concentration

of the proteins we specify corresponds to the inter-

nuclear concentration. There is a high concentration

of maternal dorsal protein in the nuclei found in the

ventral region (Roth et al., 1989). This concentration

falls rapidly in the dorsal region.

3.1 TLGRM for D-V Patterning

Since early embryonic patterning is essentially two

dimensional, we abstract the dorsal-ventral axis of

the embryo into twelve regions. This patterning takes

place in the blastoderm, which is syncytial before cel-

lularization takes place. The blastoderm has the cells

concentrated in the periphery leaving a hollow in the

center. This abstraction of the dorsal-ventral embry-

onic tissue is shown in Figure 6 (B).

Figure 6: (A): The two axes of a Drosophila. embryo. (B):

For modeling purposes the D-V section of the embryo is

assumed to be made of twelve regions.

Genes which participate in the determination of the

dorsal ventral patterning inﬂuence the expression of

each other via the proteins they synthesize. The dor-

sal (dl) protein initiates a complex gene interaction

process that culminates in the formation of well de-

ﬁned regions in the dorsal-ventral axis. Going from

ventral to dorsal, the major regions are mesoderm,

ventral ectoderm, dorsal ectoderm, and amnioserosa

(Figure 6 (A)). The nuclear gradient of the dorsal pro-

tein in the blastoderm decreases exponentially from

the ventral to the dorsal end of the embryo. The con-

centration of the dorsal protein is divided into 3 dis-

crete levels (−1, 0, 1).

We now describe the genes that are involved in the

dorsal-ventral (D-V) patterning and how they interact

with each other. The dorsal protein as mentioned ear-

lier is probably the most important protein involved

in the D-V patterning. The other genes involved are

twist(twi), snail(sna), rhomboid(rho), tolloid(tld), de-

capentaplegic(dpp) and zerkn

ullt(zen). These genes

respond to the concentration of dorsal protein in a

threshold like manner because of the presence of

high- and low-afﬁnity binding sites for the dorsal pro-

tein in their regulatory regions (Wolpert et al., 2002).

Figure 7: The gene interaction graph representing the reg-

ulatory interaction between the gene products that are re-

sponsible for the D-V patterning in Drosophila.

The interaction between these genes can be repre-

sented by a gene interaction graph (Figure 7). The

nodes of this graph represent mRNA and proteins.

The mRNA nodes are elliptical in shape and the pro-

tein nodes are rectangular. Protein nodes are la-

beled in capital letters and the mRNA nodes are la-

bels in small letters. An arrow (→) is used to rep-

resent activation and a t-connector (a) represents in-

hibition. The gray box which contains all the pro-

tein and mRNA nodes represents the boundary of the

region (one of twelve regions in the blastoderm tis-

sue shown in Figure 6 (B)). Since the gene products

usually diffuse to neighboring regions, the gene prod-

ucts in one region can interact with the gene products

of the neighboring regions (e.g: Gene products in re-

gion 1 can interact with the gene products in regions

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

216

2 and 12 and vice versa). This is represented by the

regulatory interactions (arrows and t-connectors) that

span in between the boundaries of two neighboring

regions. The complete gene interaction graph for the

D-V patterning is shown in Figure 7. This graph was

constructed by using the information about the reg-

ulatory interactions between mRNA and proteins ex-

plained previously.

Using this gene interaction graph we formulate the

threshold logic rules, using the following guidelines:

1. The weights and threshold are restricted to inte-

gers to simplify rule generation. However, this

does not limit the expressiveness of threshold

logic (Muroga, 1971).

2. Genes and gene products that act as promoters are

assigned positive weights. Inhibitors are assigned

negative weights.

3. Dependent gene products are selected from the

gene interaction graph (Figure 7). e.g: Expression

of sna is dependent on DL and TW I.

4. Since in general inhibition is stronger than activa-

tion, inhibitors have a higher absolute weight than

activators.

5. Weight assignments are modiﬁed so that the re-

sults match with biological observation.

Rule generation example:

The rule generation procedure is explained in detail

for the rho mRNA. rho is activated by DL protein in

the same region, and the DL protein in the two neigh-

boring regions (Figure 7). It is inhibited by the SNA

protein. We assign weight of +1 for each of the three

activators (DL protein in the region, and DL protein

in the neighboring regions). We set the threshold to

1 as rho is activated when any one of the activators is

present. We assign a weight of −3 to SNA (inhibitor).

