Background
During long term space missions, babies will be born and animals will be bred, perhaps for generations. Will embryos develop normally in microgravity? We focus on the very first step in embryonic development of a salamander embryo, the axolotl Ambystoma mexicanum, an event called cortical rotation, which occurs before the first cell division. The membrane and a layer of cortical cytoplasm rotate spontaneously by 30 deg. The equator of rotation becomes the midline of the embryo, breaking the cylindrical symmetry of the bottom heavy spherical egg to left/right bilateral symmetry. At this time the left/right asymmetry of the internal organs, such as the heart, is also initiated. Our problem is to link the macroscopic fluid dynamics of the flow of cytoplasm with the molecular dynamics of the cortical cytoskeleton, presumed to generate the torque.

Previous Work
As a first approximation to the yolk density gradient in the bottom heavy egg we consider two immiscible fluids [1]. At 175-180 deg rotation the denser fluid fountains downwards in an axisymmetric pattern, while it sloshes back asymmetrically at lower angles. We take these to correspond to occasional failure of inverted amphibian eggs to develop [2], and to normal development, respectively, giving a physical cause for this biological event [3]. We have previously simulated one phase sloshing in a half-filled sphere [4]. With our new two-phase ComFlo finite element code [5] we have carried out a simulation of an "egg model" containing immiscible fluids with an interfacial tension. Results so far suggest two axisymmetric flow patterns (up or down fountaining accompanied by a Rayleigh instability [6]) in addition to the sloshing mode. Under a Canadian Space Agency contract, a series of numerical experiments are being undertaken varying the viscosities and densities of the two fluids, their interfacial tension, and geometric factors such as the initial roughness of their interface, distance of the interface from the centre, and the radius of the egg, under 1G, microgravity, and simulated clinostat tumbling.

Application
The exploration and exploitation of the space environment has been hampered by our 1 G, earthbound biology. Bone loss occurs and even microtubules (the probable sources of torque for cortical rotation) behave differently in microgravity [7]. If we are to live, work, manufacture and reproduce for extended periods in this new environment, we have to understand the physics of embryos.

Questions/Problems for Workshop Participants
There are many numerical challenges to attaining an accurate model of this first step of vertebrate embryogenesis:

1. Sloshing of the yolk generates shear at the cortex, which flow aligns the cortical microtubules and alters the torque that they generate: we need to interleave the fluid dynamics simulation with steps of a molecular dynamics simulation of the microtubules. This is both a multiscale problem and one of simultaneous (computationally parallel) continuum and particulate representations of living material.

2. The cytoplasm contains a yolk gradient, which we are now approximating as two internally uniform, immiscible Newtonian fluids. We need to generalize the finite element code to encompass a nonuniform fluid.

3. The cytoplasm is better approximated as a non-Newtonian viscoelastic fluid, but stable FEA methods for such fluids have yet to be developed.

4. Empirical determination of the spatiotemporal distribution of mass density and microtubules (MTs) in the naturally or artificially rotated embryo (under normal gravity or in a clinostat), halted by high power microwaving, will be measured by 3D reconstruction from serial sections or microMRI (Magnetic Resonance Imaging) and correlated with time-lapse microscopy of the rotating surface. As yolky eggs are difficult to section without cracks appearing, new image processing algorithms for the transmitted and fluorescent light pictures may be needed. 4D correlation and error determination between simulations and reconstructions will indicate whether additional parameters, such as viscoelasticity, need to be introduced. Compression data (local and global) using a microhardness tester will indicate a combination of surface stiffness[8] and internal pressure[9] that will have to be deconvolved.

5. The bilateral asymmetry has been postulated to originate in the generation of left/right asymmetry of supercoiling of the MTs, due to their intrinsic chirality, lack of mirror self-symmetry, and interaction of their polymerization with the flow field[3]. A model for this molecular dynamics/fluid dynamics interaction needs to be developed and tested.

References

1. Flint, R.W., R. Gordon , C.C. Martin & G.W. Brodland (1989). Simulation of the inversion of amphibian eggs in a gravitational field using hollow glass spheres. In: Ubbels, G.A., H. Oser & T.D. Guyenne, Microgravity as a Tool in Developmental Biology, Selected papers from a special ESA Symposium held during the 11th International Congress of the International Society of Developmental Biologists, August 20-25, 1989, Utrecht, Paris
Cedex: European Space Agency, p. 81-83.
2. Wakahara, M., A.W. Neff & G.M. Malacinski (1985). Development of delayed gastrulae and permanent blastulae from inverted Xenopus laevis eggs. Acta Embryol. Morph. Exp. 6, 193-209. 3. Gordon, R. (1999). The Hierarchical Genome and Differentiation Waves: Novel Unification of Development, Genetics and Evolution, Singapore & London: World Scientific & Imperial College Press.
4. Loots, E. (1997). 3d dambreak in sphere computed with ComFlo, http://www.math.rug.nl/~veldman/cfd-gallery.html.
5. Wemmenhove, R., G.E. Loots, R. Luppes & A.E.P. Veldman (2005). Simulation of green water loading by a three-dimensional two-phase numerical model. In: Grue, J., 20th International Workshop on Water Waves and Floating Bodies. University Centre Svalbard, Spitsbergen, Norway, 29th May 29 - June 2005, Oslo: Department of Mathematics, University of Oslo, Norway, p. 261-264. 6. Gordon, R., N.S. Goel, M.S. Steinberg & L.L. Wiseman (1972). A rheological mechanism sufficient to explain the kinetics of cell sorting. J. Theor. Biol. 37, 43-73. 7. Portet, S., J.A. Tuszynski, J.M. Dixon & M.V. Sataric (2003). Models of spatial and orientational self-organization of microtubules under the influence of gravitational fields. Phys Rev E Stat Nonlin Soft Matter Phys 68(2 Pt 1), 021903. 8. Sawai, T. & M. Yoneda (1974). Wave of stiffness propagating along the surface of the newt egg during cleavage. J. Cell Biol. 60, 1-7. 9. Beloussov, L.V. (1998). The Dynamic Architecture of a Developing Organism: An Interdisciplinary Approach to the Development of Organisms, Dordrecht: Kluwer Academic Publishers.

