We perceive several problems related to the functions in many situations in life and develop solutions accordingly. The function shows the effect of one (or more) variable(s) on another variable. In simple terms, one factor can be called the transformation of another factor. When you send a piece of mail by cargo, you pay according to the weight of your cargo. Or imagine using an hourly Internet service in a library or airplane; the more hours you use the Internet, the more you pay.

A group of people want to rent a bus. The number of people does not affect the cost of rent for a bus. However, the cost per person could change depending on of the number of people. The more passengers, the less fee they pay for each passenger. Conversely, if the number of passengers is less, the cost of the bus fee could be more.

The examples are given above help to understand the mathematical function more easily. We perceive several problems related to the functions in many situations in life and develop solutions accordingly. The function shows the effect of one (or more) variable(s) on another variable. In simple terms, one factor can be called the transformation of another factor.

Consider shipping by mail. There are actually two situations here; the weight of the cargo sent and the amount of money paid based upon the cargo’s weight. This gives us a hint for the payable amount. For example, we may know how many pounds of cargo can be sent for $10. We also may calculate how much money is needed to ship 5 pounds of cargo. We can formulate this shipping problem:  Let x be the weight of the cargo and let y be the amount to be paid. In addition, for each parcel sent, including taxes, let us assume that it costs $3. In this case, we get an equation such as y = 2x + 3. Here, the number 2 given in front of the variable x is the money paid per pound. We calculate how many dollars we will pay for a 4-pound cargo: (2×4) + 3 = $11. If 2 dollars per pound is received, a 4-pound package shipping fee is 11 dollars.

Another example would be the use of the Internet. If 40 cents is requested per hour, $1.2 will be given for 3 hours of internet usage. Thus, the equation can be written as y=40x, where x is internet usage time per hour, y is amount to be paid.

The bus rental example is slightly different from the above two examples. Although this example specifies the function, this time there is a reverse situation. In the first two examples, like (x or) input increases, (y or) output will increase. In the case of bus rental, the bus rental fee is fixed and as the number of passengers increases, the payment per person will decrease. Let’s say $400 is required for the daily rental of a bus. 40 people pay $10 per person while 8 people pay $50 per each person.  This equation would be; y=400/x, where x is the number of people, y is the amount of money paid per person. Functions are important in understanding the relationship between two variables.

Life is a function. In mathematics first, what is the function in its simplest form? Let us give the known definition in schools.

Function: Let A and B be two sets different from the empty set and there is a relation f from set A to set B. If the relation f associates each element of the set A to one and only one element of the set B, this relation is called the function from set A to set B. It is shown as f: A→B or A→B. A is called the definition set of the function and B is called the set of values. A set of elements that match the elements of set A in set B is called the set of images of the function and is denoted by f (A).

In this case, if we clarify the definition a little, a relationship from A to B to be defined in order to function f; the following conditions must be met:

1. There should be no elements in the definition set that do not have an image. However, the element can remain exposed in the value set.
2. Each element in the definition set must not have more than one image.

Let us try to find the reflections of the definition in our lives. We assume that A is the world in which we live. If we say “World” for Set A (Definition Set), we would call Group B “the Afterlife” (Value Set). Because there is a connection from this world to the other world. We will see our values in the afterlife against what we do in this world. As someone who fits the definition of a “good” person will be of high value, the definition of a “bad” person will consequently result in a lower value. So, set A considers the world and its elements as ‘living people’. We can think of all the people in the world and include them in this group since life is finite for all of us. The mathematical meaning of this reality is the following:  all elements will be moved from set A to set B (Let A and B be two sets different from an empty set and a relation f from set A to set B.). There is such a connection (or relation) that one who leaves the world cannot go to two places at the same time. Destinations are clear: abode of reward or abode of punishment. We will all be eventually placed into one of these two groups. In some cases, the last stop of the people will be paradise, even if there are those who go to hell first and purify their sins. Consequently, in the last case, no one will be in two places at the same time (If the relation f associates each element of the set A to one and only one of the set B, this relation is called the function from the set A to the set B).

Just because everyone has one final destination does not mean that that destination can only hold one person. If the levels of Heaven and Hell are considered correctly, it is possible that there is more than one person for each level or that no one has come to that level. It is possible that there are those who have sinners at the same level. We can speculate that there may even be such a level that is inaccessible to all people. The Creator creates such places in order to show His grace, mercy, and power. (In the definition set, there should not be any elements without an image. But in the value set, there may be exposed elements.)