This weight is enough to negate the effect of all the

three activators. We denote this rule as follows:

rho

t+1

= [DL

i−1

= +1, DL

i+1

= +1, SNA

= −3; T = 1].

In this notation, the subscript refers to the cell num-

ber and the superscript refers to the time step in the

model simulation. Each region has a discrete expres-

sion value for every gene product. The cell numbers

are in the range 1 to 12. After 12, we start over with

1, as we have modeled the dorsal-ventral embryonic

section as twelve regions that are arranged in a cy-

cle. These regions are numbered in the counter clock-

wise direction from 1 to 12. Therefore we can now

specify the cells that are adjacent to cell number 1 as

(1 + 1) = 2 and (1 − 1) = 12.

Rules are generated for each gene product. These

rules are then simulated in discrete time. This is dis-

cussed in the results section.

Figure 8: Map of allocated resources for simulation of ex-

pression pattern of the dorsal-ventral patterning genes of

Drosophila melanogaster. on a 48x40 CLB FPGA.

3.2 Hardware Based Simulation

As we can observe from the topology of the interac-

tion graph, each cell in the system is represented by 14

different gene products. Most interactions take place

between products within an individual cell, with some

intercellular interactions taking place between adja-

cent cells involving the SOG and DL proteins.

To map an individual cell to the FPGA, at least 14

conﬁgurable logic blocks are required. An area of 3x5

CLBs is chosen in which to allocate the gene prod-

ucts. A rectangular cell mapping is desirable since it

allows multiple cells to ﬁt together easily on the board

without creating large “holes” of unused blocks. Min-

imization of the cell perimeter is also desirable, as it

often reduces routing distances between both blocks

within the cell and blocks of adjacent cells.

The complete system for the dorsal-ventral pat-

terning genes is modeled by 12 cells in a cyclical

structure. This system is mapped to the FPGA as

block of cells two cells wide and six cells deep. A

cyclical model is fairly easy to place and route, as

each cell in the model has only two adjacent cells.

Depending on size and complexity of both the

gene regulatory network and the FPGA being uti-

lized, there may be unused resources remaining on

the FPGA after mapping of the complete system. If

enough unused resources are available, one or more

copies of the system may be placed on the board and

simulated concurrently, decreasing the effective real

time required to simulate each time step by a factor

equivalent to the number of systems on the FPGA.

THRESHOLD LOGIC GENE REGULATORY MODEL - Prediction of Dorsal-ventral Patterning and Hardware-based

Simulation of Drosophila

217

Figure 8 shows the results of mapping to the

FPGA at the gene product, cell, and system levels. As

the ﬁgure indicates, each cell contains 14 gene prod-

ucts, each system contains 12 cells, and the 48x40

CLB board is large enough to accommodate 10 in-

dependent simulations. Complete systems are sim-

ulated independently and do not interact with each

other, therefore it makes no difference where they are

placed on the FPGA relative to each other as long as

they do not overlap.

4 RESULTS

Using the “guidelines” proposed earlier, the threshold

logic rules for each gene product are derived. The

rules for all the gene products involved with dorsal-

ventral patterning are as follows :

• rho: rho

t+1

= [DL

i−1

= +1, DL

i+1

+1, SNA

= −3;T = 1].

• twi: twi

t+1

= [TW I

= +2, DL

i−1

= +1, DL

+1, DL

i+1

= +1;T = 3].

• sna: sna

t+1

= [TW I

= +2, DL

i−1

= +1, DL

+1, DL

i+1

= +1;T = 3].

• dpp: d pp

t+1

= [SOG

i−1

= −1, SOG

−1, SOG

i+1

= −1, DL

i−1

= −1, DL

i+1

−1;T = −1].

• tld: tld

t+1

= [DL

i−1

= −1, DL

i+1

−1;T = 0].

• zen: zen

t+1

= [DL

i−1

= −1, DL

= −2, DL

i+1

−1;T = 2].