Submitted to the Fields-MITACS Industrial Problem Workshop, Toronto, Aug 14-18, 2006.

by Kamran Khan
Centre for Research on Inner City Health
Division of Infectious Diseases
St. Michael's Hospital, Toronto

Background

The relationship between global migration and the movement of infectious diseases has been well described throughout human history. However, since the advent of intercontinental air travel, an increasingly complex web of air traffic pathways has emerged across the globe. While commercial air travel has served to interconnect the world community, it has concurrently evolved into a major conduit for the global movement of infectious diseases.

The emergence of severe acute respiratory syndrome (SARS) in China and its rapid global propagation via international air travel was an abrupt reminder that infectious diseases are highly effective at breaching international and even intercontinental boundaries. Whether originating from laboratory accidents, intentional release, or natural causes, outbreaks of infectious diseases with pandemic potential constitute a significant threat to the health and economic security of the international community. Recently, the gradual movements of avian H5N1 influenza from East Asia to Western Europe and Africa have raised concerns that an influenza pandemic may be inevitable and possibly imminent.

Despite the significance of commercial air traffic and its role in the worldwide dissemination of infectious diseases, our understanding of global air traffic dynamics remains limited. Gaining insight into the nature of air traffic could help the international community better anticipate the likely movements of infectious diseases and potentially enhance the world's ability to prepare for future pandemics.

Questions for Workshop Participants:

1. Were the international movements of SARS random in nature or did cases travel across the globe in a more systematic fashion?
2. If SARS moved globally in a non-random or systematic fashion, were its movements predictable?
3. For countries affected by SARS, were there specific factors that influenced the time to importation? (i.e. why were some countries affected early on during the course of the outbreak and others much later?)
4. For countries affected by SARS, were there specific factors that influenced the number of imported cases received by a country? (i.e. why did some countries get larger numbers of imported cases relative to others)
5. Based on the knowledge of commercial air traffic patterns and flows, can one predict the likely pathways of an infectious disease from a point source location in future outbreaks or pandemic settings?

6) Modeling Mother-Child Emotional Interaction

(. pdf file) 'A conical Model for the Taxonomy of Emotional Experience'

Submitted by Jon Hunter and Bill Lancee, Mount Sinai Hospital

Background
We are interested in various known mapping rules between the affect of the mother and the affect of the child. A previous study on the structure of emotions revealed that basic emotions can be located on a 2-dimensional space.

Problem Statement
Consider a 2-dimensional affect (emotion) space:
x-dimension is valence (bad for self versus good for self) y-dimension is energy (low arousal or high arousal)
origin (0,0) is emotionally neutral
Individuals can be seen to move through this space over time, i.e., there is a continuous trajectory of points in the space over time.
At a given time t, the position of person "A" in the affect space is (Ax_t, Ay_t).
A change in position from time (t-1) to time (t), that is from (Ax_{t-1}, Ay_{t-1}) to (Ax_t, Ay_t) is a function of

(1) previous position of "A" in the affect space
(2) evaluation of current perceived context
(3) associated memory that "A" has of current perceived context
(4) other influences (not examined in this analysis)

We will be focussing on in the effects of mother on infant and infant on mother.
Effect of mother on infant:
Current perceived context states
1. mother absent
2. mother present at affect position (Mx_t, My_t)
3. presence of other stimuli

Accumulated memory (with time decay?) of interactions with mother
1. based on affect states of infant (Ax_t, Ay_t)
2. conditioned by consistency of mother's presence over time
3. conditioned by consistency of mother's affect over time

Effect of infant on mother (symmetrical to above):

Current perceived context states
1. infant absent
2. infant present at affect position (Ax_t, Ay_t)
3. presence of other stimuli

Accumulated memory (with time decay?) of interactions with infant
1. based on affect states of mother (Mx_t, My_t)
2. conditioned by consistency of infant's presence over time
3. conditioned by consistency of infant's affect over time

Questions
1. Can we define a state transition function (preferably the same one for both mother and infant) which, when applied repeatedly over time, results in three common types of infants?
Type 1: "secure" In sequences of 1-present -> 2-absent -> 3-present the infant's affect pattern in phase 3 is similar to phase 1 and 2 is different
Type 2: "anxious" In sequences of 1-present -> 2-absent -> 3-present the infant's affect pattern becomes more negative in the x-dimension and becomes higher in energy/arousal in the y direction in phase 3
Type 3: "dismissive" In sequences of 1-present -> 2-absent -> 3-present the infant's affect pattern does not change with mothers absence - phase 3 is similar to phases 2 and 1, but becomes higher in energy/arousal in the y direction.

2. Can we define the parameters of the minimal associative memory required to
generate stable types? How sensitive are the projected trajectories over
time to other influences?