Life itself is a function. God created life according to certain rules. We should do our best to explore, find, and use these rules. Seeing the beauties of life and being amazed by the astonishment and capturing happiness in both worlds is achieved by seeing the manifestations of God everywhere.

This beach is one of the starkest reminders of the extent of human impact on oceans.

Kamilo Beach is not the only junk beach in the world. Similar trashed beaches exist in the Azores in the Atlantic Ocean and Baja California. The litter on these shores and in the seas is so excessive that it is even visible to the people on land. Out towards the open seas, huge garbage patches fill the middle of the ocean, like wide loops of rotating currents, also known as gyres (www.marinedebris.noaa.gov).

Gyres

Gyres are major surface currents that circle the oceans. They are driven by persistent winds and the Coriolis Effect which is caused by the Earth’s rotation. The human impact on oceans is displayed by the gyres. They gather trash released into seas from countries around the oceans and through rivers they carry the floating objects and trash spewed from fallen shipping containers.

This was dramatically demonstrated when a cargo of sports shoes were lost in the sea in 1990. The shoes floated and drifted with the currents. Eventually thousands of them washed up on shores from Alaska to California. The manufacturer provided the serial numbers of the lost shoes. Beachcombers responded to calls by researchers regarding the time and location that they found the beached shoes. When all the data points were combined, the gyre’s circulation period over 3 years was obtained. Tracking a spill of bathtub toys provided similar results.

Outsourcing of manufacturing overseas and worldwide supply chains are made possibly by networks of container shipping lines. For reasons of economies of scale, the containers are stacked precariously high on the decks. At rough seas in the open ocean, some of these containers are washed overboard. There are about 10 million 40-foot cargo containers in use in the world. Every year a few thousand of them are washed off the decks of ships in heavy seas. The lost cargo rarely becomes news; it often stays confidential among the ship owner, the importer, the exporter and the insurer. Many of the lost containers sink to the bottom of the ocean. However some of them float and release their contents. This is the source of the flotillas of running shoes, or the toys that get carried away by the currents and winds.

The great garbage patch in North Pacific Ocean covers an area double the size of Texas. Like a conveyor belt, North Pacific Subtropical Gyre rotates clockwise, carrying with it the natural or man-made floating objects. The life span of a gyre is about three years. The floating objects may end washed up at beaches or they may be drifted to the center of the gyre where the currents are weakest. This is where the garbage patch forms—from millions of tons of plastic and other debris covering millions of square miles (www.dels.nas.edu).

Plastics

Plastic nurdles are a significant part of the pollution. These tiny beads are used as raw material in manufacturing. They are carried in container loads across the oceans and are occasionally spilled in large amounts and dispersed at sea.

Compared to organic matter that rots, decays, and gets recycled back into biomass by organisms, plastic is durable. A single plastic water bottle can last for hundreds of years. Suspended in sea water, these plastic particles absorb toxic chemicals. When marine animals consume the floating plastic, disproportionately high levels of these toxic materials are accumulated in their bodies. Some of these animals end up as food on our dinner table. Yes, that plastic tossed into the sea returns back as poisonous seasoning in our diet!

Plastics are contaminating the food chain at different levels. The animals at sea mistake trash for food. The microscopic particles get absorbed by animals filtering sea water for food. Easily mistaken for jellyfish in water, plastic bags suffocate sea animals that ingest them for food. Larger items threaten sea birds and mammals. Seabirds die when their guts get clogged with swallowed plastic items, and other animals starve because their stomachs are full of debris (www.commerce.senate.gov/pdf/marinedebris).

Fishing

A serious source of marine debris is the fishing industry. Numerous fishing nets and floats get lost and are abandoned at sea. These nets, which may extend a distance of many miles, strangle turtles and other sea mammals. Fishing industries should be inspected to keep track of their gear. There are organizations like the Monterey Bay Aquarium (www.montereybayaquarium.org) that distribute information about environmentally friendly fishing. In a free market economy where people vote with their money, consumers should inquire about the sources of seafood and support fishermen that do not leave their nets behind.