• RHO : RHO

t+1

= [rho = +1; T = 1]

• TW I : TW I

t+1

= [twi = +1; T = 1]

• SNA : SNA

t+1

= [sna = +1; T = 1]

• ZEN : ZEN

t+1

= [zen = +1; T = 1]

• DPP : DPP

t+1

= [d pp = +1; T = 1]

• TLD : T LD

t+1

= [tld = +1; T = 1]

We know the original concentration gradient of DL

(expressed ventrally and concentration decreases dor-

sally). We also know that the SOG protein is found

in the middle regions (Wolpert et al., 2002). The ex-

pression of these proteins is assumed to be constant

throughout the process of pattern formation, since

these proteins are not affected by the other gene prod-

ucts that we consider. The other gene products are

assumed to be uniformly unexpressed at t = 0.

Starting from the initial state we simulate the

threshold logic rules in discrete time steps. The steady

state is attained in the dorsal-ventral simulation af-

ter about ﬁve time steps. This steady state gene ex-

pression is shown in Figure 9. We now interpret this

steady state expression and predict if this indicates

prospective germ layer formation. The position of the

different germ layers along the dorsal-ventral axis is

shown in Figure 6.

The ventral-most stripe of cells form the meso-

derm (Wolpert et al., 2002). Twist and snail genes are

activated in this region. Both these proteins are nec-

essary for the formation of mesoderm (Wolpert et al.,

2002). Our model also predicts that both twi and sna

are expressed in the ventral regions (Figure 9 (B)).

Rhomboid is activated in the cells that form the fu-

ture neurectoderm (ventral ectoderm) (Wolpert et al.,

2002). The region in which the model predicts the ex-

pression of rho is exactly where the ventral ectoderm

develops (Figure 9 (D)).

The dpp gene is expressed in the dorsal-most region;

the gradient of this protein is crucial for the creation of

dorsal-ventral patterning (Wolpert et al., 2002). The

expression of dpp predicted by the model matches

with the biologically observed wild-type expression.

The expression of different genes observed in the

Drosophila embryo that are necessary for dorsal-

ventral patterning is shown in Figure 9 (A). Ob-

serve that the gene expression predicted by the model

matches very well with the actual wild-type expres-

sion. Correct spatial expression of these genes is re-

sponsible for the creation of the four different germ

layers. This simulation takes under one second to

complete.

Hardware simulation is expected to improve simula-

tion performance signiﬁcantly. Functionally, the sim-

ulation performed on the FPGA is no different from

the simulation performed on the CPU. Given the same

set of initial states and rules for the next state compu-

tation of the gene products, the steady states reached

are precisely the same.

A 2-GHz CPU utilizing three address code instruc-

tions with register operands requires an average of

roughly 8.6 instructions to compute the next state for

a single gene product of the genes simulated in this

system. If the processor utilizes 4 integer arithmetic

logic units, all of which operate on independent in-

structions, the effective step time of the CPU simu-

lation is approximately 180 ns. A 400-MHz FPGA

using the mapping described is able to perform 10

simulations concurrently, resulting in an effective step

time of approximately 250 ps, 720 times faster than

the general purpose CPU.

5 CONCLUSIONS

In this paper we propose a novel methodology to

model gene regulation based on threshold logic. We

demonstrate the effectiveness of this method to model

the dorsal-ventral and anterior-posterior pattern for-

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218

Figure 9: A: Biologically observed gene expression in Drosophila. blastoderm. B-E Steady state expression obtained from

simulation of the model. This matches well with the gene expression observed in actual embryos.

mation in Drosophila. The model was able to pre-

dict the role of different genes and their interaction in

determining the pattern formation in the Drosophila

embryo. The advantage of this model is that it is very

simple and makes use of only the nature of interaction

(activation and inhibition) between genes. In simula-

tion, ﬁeld-programmable gate arrays provide a great

deal of power in a relatively small package. Through

modular decomposition of gene interaction models

and highly parallel computation, FPGAs are capable

of high speed simulation that greatly surpass the com-

putational ability of a single CPU. Compared with

a large computer cluster capable of the same tasks,

GRN-application oriented FPGAs are clearly an eco-

nomical choice in terms of size, cost, and power con-

sumption.

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Simulation of Drosophila

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