Enforcement in open seas requires international cooperation. Vessels should be inspected at their ports of call. Volunteers spotting container ships may record differences in cargo and alert authorities for missing containers. Marine laws, fees, taxes, and insurance premiums can be updated to deter unsafe loading of container ships. In the long run, vessel designs, navigational routes, shipping schedules and weather monitoring should be improved for minimization of cargo loss. An international cooperation is essential to oversee these efforts. The balance sheet of shipping business should include the cost of loss-free transport of containers.

Human negligence

The seas appear to be vast, but we seem to have come to the limits of it by the sheer amounts of garbage dumped into the rivers and by the contamination by marine transportation. These are inescapable reminders that we have reached the limits of this resource. The sea often regurgitates whatever is dumped inside it. This is like a slap on the beach, where the ocean hits back at us with our own trash, not to praise us or show approval, but to bring before our eyes the chaos we have created.

The universe is granted with an internal maintenance system, recycling its own waste products, hence reflecting the absolute purity of the Divine in His creation. The responsibility of humans as “vicegerents” of the earth include using the Earth’s resources without dumping or wasting but safeguarding the environmental balance and acknowledging that every creation has its purpose in being and should be treated accordingly.

The trashed beaches and mid-ocean garbage patches are signs of the deadly and long lasting effects introduced by humans into the seas. This is totally avoidable. We should employ a zero waste approach to our consumption habits. Cost of reusing, recycling and safe disposal of products should be reflected in the price of goods. Laws and regulations should be updated and enforced to minimize the dispersion of long lasting contaminants into the environment. The shipping and fishing industries should be accountable for lost cargo and gear. The seabirds should not starve, and the turtles should not drown due to our negligence.

References

• Auman, H.J., Ludwig, J.P., Giesy, J.P., Colborn, T., (1997) “Plastic ingestion by Laysan Albatross chicks on Sand Island, Midway Atoll, in 1994 and 1995.” in Albatross Biology and Conservation, (ed by G. Robinson and R. Gales). Surrey Beatty & Sons:Chipping Norton. Pp. 239-44
• Spear, L.B., Ainley, D.G. & Ribic, C.A. (1995). “Incidence of plastic in seabirds from the tropical Pacific, 1984–91: relation with distribution of species, sex, age, season, year and body weight.” Marine Environmental Research 40: 123–146
• http://www.washingtontimes.com/news/2003/feb/26/20030226-085636-3495r/
• Ebbesmeyer, Curtis; Eric Scigliano. 2009. Flotsametrics and the Floating World: How One Man’s Obsession with Runaway Sneakers and Rubber Ducks Revolutionized Ocean Science. London: Collins.

For the sake of simplicity, the very complex – yet flawless – organization of a single cell can be portrayed as a city. Even in a very small city that has a relatively low population there are different establishments of various sizes and capabilities that work in concert. Likewise, in order to keep the cell alive the intracellular organelles and proteins of a cell complement each other’s functions. Even though the majority of cities may have the resources to supply themselves, they also need to import and export goods. In addition, some cities may have greater industrial, residential, or agricultural strengths, making it necessary for it to communicate and trade with neighboring towns to eliminate shortages. In a similar way, cells have to communicate with neighboring cells, and bring in new sustenance and get rid of waste at different stages in their life cycles. In this article we will focus on one of the key elements of the import mechanism that is used by living cells, endocytosis.

Endocytosis is the process by which cells take in substances from outside. Since living cells are surrounded by a membrane, the cargo molecules, which are too large to penetrate through the membrane, have to be internalized by different means; this can be classified as Cell eating (i.e. phagocytosis: the endocytosis of large solid materials), cell drinking (i.e. pinocytosis: the endocytosis of liquids in large amounts) and receptor mediated endocytosis (the internalization of molecules that have specific receptors on the cell membrane) (Figure 1). During the course of these events, the portion of the cell membrane that surrounds the cargo is also internalized. However, at the present time, very little is known about the mechanisms that cause the cell membrane to invaginate and eventually be pinched off during the onset of endocytosis. Here we will concentrate on receptor mediated endocytosis, which has been intriguing biologists for more than three decades now.

Receptor molecules, which are located on the outer surface of living cells, function like receiving docks for cells. These are sites where the membrane-encountering molecule is first engaged. The cargo molecules, which arrive at the receiving docks, are often known as ligands (Latin ligare = to bind). Ligand molecules are either secreted from a neighboring cell or come directly from the bloodstream and can bind to the specific receptor molecules on the cell surface. In most cases the ligand is either a signaling molecule, which transmits information to the cell from the extracellular milieu, or a nutritious substance that needs to be internalized via endocytosis.

Unlike other forms of endocytosis, receptor mediated endocytosis occurs in a very controlled way. First of all, the ligand determines whether endocytosis will occur or not. Basically, a ligand molecule that does not have a corresponding receptor on the cell membrane cannot interact with the cell and, as a result, will not be internalized. For instance, a certain virus, which can cause severe lesions in the mouths of horses, cannot infect human cells, as our cells do not have the receptors that link the virus to the cell membrane. Secondly, the ligand cell determines how much intake will take place. That is, cells can control the amount of receptors on their membranes and thus how much the ligands that are recognized by these receptors will be internalized. By acting in this way, cells can adapt to different conditions by controlling the rate of endocytosis through the receptor molecules. The third way of control is concerned with the size of the molecules that are to be internalized. As we will discuss below, receptor mediated endocytosis is performed via some scaffold proteins that form cages (i.e. coats) around the membrane. These cages are fairly small in size, and molecules that are larger than 100 nanometers (100 nanometer is one-thousandth of the thickness of a human hair) cannot fit into them nor enter the cells along this pathway.

The conformation of the plasma membrane changes substantially during endocytosis. Given that the cell membrane has a fluidic nature, it must be accompanied by a firmer construction in order to perform these structural modifications. In the case of receptor mediated endocytosis, this function is carried out by clathrin cages (or clathrin coats), which are formed adjacent to the membrane. The clathrin protein is a three-legged molecule which can freely diffuse inside the cell. However, when a ligand molecule binds to a receptor on the membrane then multiple clathrins aggregate into that region and polymerize into cages composed of pentagons and hexagons (Figure 2a). These cages surround the cell membrane, compelling it to engulf the cargo (ligand) by invagination. When the cargo is entirely enclosed by the membrane a scission protein seals off the enclosed cargo from the plasma membrane (Figure 2b). The completely internalized membrane pouch rapidly casts away its clathrin coat with a quick uncoating reaction. After this point, the membrane pouch, including the cargo, can be transported to the inner compartments of the cell. In living cells, this process takes a little less than a minute. Even though the lifetimes of the clathrin coats are not very long, scientists have found ways of ascertaining the detailed organization of these molecules.

One of the most intriguing findings about the clathrin scaffold (or coat) is its breathtaking geometry. X-ray crystallography studies showed that the cages formed by the polymerization of multiple clathrin proteins are composed of pentagons and hexagons (Figure 3a). It has been also shown that in the majority of the cases the three dimensional structures of the cages are very similar to a soccer ball, where 12 pentagons are accompanied by 20 hexagons in order to create a curve (Figure 3b). However this is not the only geometry that clathrin coats can have. Electron microscopy images taken on frozen cells reveal that clathrins can also form some large flat arrays on the membrane which look exactly like honeycombs. In this configuration the pentagons are missing and all the clathrins align in the hexagon geometry to preserve a flat surface (Figure 3c).

The function of the honeycomb-shaped flat clathrin arrays is still a mystery. All we know is that they are more durable than the cages. In other words, although they do not have definite sizes like cages, they last much longer. Some scientists believe that these structures might be functional in the internalization of huge cargo molecules, such as bacteria, however, these claims still need to be proved. More research is needed if scientists are to be able to better understand clathrin-based endocytosis. Nevertheless, the fact that a simple geometrical modification at the molecular level can alter the entire mechanism of a biological process is on its own a very fascinating discovery.

Hamdi Sener is a biophysicist living in Boston. He can be contacted at hamdisener@gmail.com.

Note

1. Even though bacteria are the smallest cells known, compared to the regular cargo molecules they are gigantic structures.

Figure 1. Taken from Wikipedia.org

Figure 2. A) A clathrin coat (or cage) formed by the polymerization of multiple three-legged clathrin molecules. A single clathrin molecule is shown died with cyan. Adapted from RCSB protein data bank website (http://www.rcsb.org/pdb). B) The different stages of receptor-mediated endocytosis. The entire process takes a little less than a minute.

Figure 3. Taken from Wikipedia.org